Toolbox for reconstructing quantum theory from rules on information acquisition
aa r X i v : . [ qu a n t - ph ] M a r Toolbox for reconstructing quantum theory fromrules on information acquisition
Philipp Andres Höhn
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5March 14, 2018
We develop an operational approachfor reconstructing the quantum theory ofqubit systems from elementary rules on in-formation acquisition. The focus lies on anobserver O interrogating a system S withbinary questions and S ’s state is taken as O ’s ‘catalogue of knowledge’ about S . Themathematical tools of the framework aresimple and we attempt to highlight all un-derlying assumptions. Four rules are im-posed, asserting (1) a limit on the amountof information available to O ; (2) the mereexistence of complementary information;(3) O ’s total amount of information to bepreserved in-between interrogations; and,(4) O ’s ‘catalogue of knowledge’ to changecontinuously in time in-between interroga-tions and every consistent such evolutionto be possible. This approach permits a constructive derivation of quantum theory,elucidating how the ensuing independence,complementarity and compatibility struc-ture of O ’s questions matches that of pro-jective measurements in quantum theory,how entanglement and monogamy of en-tanglement, non-locality and, more gener-ally, how the correlation structure of arbi-trarily many qubits and rebits arises. Therules yield a reversible time evolution anda quadratic measure, quantifying O ’s infor-mation about S . Finally, it is shown thatthe four rules admit two solutions for thesimplest case of a single elementary sys-tem: the Bloch ball and disc as state spacesfor a qubit and rebit, respectively, togetherwith their symmetries as time evolution Philipp Andres Höhn: [email protected], Present address:Institute for Quantum Optics & Quantum Information, Aus-trian Academy of Sciences; and Vienna Center for Quan-tum Science and Technology, University of Vienna, Boltz-manngasse 3, 1090 Vienna, Austria groups. The reconstruction for arbitrarilymany qubits is completed in a companionpaper [1] where an additional rule elimi-nates the rebit case. This approach is in-spired by (but does not rely on) the re-lational interpretation and yields a novelformulation of quantum theory in terms ofquestions.
Tools and concepts from information theory haveseen an ever growing number of applications inmodern physics, often proving useful for under-standing and interpreting specific physical phe-nomena. Among a vast number of examples,black hole entropy, or more generally space-timehorizon entropies, can be understood in termsof entanglement entropy [3–7], thermodynamicsnaturally adheres to entropic and thus informa-tional perspectives [8–12], and, above all, the en-tire field of quantum information and computa-tion is the natural physical arena for applicationsof information theoretic tools [13].This manuscript, by contrast, is motivated bythe question whether information theoretic con-cepts, apart from their useful applications to con-crete physical situations, can also tell us some-thing deeper about physics, namely about thephysical content and architecture of theories. Theoverriding idea is that elementary rules or restric-tions of certain informational activities, e.g. infor-mation acquisition or communication, should bedeeply intertwined with the structure of the ap-propriate theory. In this article, we shall addressthis question by means of the concrete exampleof quantum theory. Our ambition is to developa novel informational framework for deriving theformalism and structure of quantum theory forsystems of arbitrarily many qubits from elemen-
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Quantum2017-11-27, click title to verify ary operational postulates – a task which is com-pleted in the companion paper [1]. While neitherthis question nor the fact that one can recon-struct quantum theory from elementary axioms isnew and has been extensively explored before invarious contexts [14–25], we shall approach bothfrom a novel constructive perspective and witha stronger emphasis on the conceptual contentof the theory. The ultimate goal of this work istherefore very rudimentary: to redo a well estab-lished theory – albeit in a novel way which is es-pecially engineered for exposing its informationaland logical structure, physical content and dis-tinctive phenomena more clearly. In other words,we shall attempt to rebuild quantum theory forqubit systems from scratch.In such an information based context it is nat-ural to follow an operational approach, describingphysics from the perspective of an observer. Ac-cordingly, we shall work under the premise thatwe may only speak about the information an ob-server has access to in an experiment. Our ap-proach will thus be purely operational and epis-temic (i.e. knowledge based) by construction andshall survive without ontic statements (i.e. ref-erences to ‘reality’). We shall thus say nothingabout whether or not ‘hidden variables’ could giverise to the experiences of the observer. Underthese circumstances we adopt an ‘inside view ofphysics’, holding properties of systems as being relationally , rather than absolutely defined.Indeed, more generally the replacement of ab-solute by relational concepts goes in hand withthe establishment of universal (i.e. observer in-dependent) limits . For instance, the crucial stepfrom Galilean to special relativity is the realiza-tion that the speed of light c constitutes a univer-sal limit for information communication amongobservers. The fact that all observers agree onthis limit is the origin of the relativity of spaceand time. Similarly, the crucial step from classi-cal to quantum mechanics is the recognition thatthe Planck constant ~ establishes a universal limit on how much simultaneous information is acces-sible to an observer. While less explicit than inthe case of special relativity, this simple observa-tion suggests a relational character of a system’squantum properties. More precisely, the processof information acquisition through measurementestablishes an informational relation between theobserver and system . Only if there was no limit on the acquisition of information would it makesense to speak about an absolute state of a sys-tem within a purely operational approach (unlessone accepts the existence of an omniscient andabsolute observer as an external standard). Butthanks to the existence of complementarity, im-plied by ~ , an observer may not access all conceiv-able properties of the system at once. Further-more, the observer can choose the experimentalsetting and thereby which property of the systemshe would like to reveal (although, clearly, shecannot choose the experimental outcome). Underour purely operational premise, we shall treat thesituation as if the system does not have any otherproperties than those accessible to the observer atany moment of time. In particular, the system’sstate is naturally interpreted as representing theobserver’s state of information about the system.These ideas are in agreement with earlier propos-als in the literature [26,27] and, most specifically,with the relational interpretation of quantum me-chanics [28, 29].Of course, in order for different observers whomay communicate (by physical interaction) tohave a basis for agreeing on the description ofa system, some of its attributes must be observerindependent such as its state space, the set ofpossible measurements on it and possibly a limiton its information content. But without adheringto an external standard against which measure-ment outcomes and states could be defined, it isas meaningless to assert a system’s physical stateto be independent of its relations to other sys-tems as it is to relate a system’s dynamics to anabsolute Newtonian background time.It is worthwhile to investigate what we canlearn about physics from such an operationaland informational approach. For this endeavourwe shall adopt the general conviction, which hasbeen voiced in many different (even conflicting)ways before in the literature [26, 28, 30–41], thatquantum theory is best understood as an opera-tional framework governing an observer’s acquisi-tion of information about a system. While mostearlier works take quantum theory as given andattempt to characterize and interpret its physicalcontent with an emphasis on information infer-ence, here and in [1] we take a step back and showthat one can actually derive quantum theory fromthis perspective. This will require a focus on theinformational relation between an observer and a Accepted in
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Quantum2017-11-27, click title to verify ystem and the rules governing the observer’s ac-quisition of information. More precisely, our ap-proach will be formulated in terms of the observerinterrogating a system with elementary questions.While the present work has been inspired by re-lational ideas, we emphasize that the sequel does not actually rely on them so this should not dis-courage a reader unsympathetic with relationalinterpretations of quantum theory. Our approachwill reconstruct and produce a novel formulationof quantum theory, but will clearly not single outthe relational interpretation as ‘the right one’.However, it will support a partial interpretation,namely that quantum theory is a law book gov-erning an observer’s acquisition of informationabout physical systems.This is clearly not the only physical situationto which such a relational approach applies; itlikewise opens up a novel perspective on elemen-tary space-time structure which is encoded in theinformational relations among different observersand can be exposed by a communication game.For instance, without presupposing a particularspace-time structure – and thus without assuming an externally given transformation group betweendifferent reference frames – one can also derivethe Lorentz group as the minimal group trans-lating between different observer’s descriptionsof physics from their informational relations, es-tablished by communication with quantum sys-tems [42].More fundamentally, relational ideas are actu-ally required and commonly employed in the con-text of background independent quantum gravityapproaches where the notion of coordinates dis-appears together with a classical notion of space-time within which a dynamics could be defined.Instead, one has to resort to dynamical degreesof freedom to define physical reference frames(i.e., dynamical ‘rods’ and ‘clocks’) relative towhich a meaningful dynamics can be formulatedin the first place. This constitutes the relationalparadigm of dynamics [43–51] but goes beyonda purely informational and operational approachand thus clearly beyond the scope of this work.The remainder of this manuscript is organizedas follows. Contents S relative to O . . . . . . . . . . Σ and Q . . . . . . . . . . . . . . . . . . . . . . . . . . S ’s state and tomography . . . . . . . . . . . . . . . . . . . S ’s state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accepted in
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Quantum2017-11-27, click title to verify .2.2 Independence, complementarity and entanglement . . . . . . . . . . . . . . . . N = 2 qubits . . . . . . . . . . . . . . . . . N = 2 rebits . . . . . . . . . . . . . . . . . local hidden variables . . . . . . . . . N = 2 gbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . N = 2 . . . . . . N > gbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N > qubits . . . . . . N > rebits . . . . . . . SO(D N ) . . . . . . . . . . . . . . . . . . N = 1 case and the Bloch ball 68 Acknowledgements 74References 74
The first part of the article up to and includingsection 4 includes a substantial amount of con-ceptual elaborations. The second part, by con-trast, will become more technical upon puttingthe novel postulates to use in sections 5-7. Thereconstruction for arbitrarily many qubits is per-formed in the companion article [1] where an ad- ditional postulate eliminates rebits (two-level sys-tems over real Hilbert spaces) in favour of qubitquantum theory. The rebit case is considered sep-arately in [2].
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Quantum2017-11-27, click title to verify Why a(nother) reconstruction ofquantum theory?
Given that we have a beautifully working theory,one may wonder why one should bother to re-construct quantum theory from operational state-ments. There are various motivations for this en-deavour:1. To equip the standard, physically obscuretextbook axioms for quantum theory withan operational sense. In particular, in ad-dition to its empirical success, a derivationfrom operational statements can conceptu-ally justify the formulation of the theory interms of Hilbert spaces, complex numbers,tensor product rule for composite systems,etc.2. To better understand quantum theory withina larger context. By singling out quan-tum theory with operational statements onecan answer the question “what makes quan-tum theory special?”, thereby establishing abird’s-eye perspective on the formalism andconceivable alternatives.3. It may help to understand why or why notquantum theory in its present form shouldbe fundamental and thus why it should orshould not be modified in view of attempt-ing to construct fundamental theories. Bydropping or modifying some of its definingphysical principles, one obtains a handle forsystematic generalizations of quantum the-ory. This may also be interesting in view ofquantum gravity phenomenology (away fromthe deep quantum regime). More fundamen-tally, the question arises whether an infor-mational perspective could be beneficial forquantum gravity in general.4. The hope has been voiced that a clear inter-pretation of the theory may finally emergefrom a successful reconstruction, in analogyto how the interpretation of special and gen-eral relativity follows naturally from its un-derlying principles [28, 31, 36].It is fair to say that the hope alluded to un-der point 4 has not been realized thus far becausethe existing successful reconstructions [14–25] are fairly neutral as far as an interpretation is con-cerned. While most of them emphasize the op-erational character of the theory, a particular in-terpretation of quantum theory is not stronglysuggested.The language and concepts of the present re-construction are different. It will emphasize andconcretize the view (or partial interpretation)that quantum theory is a framework governingan observer’s acquisition of information about theobserved system [26, 28, 30–41]. The new postu-lates are simple and conceptually comprehensible,concerning only the relation between an observerand the system. Due to the simplicity, the ensu-ing derivation is mathematically quite elementaryand, in contrast to previous derivations, yieldsthe formalism, state spaces and time evolutiongroups explicitly and in a more constructive man-ner. The disadvantage, compared to other recon-structions, is that a large number of detailed stepsis required. The advantage, on the other hand, isthe simplicity of the principles and mathematicaltools, and the fact that the reconstruction affordsnatural explanations for many quantum phenom-ena, including entanglement and monogamy, andelucidates the origin of the unitary group.
The ambition of a (re-)construction of quantumtheory is to derive its formalism, state spaces,time evolution groups and permissible operationsfrom physical principles – in some rough analogyto the construction of relativity theory from theprinciple of relativity and the equivalence prin-ciple. But in order to formulate physical postu-lates, we clearly have to presuppose some mathe-matical structure within which a precise meaningcan be given to them. (For example, also theconstruction of special and general relativity cer-tainly presupposed a substantial amount of me-chanical structure.)The procedure is thus to firstly define some landscape of theories , which hopefully containsquantum theory and classical information theory,but within which theories are generally not for-mulated in terms of the usual complex Hilbertspaces, tensor product rules, etc. The mathe-matical formulation of the landscape must there-fore be more elementary and, in particular, op-erational. That is, for the time being, we have
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Quantum2017-11-27, click title to verify o forget about the usual – mathematically crispbut physically rather obscure – textbook axiomsof quantum theory. While different theories willhave the mathematical and physical structure ofthe landscape in common, they may have oth-erwise very different physical and informationalproperties; e.g., they may admit much strongercorrelations than quantum theory [52], or weakercorrelations as classical probability theory, theymay allow exotic communication and informationprocessing tasks to be accomplished [53, 54], andso on. Secondly, given the language of this land-scape, one can attempt to formulate comprehen-sible physical statements which single out quan-tum theory from within it. Going to a larger the-ory landscape and beyond the language of Hilbertspaces is precisely what allows us to ask the ques-tion “what makes quantum theory special?” and,ultimately, to find an operational and physicaljustification for the usual textbook axioms andthe standard Hilbert space formulations.The goal of this section is precisely to buildsuch an appropriate landscape of inference theo-ries both conceptually and mathematically fromscratch within which we shall subsequently for-mulate those elementary rules, governing an ob-server’s acquisition of information about a sys-tem, that single out qubit quantum theory. Thisnovel landscape of inference theories employs adifferent language and is conceptually distinctfrom the by now standard landscape of general-ized probabilistic theories (GPTs) which are com-monly employed for characterizations (or gener-alizations) of quantum theory. For contrast andto put the new tools into a larger perspective, webegin with a brief synopsis of the GPT languagebefore we establish the landscape of inference the-ories underlying this manuscript. It has become a standard in the literature toemploy the formalism of generalized probabilistictheories (GPTs) for operational characterizationsor derivations of quantum theory [14–23, 55–61].The setup of GPTs is exclusively operational andone considers three kinds of operation devices (seefigure 1): (1) a preparation device which can spitout systems in some set of states defined by avector of probabilities for the outcome of fiducialmeasurements. The state spaces of the systems are necessarily required to be convex to permitconvex mixtures of states. (2) A transformationdevice can perform physical operations on theprepared systems (e.g., a rotation) which maychange the state of the system (e.g., by somegroup action on the state vector), but must al-low it to continue its journey to (3), a measure-ment device, which detects certain experimentaloutcomes. The measurement devices are math-ematically described by so-called ‘effects’ whichare assumed to be dual to, i.e. linear functionalson, the states. (For the interested reader we note,however, that Holevo has shown how the math-ematical incarnation of measurements as linearfunctionals on states follows from other simpleoperational assumptions [62].)
PSfrag replacements preparation transformation measurement systemsconvex state spaces cbit square bit rebit qubit ‘effects’ dual to states
Figure 1: The standard operational setup of generalizedprobabilistic theories with examples of allowed convexstate spaces of elementary two-level systems.
Measurements and states are at the heart ofGPTs; ‘effects’ directly determine the outcomeprobabilities of measurements and thereby, whena complete set is engaged, reveal the probabilisticstate of a system. An observer assumes a support-ing role, giving intuitive meaning to the notionof preparation, transformation and measurementsof systems. The reconstructions of quantum the-ory within the GPT formalism depart from ratherglobal operational axioms, restricting the sets ofpossible preparations (i.e. state spaces), transfor-mations and measurements. The concrete acqui-sition of information of the observer about thesystems is otherwise not accentuated. In par-ticular, since the outcome probabilities are theprimary concept of GPTs, these axioms say lit-tle about what an observer experiences in indi-vidual experimental runs, but instead focus onthe totality of a large number of experimentalruns. This has led to a whole wave of successful
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Quantum2017-11-27, click title to verify uantum theory reconstructions, employing GPTconcepts in one way or another [14–23], most ofwhich, however, are fairly abstract on account ofthe rather global axioms.It is perhaps one of the great strengths of GPTsthat they constitute a functional and purely op-erational framework which is interpretationallyfairly neutral. It is not the ambition of this frame-work to elucidate the measurement problem, toclarify what happens to a state during a mea-surement, what probabilities are or, ultimately,how to interpret quantum mechanics (except thatit highlights its operational character). As such,this framework is compatible with most interpre-tations of quantum theory. The success of GPTs notwithstanding, we shallnow change semantics and perspective to define anew landscape of theories and a novel frameworkfor (re)constructing and understanding quantumtheory. Henceforth, we shall fully engage theobserver and give primacy to his acquisition offundamentally limited information from observedsystems. This will include being more explicitabout what general sort of information will beavailable to the observer even in individual exper-imental runs. Probabilities, on the other hand,can be viewed as secondary and as a consequenceof the limited information available to the ob-server – although, clearly, probabilities will as-sume a pivotal role too (after all we want to re-construct quantum theory).
As schematically depicted in figure 2, we shallconsider an observer O who can only interact witha system S through interrogation via questions Q i from some set of questions Q which we shallfurther constrain below. (At this stage we makeno assumption about whether Q is continuous ordiscrete.) The only information which we allow O to acquire about S is by asking questions fromthis set Q .In principle, of course, O could conceive of andask S all kinds of questions, but S could not al-ways give a meaningful answer; S may simplynot have the desired properties or carry the in-formation O is inquiring about (e.g., S may notbe complex enough, or S does not interact with other systems carrying the information in ques-tion), or possibly the question is not a sensefulone in the first place. We shall call a question Q physically implementable on S if O can acquire a‘meaningful’ answer from S to Q . An elementaryrestriction on Q is that any question Q i from thisset be implementable on S and that, whenever O asks Q i to S , S will give an answer to O . Clearly, Q depends on S .But how does O know whether S will give ananswer and how can he judge whether the latteris ‘meaningful’ ? This requires O , like any experi-menter, to have developed, from previous experi-ences, a theoretical model by means of which hedescribes and interprets his interactions with S .An answer can only be ‘meaningful’ in the con-text of this model such that our notion of imple-mentability is actually dependent on O ’s model.We shall come back to this model frequently.The central ingredients of this framework willthus be questions and answers – and O ’s informa-tion about answer outcomes and their relations.This will lead to a novel question calculus fromwhich many crucial quantum properties will bederived. That is, rather than focusing mostlyon probabilistic properties as in GPTs, this novelframework will connect more directly to what canbe measured in experiments through its empha-sis on questions and their relations. As a conse-quence, this framework will also produce opera-tionally more compelling explanations of typicalquantum phenomena.In the sequel, we shall solely speak about the information O has about S and, correspondingly,about the state that O assigns to S based onthis information. Such a state of S is then de-fined relative to O (whether or not some hiddenvariables give rise to this state is a question weshall not address). The act of information ac-quisition establishes a relation between O and S and this will be the center of our attention. (Apriori, a different observer O ′ may establish adifferent relation with S .) Although this frame-work will also not give rise to a unique interpre-tation, it will support the partial interpretationthat quantum theory is a law book governing anobserver’s acquisition of information about phys-ical systems and might therefore be consideredinterpretationally less neutral than GPTs. Whilethis connects with general ideas underlying, e.g.,the relational [28, 29], Brukner-Zeilinger informa- Accepted in
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Quantum2017-11-27, click title to verify Sfrag replacementsPreparation Interrogation
S O Q i ? Figure 2: Schematic representation of an observer O interrogating a system S . tional [31–35], or QBist [36–39] interpretations ofquantum mechanics, we emphasize that this re-construction does not rely on them and shouldthus not discourage readers uncomfortable withthese specific interpretations.We shall not explicitly deal with transforma-tion and measurement (‘effect’) devices as inGPTs; instead, these will be replaced more gener-ally by time evolution and questions, respectively.The set of all possible (information preserving)operations that O could perform on S can laterbe identified with the set of all possible time evo-lutions of S . However, in analogy to GPTs, wewill assume that O has access to some method ofpreparing S in different ways such that S ’s an-swers to O ’s questions depend on the precise wayof preparation. ( O could either control himself apreparation device or have a distinct observer O ′ prepare systems for him.)When constructing the new landscape L of the-ories describing O ’s acquisition of informationabout S within which we shall later, in section 4,formulate our postulates, we will make a numberof restrictions and assumptions. In order to fa-cilitate future generalizations and improvementsof the present construction of quantum theory –which constitutes a proof of principle – we shallattempt to be as clear as possible about the as-sumptions made throughout this work.As a starter, we would like to keep Q as sim-ple as possible, while still having non-trivial ques-tions. In particular, we do not wish to considertrivial propositions which are always true or al-ways false. We shall therefore assume the follow-ing. Assumption 1.
The set of questions Q which we shall permit O to ask S only contains binary questions Q i . Any Q i ∈ Q is a non-trivial ques-tion such that S ’s answer (‘yes’ or ‘no’) is not independent of its preparation. Furthermore, any Q i ∈ Q is repeatable such that O , by asking thesame S the same Q i m times in succession willreceive m times the same answer. The restriction to elementary ‘yes-no’-questions greatly simplifies the discussion and,ultimately, will give rise to the quantum theoryof qubit systems. For instance, in quantumtheory, a binary question could be ‘is the spin ofthe qubit up in x -direction?’ However, it will notbe too difficult to generalize Q to also consist ofternary, quaternary, quinary, etc. questions, butwe shall not attempt to do so here. Since trivialquestions are not considered, we already seethat not even all implementable questions willbe taken into account. We shall impose furtherrestrictions on Q such that, ultimately, it will bea strict subset of all possible binary questionswhich O could, in principle, ask S .Since this is an operational approach, it is fairto assume that O can record the answers to hisquestions asked to any system (e.g., by writingthem on a piece of paper) and that he can dostatistics over the outcomes (e.g., by counting thefrequency of outcomes). We shall require thatevery possible way of preparing S will give riseto a particular statistics over the answers to all Q i ∈ Q ; O could test these statistics by interro-gating a large number n of identically preparedsystems S a , a = 1 , . . . , n , sufficiently often with Given the present structure, two systems S and S could only be considered as distinct in nature if either the(maximal) set of questions Q which O can ask the sys- Accepted in
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Quantum2017-11-27, click title to verify at least ideally) all Q i ∈ Q . In fact, this is pre-cisely how O will operationally distinguish differ-ent preparations of systems.By having interrogated, in this manner, the n always identically prepared S a for all possibleways of preparation, we shall assume O to havegained a ‘stable’ knowledge of the set Σ of all pos-sible answer statistics for S over Q , i.e., the an-swer statistics for all Q ∈ Q and all possible waysof preparing a system S . While ideally n → ∞ isnecessary, ‘practically’ n should be large enoughfor O to develop a theoretical model of both Q and Σ up to some accuracy which agrees withhis observations. It is not our ambition to clarifyfurther what n is nor how precisely O has de-veloped his theoretical model, instead, we shallhenceforth just assume that O has puzzled outthe pair ( Q , Σ) . As any experimenter in an ac-tual laboratory, O shall interpret the outcomesof his interrogations by means of his model for ( Q , Σ) and he can decide whether a given ques-tion is contained in the set Q or not. Henceforth,we assume that O only asks questions from Q .Our task will be to establish what this model is,based on the ensuing assumptions and postulates. S relative to O The previous subsection did not refer to the no-tion of probabilities. But it defined two prepa-rations as being identical if they give identicalstatistics. This permits us to regard Σ equiva- tems or the totality of answer statistics for all possiblepreparations were distinct for S and S (if always thesame two S , S are interrogated and thereafter freshlyprepared again). If O can not distinguish S and S in thisway for sufficiently many trials (ideally infinitely many),we shall call them identical. Let S and S be identi-cal and O prepare both systems with the same procedure(for instance, the setting of the preparation device is thesame for both systems). If the answer statistics (frequen-cies) for S , S for all Q i ∈ Q become indistinguishable af-ter sufficiently many trials (of preparing and interrogatingthe same systems with the same procedure), we considerthese systems as ‘identically prepared’. (After completionof this work, the author was made aware that this notionis similar to the definition of ‘operational equivalence’ putforward in [63].) We assume the preparation method to be ideal in thesense that it accounts for any possible answer statisticswhich S can admit in O ’s world for questions in Q . Thatis, there do not exist other methods which can prepare S in ways that O ’s method does not encompass. lently as the set of all possible answer statisticsor as the set of all (distinct) preparations.Based on this notion, we shall now identifyprobabilities as degrees of belief. The ‘knowledge’of what Σ is for a given S will permit O to assignprobabilities to the outcomes of his questions. Itis very natural for O to assign probabilities toquestions because he deals with statistical fluctu-ations and furthermore, as we shall see later, withsystems about which he always has incomplete in-formation in the sense that the corresponding Σ is such that he can never know the answers to all Q i ∈ Q at the same time.More precisely, for a specific S and any Q i ∈ Q that he may ask the system next, O can assign aprobability y i that the answer will be ‘yes’ (or aprobability n i that the answer will be ‘no’), ac-cording to(i) O ’s knowledge of Σ , and(ii) any prior information that O may have aboutthe specific S .Given our setup, the only prior information that O may have about the particular S (apart fromwhat the associated Σ and Q are) must result ei-ther from having interrogated some ensemble ofsystems identically prepared to S with some sub-set of Q beforehand and from the correspondingaccumulated statistics of the asked Q j ∈ Q (e.g.,that O may have recorded on a piece of paper),or from any other prior information about themethod of preparation. In particular, this previ-ous ensemble could have been empty.For instance, if every time that O asked thespecific Q i to any of the identically prepared sys-tems gave a ‘yes’ answer before, he will assignthe prior probability y i = 1 to Q i and to the nextidentically prepared S that he will interrogate. If,on the other hand, the number of ‘yes’ and ‘no’answers to some other Q j was equal for the previ-ously identically prepared systems, O will assign,as a best guess, the prior probability y j = tothis Q j and to the next S . Similarly, for anyother answer statistics, O would assign y i to thenext S depending on the recorded frequencies of‘yes’ answers. But, thanks to his knowledge of Σ and therefore of any possible relations in the If O asks more than one question to any S , the order-ing of the questions may matter. But then O could askthe questions for any S always in the same order. Accepted in
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Quantum2017-11-27, click title to verify nswer statistics, O can also assign prior proba-bilities y k to questions Q k that he did not ask theprevious set of identically prepared systems. Forexample, Σ may be such that whenever S givesa ‘yes’ answer to Q i , it will give a ‘no’ answer toan immediately following Q k . Accordingly, if O assigns a prior probability y i = 1 to Q i as above,he will also assign a prior y k = 0 to Q k withoutpreviously having asked Q k . But other relationsbetween questions will be permitted too. In par-ticular, it may be that the information gainedfrom the questions he previously asked the iden-tically prepared systems and the structure of Σ make it equally likely that the answer to Q k askedto the next S will be ‘yes’ or ‘no’. In this case, O will assign y k = to Q k that he may ask thenext S . This will become more precise along theway.We therefore take a broadly Bayesian perspec-tive on probabilities: O assigns probabilities toquestions according to his ‘degree of belief’ about S . These probabilities y i are thereby relative tothe observer O . A different observer O ′ may havedifferent information about S and thereby assigndifferent probabilities to the various outcomes ofquestions posed to S (for a discussion, withinquantum theory, of the consistency of differentobservers having different information about asystem, see [28, 29, 64, 65]). E.g., O ′ could be theone preparing S . She could ‘know’ the statisticsfor the specific preparation setting (from previoustests) and then send O the specifically prepared S without informing him about her knowledge.For consistency, we tacitly assume the set Σ ofall possible answer statistics to coincide with theset of possible ‘beliefs’. That is, to every equiva-lence class of preparations of S (with ‘identicallyprepared’ defining the equivalence relation) thereshall correspond a unique ‘belief’ { y i } Q i ∈Q andvice versa. Since the only way for O to acquire in-formation about S is by interrogation with ques-tions in Q , the prior probabilities y i that O as-signs to every Q i ∈ Q encode the entire informa-tion that O has about S . Hence, we shall makethe following identification. Definition 3.1. (State of S relative to O ) The collection of all probabilities y i ∀ Q i ∈ Q is the state of S relative to O . Accordingly, the set We add the qualifier ‘broadly’ here since we also allowfor the typical laboratory situation of ensembles of systems(which may or may not contain more than one element). Σ of all possible answer statistics on Q which S admits is the state space of S . Of course, ultimately not all y i will be inde-pendent such that the full collection of probabili-ties will yield a redundant parametrization of thestate. However, this is not important for the mo-ment and we shall come back to this shortly.This definition of the state of a system S ex-plicitly identifies it with the ‘state of informa-tion’ that O has acquired about S ; O assignsthis state to S according to his information aboutthe Q i ∈ Q . As such, the state of system S is epistemic (i.e. a ‘state of knowledge’) and apriori only meaningful relative to the observer O . The interpretation of the quantum state asa ‘state of information’ is certainly not new andhas been proposed in various ways before (seealso, e.g., [36–38, 40]). However, the above def-inition is closest in spirit to the ideas underly-ing Relational Quantum Mechanics [28, 29] andthe Zeilinger-Brukner interpretation [15, 31–34]and thereby generalizes them to the landscape oftheories describing O ’s acquisition of informationwhich we are in the process to establish.While the state of S is thus a priori only mean-ingful relative to O , we emphasize that both theset of questions Q which O may ask and the statespace Σ are to be intrinsic to the system S . Oth-erwise, it would be difficult for two observers toagree on the description of a given S . At this stage it is important to distinguish single from multiple shot interrogations . In a single shot interrogation O interrogates asingle system S , in some prior state, witha number of questions from Q without in-termediate re-preparations of S . The defi-nite answers to these questions give O def-inite information about this specific S afterthe interrogation. Furthermore, his knowl-edge of Σ and any prior knowledge of S (acquired through previous interrogations ofidentically prepared systems) give him statis-tical information about any questions he did not ask S . In conjunction, the new answersand his prior knowledge thus determine thestate of S after the interrogation. This willconstitute a posterior state update rule , andthereby a ‘belief’ update form a prior to a Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify osterior state for a single system which weshall turn to shortly (and which clearly de-pends on the specific way O interrogates S ).This posterior state of S will reflect O ’s defi-nite information about every asked question Q i by featuring either y i = 0 or y i = 1 due torepeatability, depending on whether the an-swer was ‘no’ or ‘yes’, respectively (assumingfor now, of course, that Σ is such that theanswers to the selection of questions that O asked can be known simultaneously).If this posterior state does not coincide withthe prior state that O assigned to S beforethe interrogation, based on his prior infor-mation about S , then S ’s state has ‘col-lapsed’ relative to O during the interroga-tion. Hence, a state ‘collapse’ only occursif O ’s posterior information about S doesnot coincide with his prior information about S , i.e. if O experienced an information gain about S via the interrogation. We shalltherefore view a state ‘collapse’ as O ’s infor-mation gain about this specific S rather thana ‘disturbance’ of S (we refer the reader alsoto [26, 60, 66, 67] for a related discussion). multiple shot interrogation O interrogatesan ensemble of identically prepared systems S a , a = 1 , . . . , n , where the interrogationof every S a is a single shot interrogation. O will carry out such a multiple shotinterrogation to do state tomography , i.e.to estimate the state of the ensemble { S a } for the specific setting of preparation or, inother words, the state of any of the systems prior to being interrogated by O .This will also be a ‘belief’ updating, how-ever, a prior (or ensemble) state updating .After every interrogation of a system in theensemble, O will assign probabilities y i to the Q i in the manner described above. This willthen define the prior state of the next sys-tem in the ensemble to be interrogated. Byinterrogating more and more systems, O willgain more and more information about the A ‘disturbance’ of the system S is only meaningful ifthere was an underlying ontic state (i.e. ‘state of reality’)to which, however, O would have no access. Here we shallmerely speak about the information that O has access toand therefore not make any ontic statements, regardingthem as excess baggage for our purposes. ensemble state such that his assignments ofthe y i will fluctuate less the larger the num-ber of interrogated systems. This processgives rise to an ensemble state updating. In-dependent of this updating, the prior statethat O assigns to any individual system mayexperience a ‘collapse’ during the interroga-tion of that specific system because his infor-mation about the specific system may havechanged. Accordingly, O will have to distin-guish the ensemble state from the posterior state of any system in the ensemble. But thecollection of posterior states determines theensemble state. Σ and Q The present structure is still too rudimentary fora quantum (re)construction. We therefore needmore.Firstly, as in GPTs we will need O to be able toassign a single prior state to any pair of identicalsystems whenever he flips a biased coin in orderto decide which of the two systems he will in-terrogate. (Equivalently, the preparation methodcould involve a biased coin toss, the outcome ofwhich determines the preparation setting.) Thatis, O will be permitted to build convex combina-tions of states. Assumption 2.
The state space Σ of S is aclosed convex set. Closure of Σ is assumed because points lying inthe boundary of Σ can be arbitrarily well approx-imated by states in the interior. Perfect prepa-ration and arbitrarily good preparation are op-erationally indistinguishable and so adding theboundary does not change the physical predic-tions [16].Next, we need to define some elementary struc-ture on Q in order to meaningfully speak aboutrelations among questions (and answers). To thisend, we shall establish additional structure onthe pair ( Q , Σ) . We declared in assumption 1that there are to be no trivial questions in Q theanswers to which would be independent of S ’spreparation. We also insisted before that O willdistinguish the different ways of preparing a sys-tem S – and thus the different states he can assign– by the particular answer statistics. We shallnow strengthen these requirements, by asserting Accepted in
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Quantum2017-11-27, click title to verify hat there exists a distinguished state of ‘no infor-mation’, corresponding to the situation that O ’sprior knowledge about S makes it equally likelyfor him that the answer to any Q ∈ Q is ‘yes’ or‘no’. We note that this assumption is a restric-tion on O ’s model for the pair ( Q , Σ) . Assumption 3.
There exists a special state in Σ , called the state of no information , which isgiven by y i = , ∀ Q i ∈ Q . This state shall be the prior state that O as-signs to S (or an ensemble thereof) in a beliefupdating whenever he has ‘no prior information’.For example, a distinct observer O ′ may preparea system S and send it to O in such a way thatthe latter knows only the associated Σ but notthe preparation setting. In this case, as a prior, O will assign the state of no information to S ,i.e. y i = , ∀ Q i ∈ Q . Similarly, there will exista special preparation setting which is such that amultiple shot interrogation on an ensemble pre-pared in this setting will give totally random an-swers to O such that he will assign the state of noinformation to the ensemble. But note that thestate of any individual system after the interroga-tion of that system will not be the state of no in-formation because, through the interrogation, O will have acquired information about that specificsystem (see the discussion of the state ‘collapse’above).Given the state of no information, we shall pre-liminarily quantify the amount of information α i that the definite answer to any Q i ∈ Q defines asone bit . Similarly, whenever y i = , as in thestate of no information, we shall say that O has α i = 0 bits of information about Q i . In general,under the premise that information can neitherbe negative nor complex, O ’s information about We emphasize that GPTs are more general by, in prin-ciple, permitting state spaces which do not contain sucha distinguished state. However, most operationally inter-esting GPTs do possess such a state. For instance, we note that on account of the exis-tence of binary POVMs with an inherent bias, such as( E = 2 / · , E = 1 / · ), the pair given by Q = { binary POVMs } and Σ = { unit trace density matrices } cannot satisfy this condition because no unit tracedensity matrix exists which yields probability 1 / E , E ). This pair will therefore ultimately not be thesolution of this reconstruction. However, a subset of { binary POVMs } together with the full quantum statespace will be the solution. Q i should satisfy bit ≤ α i ≤ bit . (1) We shall not propose an explicit information mea-sure α i (as a function on Σ ) here because thismust follow from the rules on information acqui-sition postulated below and we shall indeed deriveit therefrom later in section 6.8. Until then it willbe sufficient to work with this implicit notion ofquantifying O ’s information about any Q i .But the questions O can ask, and the informa-tion that the corresponding answers define, maynot be independent. However, a priori the no-tion of independence of questions, in the sense ofstochastic independence, is state dependent. Forexample, for a pair of qubits in quantum theorythe questions Q , ‘is the spin of qubit 1 up in x -direction?’, and Q , ‘is the spin of qubit 2 up in x -direction?’, are stochastically independent rel-ative to the completely mixed state, but fully de-pendent relative to an entangled state (with cor-relation in x -direction). As a result of the statedependence, the independence of questions mayalso be viewed as observer dependent. For in-stance, in quantum theory an observer O ′ couldsend O an entangled pure state (with correlationin x -direction) and refuse to tell O which state itis. Relative to O ′ , Q and Q will be dependent,but they will be independent relative to O be-cause the latter will assign the completely mixedstate to the pair prior to measurement.Since the notion of independence of questionsis state dependent, we need a distinguished statein order to unambiguously define it – this is thesecond purpose of assumption 3 and brings us incontact with a state update rule. Indeed, suppose O acquires S in the state of no information andposes the question Q i ∈ Q . By assumption 1,any Q i ∈ Q is repeatable such that, upon receiv-ing either the answer ‘yes’ or ‘no’, O will assign y i = 1 or y i = 0 , respectively, as the probabilityfor S giving the answer ‘yes’ if posing Q i again.The posterior state update rule, which enables O to update his information about a specific S incompliance with the given answers, must respectthis repeatability. It depends on the details of thisupdate rule what the probabilities y j for all other Q j ∈ Q is after having asked only Q i . Ratherthan fully specifying at this stage what this ruleis, we shall simply assume that O employs onewhich is consistent. Whatever this posterior stateupdate rule, we shall refer to Q i , Q j ∈ Q as Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify ndependent if, after having asked Q i to S inthe state of no information, the probability y j = . That is, if the answer to Q i relativeto the state of no information tells O ‘noth-ing’ about the answer to Q j . We shall requirethis relation to be symmetric, i.e., Q i is in-dependent of Q j if and only if Q j is indepen-dent of Q i . This is equivalent to saying that Q i , Q j are stochastically independent withrespect to the state of no information, i.e. thejoint probabilities factorize relative to thelatter, p ( Q i , Q j ) = y ∗ i · y ∗ j = · = . Here p ( Q i , Q j ) = p ( Q j , Q i ) denotes the probabil-ity that Q i and Q j give ‘yes’ answers if askedin sequence to the same S arriving in thestate of no information and y ∗ i , y ∗ j are theindividual ‘yes’-probabilities in this distin-guished state. dependent if, after having asked Q i to S inthe state of no information, the probability y j = 0 , . That is, if the answer to Q i rel-ative to the state of no information impliesalso the answer to Q j . Again, we require thisrelation to be symmetric. This is equivalentto saying that, relative to the state of no in-formation, Q i , Q j are stochastically fully de-pendent as either p ( Q i , Q j ) = y ∗ i = y ∗ j = = p ( ¬ Q i , ¬ Q j ) or p ( Q i , ¬ Q j ) = y ∗ i = y ∗ j = = p ( ¬ Q i , Q j ) , where ¬ Q is the negation of Q . partially dependent if, after having asked Q i to S in the state of no information, the prob-ability y j = 0 , , . That is, if the answerto Q i relative to the state of no information It should be noted that, while this is true for projec-tive measurements in quantum theory, it does not holdfor generalized measurements. I thank Tobias Fritz forpointing this out. We emphasize that this is a definition of maximal inde-pendence. For example, for a qubit two linearly indepen-dent directions ~n , ~n in the Bloch sphere with ~n · ~n = 0would define two spin observables ~n · ~σ and ~n · ~σ whosecorresponding projectors would not be maximally inde-pendent according to this definition. The correspondingquestions would be partially dependent (see below) be-cause whenever the observer knows the answer to onequestion, the probability for a ‘yes’ answer to the otherwould be distinct from . The most trivial example of a pair of dependent ques-tions are clearly Q and ¬ Q . But there exist less trivialones. E.g., see theorem 5.7 below. gives O partial information about the answerto Q j . Again, this relation is required to besymmetric. This is equivalent to saying thatthe joint probabilities p ( Q i , Q j ) relative tothe state of no information do not factorizeand the answer to one question does not fullyimply the answer to the other.We shall henceforth assume that Σ is such thattwo questions Q i , Q j which are fully or partiallydependent relative to the state of no informationare also fully or partially dependent relative toany other state in Σ . These definitions of (in-)dependence depend a priori on the update rulethrough the joint probabilities. For our purposesit will turn out not to be necessary to fully specifythis update rule.Next, we need a notion of compatibility andcomplementarity. Q i , Q j are maximally compatible if O may know the an-swers to both Q i , Q j simultaneously, i.e. ifthere exists a state in Σ such that y i , y j canbe simultaneously or . maximally complementary if maximal infor-mation about the answer to Q i forbids O tohave any information about the answer to Q j at the same time (and vice versa). That is,every state in Σ which features y i = 0 , willnecessarily have y j = (and vice versa).Consequently, complementary questions are in-dependent, but independent questions are notnecessarily complementary. Finally, Q i , Q j arepartially compatible (or complementary) if max-imal information about one precludes maximal,but permits non-maximal information about theother.This permits us to further constrain the up-date rule. Firstly, maximal complementarity hasan obvious consequence for an update rule. Sec-ondly, we shall assume the following. Assumption 4. If Q i , Q j are maximally com-patible and independent then asking either shallnot change O ’s information about the other, re-gardless of S ’s state. That is, asking Q i mustleave y j invariant – and vice versa for i, j inter-changed. This is to prevent O from losing or gaining in-formation about some question by asking another Accepted in
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Quantum2017-11-27, click title to verify uestion which is compatible with but indepen-dent of the first. These constraints on the updaterule turn out to be sufficient for our reconstruc-tion.Assumption 4 leads to a first result which weshall use at times: it implies what sometimes isreferred to as ‘Specker’s principle’ [68] (see also[69–72]). We shall call Q , . . . , Q n ∈ Q mutuallymaximally compatible if there exists a state of S where the answers to all of Q , . . . , Q n are knownsimultaneously to O . Theorem 3.2. (‘Specker’s principle’) If Q , . . . , Q n ∈ Q are pairwise maximally compat-ible and pairwise independent then they are alsomutually maximally compatible.Proof. Let Q , . . . , Q n ∈ Q be pairwise indepen-dent and pairwise maximally compatible. On ac-count of repeatability, after asking S the ques-tion Q , O will assign the probability y to beeither 0 or 1, depending on the answer. O maysubsequently ask S the question Q upon whichthe probability y will likewise be either 0 or 1.According to assumption 4, this will not change y because, by assumption, both are maximallycompatible and independent. By the same ar-gument, upon next asking and receiving an an-swer to Q , O will assign either y = 1 or y = 0, while both y , y are unchanged. Re-peating this argument recursively, it is clear that O can thereby generate a state of S in which all y i , i = 1 , . . . , n , are either 0 or 1 which representsa state of S where O knows the answers to all of Q , . . . , Q n . Notice that this property is satisfied for classi-cal bit theory and in quantum theory for projec-tive, however, not for generalized measurements.
Given multiple questions, nothing, in principle,stops O from composing them via logical con-nectives to all kinds of propositions which them-selves constitute questions. The issue is, however,whether he will be able to get an answer from S tosuch a question, i.e. whether it is implementableon S and thus whether it could be contained in Q (see section 3.2.1). For example, given any twoquestions Q i , Q j ∈ Q , O could consider the cor-relation question Q ij , ‘are the answers to Q i , Q j the same?’. Do we allow Q ij to also be imple-mentable on S ? Clearly, if Q i , Q j are maximallycompatible, then Q ij is implementable because O can always find the answer to the latter by asking Q i , Q j . In this case, since Q i , Q j are simultane-ously defined relative to O , we can also write Q ij := Q i ↔ Q j , (2) where ↔ is the logical biconditional or(XNOR). Q ij will then automatically be maxi-mally compatible with both Q i , Q j .Let us now contrast this with the situation inwhich Q i , Q j are (partially or maximally) comple-mentary. The structure introduced thus far doesnot preclude the ‘correlation question’ Q ij to alsobe implementable and, ultimately, contained in Q even if Q i , Q j are complementary. In fact, it is not possible to decide without a theoretical model fordescribing and interpreting ( Q , Σ) whether Q ij isimplementable on S in the case that Q i , Q j are atleast partially complementary. What, however, isunambiguously clear is that for all practical pur-poses Q ij will not be implementable in this case.Namely, Q ij would be a statement about the cor-relation of two complementary questions Q i , Q j and O could say ‘the answers are the same’, buthe can never directly test them individually andsee that they are actually ‘the same’ at the sametime. Indeed, Q i , Q j , Q ij would need to forma mutually (partially or maximally) complemen-tary set. Thus, from a purely operational per-spective alone, O could never tell, even in princi-ple, that Q ij was implementable; he simply can-not get an answer to Q ij which he could inter-pret, based on operationally accessible informa-tion alone, as a correlation of the complementary Q i , Q j .It thus depends on the model for ( Q , Σ) whether Q ij is implementable when Q i , Q j arecomplementary. Since it is impossible for O tosettle this question operationally, it would re-quire a model which employs hidden and opera-tionally inaccessible ontic information and whichis devoid of complementarity at an ontic levelto conclude that Q ij ∈ Q in this case. For ex-ample, Spekkens’ elegant toy model [40] and the‘black boxes’ of [54] employ ontic states, satisfythe structure established thus far (at least at the That is Q ij = ‘yes’ if Q i = Q j = ‘yes’ or ‘no’ and Q ij = ‘no’ otherwise. Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify pistemic level and modulo restrictions on the no-tion of convexity), and explicitly feature such atriple of questions. However, such a model isin conflict with our premise of following a purelyoperational approach which only speaks about in-formation that O has access to via direct inter-rogation. The model by means of which O in-terprets the answers that he gets from S – and,hence, the information he can acquire about S –shall be based entirely on operational statementsthat O can, in principle, check through interroga-tion, and not on propositions that require hiddenand inaccessible ontic information. Accordingly, O ’s theoretical model for Q should not contain(a question which is logically equivalent to) Q ij whenever Q i , Q j are at least partially complemen-tary.Of course, the ‘correlation’ (represented by theXNOR connective ↔ ) is only one of many possi-ble logical connectives. More generally, we shallrequire that O ’s model for Q does not contain any logical connectives of complementary questions.He can only logically connect questions which aremaximally compatible – and thus are simultane-ously defined with respect to him – such that hecould meaningfully write down a truth table forthe questions to be connected and the connectivequestion. If he cannot connect questions, he canalso not ask for the connective. Assumption 5.
Let Q i , Q j ∈ Q and ∗ be a logi-cal connective. According to O ’s theoretical modelfor Q , Q i ∗ Q j is implementable if and only if Q i , Q j are maximally compatible. The reader familiar with Spekkens’ toy model [40] willrecall the simplest (epistemic) 1-bit system which has fourontic states ‘1’, ‘2’, ‘3’ and ‘4’. An epistemic restrictionforbids an observer to know the ontic state. Instead, theepistemic states of maximal knowledge correspond to ei-ther of the following three questions (and their negations) Q : “1 ∨ , Q : “2 ∨ ,Q : “2 ∨ , where ∨ is to be read as ‘or’. Q , Q , Q are mutuallycomplementary and it can be easily checked that Q coin-cides with the ‘correlation’ Q of Q and Q : it gives ‘yes’when the (ontic) answers to Q , Q are equal and ‘no’ oth-erwise. This relation is cyclic: Q is also the ‘correlation’ Q of Q , Q and Q is the ‘correlation’ Q of Q , Q .Accordingly, there are three complementary questions forthis 1-bit system, in principle the correct number for aqubit. Later in section 5.2.3, we will develop a differentapproach, without ontic states, in order to reason for thethree-dimensionality of the Bloch-sphere. We recall from section 3.2.1 that O ’s theoreticalmodel for Q only contains implementable ques-tions (but not necessarily all implementable ques-tions).Since ∗ is only applied to connect maximallycompatible questions which have also opera-tionally simultaneous truth values relative to O ,we shall allow ∗ to be any of the 16 binary connec-tives (or binary Boolean functions) ¬ , ∨ , ∧ , ↔ , . . . of classical Boolean or propositional logic.We stress that assumption 5 is a restriction on O ’s model for Q ; this assumption clearly doesnot rule out that there may exist other (namely,ontic) models which also yield a consistent de-scription of O ’s experiences with the systems heis interrogating, yet which ascribe Q i ∗ Q j to be(logically equivalent to a question) contained in Q regardless of whether Q i , Q j are compatibleor not. However, we shall not worry about suchmodels here.We note that assumption 5 does not only ap-ply to logical connectives of two questions, butarbitrarily many. For example, if Q i ∗ Q j is im-plementable (and contained in Q ), according to O ’s model, then ( Q i ∗ Q j ) ∗ Q k is implementabletoo iff Q i ∗ Q j and Q k ∈ Q are compatible, andso on. It is also important to note that assump-tion 5 is a statement about which questions canbe directly connected logically. It does not entailthat O , according to this model, can only formmeaningful logical expressions containing exclu-sively questions which are mutually maximallycompatible. Namely, questions which are com-patible might be logically further decomposableinto other questions for which different compat-ibility relations hold. For instance, it might bethat Q k in the expression ( Q i ∗ Q j ) ∗ Q k is onlycompatible with Q i ∗ Q j , yet not with Q i or Q j in-dividually. In this case, there is no harm in never-theless forming the expression ( Q i ∗ Q j ) ∗ Q k eventhough, say, Q j and Q k might be complemen-tary such that O cannot write down a truth tablewith all three Q i , Q j , Q k separately. However, itis clear that in this case there is a hierarchy inwhich the logical connectives are to be executed.More specifically, the connective ∗ in ( Q i ∗ Q j ) ∗ Q k cannot be associative since Q i ∗ ( Q j ∗ Q k ) is notimplementable according to O ’s model because Q j , Q k are, by assumption, complementary andcan therefore not be directly connected. That isto say, the left ∗ in ( Q i ∗ Q j ) ∗ Q k must be executed Accepted in
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Quantum2017-11-27, click title to verify rst and Q k can only be logically connected withthe result. Thus, assumption 5 does admit logicalexpressions containing also mutually complemen-tary questions as long as the latter are not directly connected through a logical connective. What as-sumption 5 entails is that the ordering of the exe-cution of the various logical connectives in a com-position has to respect a hierarchy which ensuresthat at every step two compatible subexpressionsare connected. This will become important laterin the reconstruction where we shall encounterconcrete such examples.This brings us to the rules of inference bymeans of which O may transform or evaluatelogical expressions of questions and derive logi-cal identities, e.g. to establish possible logical re-lations among various question outcomes. Thestate update rules need, in particular, respectthese rules of inference. Again, the rules of in-ference which O may employ depend on his the-oretical model for ( Q , Σ) . For instance, in onticmodels all questions will typically have simultane-ous values, in contrast to purely operational ones,which will require different rules of inference.In line with our operational premise, we shallrequire that the rules of inference of O ’s modelmust not only be consistent with O ’s experienceswith the systems he is interrogating, but alsotestable. In particular, any logical identities de-rived from these rules must be testable throughinterrogations. It is thus appropriate to call them operational rules of inference .Classical rules of inference require that thequestions or propositions in a logical expressionhave truth values simultaneously. In O ’s modelany truth value must be operational such thatonly mutually maximally compatible questionshave simultaneous meaning. Nothing stops O from applying classical rules to such kind of ques-tions. We shall thus require that it is appropri-ate for O to employ classical rules of inference,according to Boolean logic, – and only those –for any logical expression or subexpression whichonly contains questions that are mutually maxi-mally compatible. In all other cases (which stillmust abide by assumption 5), any possible rule ofinference transforming the composition of ques-tions will directly involve mutually complemen-tary questions whose truth values, however, haveno simultaneous operational meaning to O . Ac-cordingly, classical rules will be operationally in- appropriate in such cases and we shall demandmore precisely the following. Assumption 6. O ’s model for Q allows him toapply exclusively classical rules of inference, ac-cording to the rules of Boolean logic, to (trans-form or evaluate) any logical expression (orsubexpression of a larger expression) which iscomposed purely of mutually maximally compati-ble questions. This holds regardless of whetherthe mutually compatible questions can be logi-cally decomposed into other questions which fea-ture different compatibility relations. In all othercases, no classical rules of inference (and thus noclassical logical identities) are valid. In consequence, O can take any set of mutuallycompatible questions and treat them, with clas-sical logic, as a Boolean algebra. The assump-tion states what is possible for compositions ofcompatible questions and what is not possible forcompositions involving also complementary ques-tions. This will become useful later in section5.2.7 where we shall also see what is possible forcompositions of complementary questions.We emphasize that assumption 6 by itself does not severely constrain the nature of any conceiv-able ‘hidden variable model’ which could also con-sistently describe O ’s world. However, in con-junction with two of the subsequent quantumprinciples, it will rule out local ‘hidden variables’in section 5.2.7. S ’s state and tomogra-phy Now that we have a notion of independence on Q we can say more about the parametrization andthus representation of S ’s state relative to O . Notall Q i ∈ Q will be necessary to describe the state;the pairwise independent questions shall be thefundamental building blocks of the landscape L of inference theories.Suppose there is a maximal set Q M = { Q , . . . , Q D ∈ Q} of D pairwise independent(but not necessarily compatible) questions, suchthat no further question Q ∈ Q\Q M exists whichis pairwise independent from all members of Q M too. Then, every other Q ∈ Q \ Q M is either(i) dependent on exactly one Q j ∈ Q M and in-dependent of all other Q M ∋ Q l = Q j (if Q was dependent on Q j ∈ Q M and partially de-pendent on Q M ∋ Q l = Q j , then Q j , Q l could Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify ot be independent), (ii) partially dependent onsome and independent of the other questions in Q M , or (iii) partially dependent on all Q j ∈ Q M .While O will not be able to infer maximal infor-mation about the answers to questions of cases(ii) and (iii) from his information about individ-ual (or even subsets of) members of Q M alone,the question is whether his information about thefull set Q M will be sufficient to do so. Definition 3.3. (Informational Complete-ness)
A maximal set Q M = { Q , . . . , Q D ∈ Q} of pairwise independent questions is said to be in-formationally complete if O ’s information aboutthe questions in Q M determines his informationabout all other Q ∈ Q \ Q M in such a waythat the probabilities y i which O assigns to every Q i ∈ Q M are sufficient in order for him to com-pute the probabilities y j ∀ Q j ∈ Q for all prepa-rations of S . In this case, the set of probabilities { y i } Di =1 of the Q i ∈ Q M parametrizes the statethat O assigns to S and thereby yields a com-plete description of the state space Σ . We shallcall D the dimension of Σ . We emphasize that the dimension D of Σ willultimately not be the Hilbert space dimension inquantum theory, but the dimension of the set ofdensity matrices.If Q M was not informationally complete, O would require further questions that are partiallydependent on at least some of the elements in Q M in order to fully describe the system S andits state. This situation cannot be precluded,given the structure we have devised so far. How-ever, we deem it undesirable, given that we wouldlike to employ pairwise independent questions asbuilding blocks for system descriptions. We shalltherefore require that no more independent infor-mation about S can be learned from any questionin addition to a maximal set Q M . Assumption 7.
Every maximal set Q M of pair-wise independent questions is informationallycomplete. There may exist (even continuously) many suchinformationally complete sets of questions on Q which, at this stage, may still be either discrete orcontinuous. However, their dimensions are equal. Lemma 3.4.
The dimensions of all maximal sets Q M on Q are equal (if finite). Proof. Consider any Q M . On account of pairwiseindependence, for each Q i ∈ Q M there must exista state such that O only knows the answer tothis Q i with certainty, but nothing about anyother Q j = i ∈ Q M , i.e. y i = 0 or 1 and all other y j = i = 1 /
2. (Namely, O could have asked Q i to S prepared in the state of no information.) Butsince also in the state of no information y j = i =1 / y i cannot in general be computed from the y j = i . Hence, given an informationally completeset Q M , all associated probabilities { y i } Di =1 arenecessary to parametrize the full state space Σ.Now Σ is a convex set (see assumption 2).Given the lack of redundancy in { y i } , a giveninformationally complete Q M with D elementstherefore encodes Σ as a D -dimensional convexsubregion of R D . Suppose there is a second Q ′ M with D ′ elements. In terms of Q ′ M , Σ would bedescribed as a D ′ -dimensional convex subregionof R D ′ . But clearly, these two descriptions of Σmust be isomorphic which, for D finite, is onlypossible if D ≡ D ′ . Finiteness of D will later be established in sec-tion 5 with the help of the principles.Any such Q M establishes a question referenceframe on Q (see also [73]) and thereby also a ‘co-ordinate system’ on Σ . In particular, in orderto do state tomography with a multiple shot in-terrogation, as outlined in section 3.2.3, it willbe sufficient for O to interrogate an ensemble ofidentically prepared S with the questions withina given Q M only. Given a specific Q M , thereare now three equivalent ways for O to describe S ’s state: he could represent it by either the D -dimensional yes- or no-vector ~y O → S := y y ... y D , ~n O → S := n n ... n D , (3) of probabilities y i and n i , i = 1 , . . . , D , that theanswers to question Q i ∈ Q M are ‘yes’ and ‘no’,respectively. That is, ~y O → S + ~n O → S = p~ , (4) where ~ is a D -dimensional vector with a in eachof its entries. Here we have introduced a new pa-rameter: p is the probability that S is presentat all. (For example, the method of preparation Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify ould be such that O cannot always tell with cer-tainty when a system is spit out by the prepara-tion device.) This probability will rescale all y i and n i simultaneously. However, it only becomesrelevant in subsection 3.2.8 and otherwise is usu-ally taken to be p = 1 . Evidently, the assignmentof which answer to Q i is ‘yes’ and which is ‘no’ isarbitrary, but any consistent such assignment isfine for us.But he could also represent the state redun-dantly as a D -dimensional vector ~P O → S = ~y~n ! (5) which will turn out to be convenient especiallywhen p < . We shall write a state with a sub-script O → S to emphasize that it is the state of O ’s information about S .Lastly, this structure also puts us into the po-sition to specify O ’s total amount of informa-tion about S . Clearly, the total amount of in-formation must be a function of the state. Let Q M = { Q , . . . , Q D } be an informationally com-plete set of questions in Q . Given that these ques-tions carry the entire information O may knowabout S , we define the total information I O → S asthe sum of O ’s information about the Q i ∈ Q M ,as measured by the α i (1): I O → S ( ~y O → S ) := D X i =1 α i . (6) (We imagine O as an agent who can write downresults on a piece of paper and add these up.) Thespecific relation between α i and ~y O → S will be de-rived later in section 6.8; it will not be the Shan-non entropy which, as discussed in section 6.1,describes average information gains in repeatedexperiments rather than the information contentin the state ~y O → S . For later purpose we need to clarify the notionof a composite system. Since we are pursuing apurely operational approach, the notion of a com-position of systems must be defined in terms ofthe information accessible to O through interro-gation. O should, in principle, be able to tella composite system apart into its constituents.Accordingly, we require that the set of questions which can be posed to the composite system con-tains all questions about the individual subsys-tems and that the remaining questions are lit-erally composed of these individual questions oriteratively composed of compositions of them. Definition 3.5. (Composite System)
Let Q A , Q B denote the full question sets associatedto S A , S B , respectively. Two systems S A , S B aresaid to form a composite system S AB if any Q a ∈ Q A is maximally compatible with and in-dependent of any Q b ∈ Q B and if Q AB = Q A ∪ Q B ∪ ˜ Q AB , (7) where ˜ Q AB only contains composite ques-tions which are iterative compositions, Q a ∗ Q b , Q a ∗ ( Q a ′ ∗ Q b ) , ( Q a ∗ Q b ) ∗ Q b ′ , ( Q a ∗ Q b ) ∗ ( Q a ′ ∗ Q b ′ ) , . . . , via some logical con-nectives ∗ , ∗ , ∗ , · · · , of individual questions Q a , Q a ′ , . . . ∈ Q A about S A and Q b , Q b ′ , . . . ∈ Q B about S B . For an N -partite system, we use thisdefinition recursively. Of course, thanks to assumption 5, O can onlydirectly compose questions with a logical connec-tive ∗ if they are compatible.There are further repercussions: let Q M A , Q M B be informationally complete sets for S A , S B , re-spectively. Then an informationally complete set Q M AB for a composite S AB can be formed itera-tively by joining (in a set union sense) Q M A , Q M B and adding to it a maximal pairwise independentset of questions which are the logical connectivesof members of Q M A with elements of Q M B and/oriterative compositions of such compositions andpossibly questions from Q M A , Q M B . In section5.2.1 below, we shall determine which logical con-nectives ∗ we may employ to build up Q M AB from Q M A , Q M B . S ’s state We shall assume that O has access to a clock andbegin with a definition: static states: A state ~P O → S which is constant in time – corresponding to the situation that For the GPT specialists, we emphasize that this defi-nition has nothing to do with the usual requirement of localtomography [15–18,20,55,59] in GPTs. To give a concreteexample, we shall see later that two-level systems over realHilbert spaces (rebits) satisfy this definition while violat-ing local tomography.
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Quantum2017-11-27, click title to verify will always assign the same probabilityto each of his question outcomes – is called static .If all the states O would assign to any system S were static – according to his information – O ’s world would be a rather boring place. In or-der not to let O die of boredom, we allow theprobability vector ~P O → S to change in time. (Butwe require that Q and Σ are time independent .)We shall now make use of operational reasoning,to briefly consider how ~P O → S evolves under timeevolution. The following argument bears someanalogy to an argument typically employed inthe GPT framework [14–18, 55] concerning con-vex mixtures and measurements, however, is dis-tinct in nature as we instead consider states andtheir mixtures under time evolution.Let O have access to two identical non-interacting systems S and S . O ’s informationabout S and S can be different such that theyare allowed to be in distinct states ~y O → S and ~y O → S relative to O . Now let O perform a (bi-ased) coin flip which yields ‘heads’ with probabil-ity λ and ‘tails’ with probability (1 − λ ) . Giventhat S and S are identical, O can ask the samequestions to both systems. If the coin flip yields‘heads’, O will interrogate S , if it yields ‘tails’ O will interrogate S .PSfrag replacements OS S λ − λ In particular, before tossing the coin, say attime t , the probability he assigns to receiving a‘yes’ answer to question Q i (asked to either S or S depending on the outcome of the coin flip)is simply the convex sum y i ( t ) = λ y i ( t ) +(1 − λ ) y i ( t ) . ( O is allowed to build this con-vex combination thanks to assumption 2.) Butthis holds at any time t before O tosses the coin(which O can also choose never to do). As will That is, both S and S carry the same Q and Σ. Equivalently, O may use another system to which hehas assigned a stable state vector by repeated interroga-tions on identically prepared systems. Given an arbitraryelementary question Q , he can use the probability that theanswer is ‘yes’ as λ and the probability that the outcomeis ‘no’ as (1 − λ ) and thus use Q as the ‘coin’. We assume λ to be constant in time. become clear shortly, it is convenient to considerthe states ~P O → S with ≤ p ≤ for now. Weshall return to the yes-vector later in section 6.We then have for all t~P O → S ( t ) = λ ~P O → S ( t ) + (1 − λ ) ~P O → S ( t ) . (8) This convex combination is the state of an ‘effec-tive’ system S (identical to S , S ) describing O ’s information about the coin flip scenario. Notethat S is not a composite system according todefinition 3.5 because Q = Q = Q .Denote by T k the time evolution map of thestate of system S k , k = 1 , , , from t = t to t = t . Under the assumption that ~P O → S , ( t ) evolve independently of each other, given that S and S do not interact, equation (8) implies T [ ~P O → S ( t )] = T [ λ ~P O → S ( t )+ (1 − λ ) ~P O → S ( t )]= λ T [ ~P O → S ( t )] (9)+ (1 − λ ) T [ ~P O → S ( t )] . All three states ~P O → S k , k = 1 , , , are elementsof the same Σ . As such, we assume the following. Assumption 8.
Every time evolution map canact on any state in Σ . Next, we restrict to the situation where O ex-poses both S , S to evolve under the same timeevolution T := T = T . Thanks to assumption8, (9) becomes in this case T [ ~P O → S ( t )] = λ T [ ~P O → S ( t )]+ (1 − λ ) T [ ~P O → S ( t )] and T is convex linear . This is clear since O may,in particular, prepare S , S in identical states ~P O → S ( t ) = ~P O → S ( t ) = ~P O → S ( t ) . This spe-cial case, when inserted into (9), yields T = T .This scenario can be easily generalized to ar-bitrarily many identical systems. Remarkably, itcan be shown that convex linearity of operationson probability vectors, in fact, implies full lin-earity of the operation [14, 55]. Hence, the timeevolution of the state must be linear : ~P O → S ( t ) = T [ ~P O → S ( t )]= A ( t , t ) ~P O → S ( t ) + ~V ( t , t ) , where A ( t , t ) is a D × D matrix and ~V some D -dimensional vector. Given our assumptionabove that O ’s world is not static , A ( t , t ) will Accepted in
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Quantum2017-11-27, click title to verify enerally be a non-trivial matrix. Requiring thatthe last relation also holds at t = t for allinitial states, it follows that ~V ( t, t ) = 0 and A ( t, t ) = . Demanding further that the spe-cial state ~P O → S = ~ , corresponding to p = 0 and the absence of a system, is invariant un-der time evolution (such that no system or anyinformation is created out of ‘nothing’), finallyyields ~V ≡ ~ , ∀ t , t . Given that ~P O → S ( t ) is aprobability vector for all t and (4) must alwayshold, A ( t , t ) must be a nonnegative (real) ma-trix which is stochastic in any pair of components i and i + D of ~P O → S .Consequently, the elementary and natural as-sumption 8 rules out non-linear and state de-pendent time evolution, such as in Weinberg’snon-linear extension of quantum mechanics [74],and thereby also eliminates the many operationalproblems that arise with it (e.g., superluminalsignaling) [75–77]. For later purpose, we impose a further condi-tion on the evolution of S ’s state. Assumption 9. (Temporal Translation In-variance)
There exist no distinguished instantsof time in O ’s world such that O is free to setany instant he desires as the instant t = 0 . Timeevolution, as perceived by O , is therefore (tempo-rally) translation invariant A ( t , t ) = A ( t − t ) . Since the time evolution matrix A can only de-pend on the duration elapsed, but not on the par-ticular instant of time, we can collect the aboveresults in the simple form: ~P O → S ( t ) = A (∆ t ) ~P O → S ( t ) , ∆ t = t − t . (10) We do not yet have sufficient evidence to con-clude that a given time evolution will be describedby a continuous one-parameter matrix group be-cause we neither know (i) that A is invertible for A similar argument would not work for the evolutionof ~y O → S because, in principle, ~n O → S = ~ ~y O → S = ~ ~ D -dimensional zero vector).This is the reason why here it is more convenient to workwith ~P O → S , which contains the information about both ~y O → S , ~n O → S , rather than ~y O → S alone. In fact, we shallsee later in section 6 that the evolution of the latter does involve an affine D -dimensional vector ~V ′ = ~ It should be emphasized that assumption 8 is also tac-itly made in the GPT framework [14–18, 55] for any kindof transformations on states, such that the present setupis in this regard not less potent. all ∆ t , (ii) that the evolution is actually continu-ous, nor (iii) that every composition of evolutionmatrices is again an evolution matrix. We shallreturn to this question in section 6 with the helpof the set of principles for quantum theory andalso defer the discussion of the time evolution of ~y O → S until then.For now we note, however, firstly, that any A (∆ t ) acting on ~P O → S implies a unique map T ∆ t ( ~y O → S ) acting on the yes-vector thanks to (4,5). Secondly, a multiplicity of time evolutionsof S is possible, depending on the physical cir-cumstances (interactions) to which O may sub-ject S . The set of all possible time evolutionswhich O shall henceforth be able to implementwill be denoted by T . This set need not containall physically possible time evolutions. Indeed,upon imposing the quantum principles, T will be-come the set of unitaries (rather than of arbitrarycompletely positive maps). Clearly, T is part of O ’s model for describing S ; his model is thus atriple ( Q , Σ , T ) . L The landscape L of theories describing O ’s acqui-sition of information about a physical system S is now the set of all theories which comply withthe structure and assumptions established in thissection.In the sequel, we shall restrict O ’s attentionsolely to composite systems of N ∈ N general-ized bits (gbits), where a single gbit is character-ized by the fact that O can maximally know theanswer to a single question at once such that itcan carry at most one bit of information. Ev-ery inference theory specifies for every system of N gbits a triple ( Q N , Σ N , T N ) . We shall be con-cerned with the landscape of gbit theories L gbit which contains all gbit theories, satisfying as-sumptions 1–9. To name concrete examples atthis stage, L gbit contains, among a continuum ofother theories, classical bit, rebit and qubit the-ory for all N ∈ N . We shall see more of this later,but for now we summarize their characteristicsfor N = 1 , classical bit theory gives Q = { Q, ¬ Q } , Σ ≃ [0 , (for normalized states) with ex-tremal points corresponding to Q = ‘yes’ and‘no’, respectively, and T ≃ Z . There are precisely two states of maximal information
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Quantum2017-11-27, click title to verify ebit theory (two-level systems on real Hilbertspaces) yields Q ≃ S and every two maxi-mally complementary questions are informa-tionally complete. Σ ≃ D (for normalizedstates) and T ≃ SO(2) . qubit theory has Q ≃ S and every tripleof mutually maximally complementary ques-tions are informationally complete. Σ ≃ B (for normalized states) and T ≃ SO(3) .With all the assumptions made along the way,the theory landscape devised here is perhaps notquite as general as the commonly employed land-scape of generalized probability theories [14–18,23, 55, 60, 61]. In particular, GPTs easily handlearbitrary finite dimensional systems, while thisremains to be done within the present frame-work. Nevertheless, the new landscape L gbit islarge enough and provides new tools for a non-trivial and instructive (re)construction of qubitquantum theory that approaches the latter from aconceptually and technically new angle. To facil-itate future generalizations of this work, we havealso attempted to spotlight the assumptions un-derlying L gbit as clearly as possible in this section. We shall now use the landscape L gbit of gbit theo-ries and formulate the principles for the quantumtheory of qubit systems on it as rules on an ob-server’s information acquisition. These principlesconstitute a set of ‘coordinates’ of qubit quantumtheory on L gbit . that O can assign to a classical bit ~P O → S = (cid:18) (cid:19) , ~P O → S = (cid:18) (cid:19) , corresponding to ‘yes’ and ‘no’ answers to Q . Time evo-lution is described by the abelian group Z , given by = (cid:18) (cid:19) , P = (cid:18) (cid:19) , where P swaps the two states. For classical bit theory, thepermitted time evolution is therefore discrete . As we are reconstructing a technically and empir-ically well-established theory, we are in the fortu-nate position to avail ourselves of empirical ev-idence and earlier ideas on characterizing quan-tum theory in order to motivate a set of basicpostulates. However, the ultimate justificationfor these postulates will be their success in sin-gling out qubit quantum theory within the infer-ence theory landscape L gbit which will be com-pleted in [1]. As ‘coordinates’ on a theory spacethese principles will not be unique and one couldfind other equivalent sets. (As usual, at leastmany roads lead to Rome.) But we shall takethe below ones as a first working set which re-stricts the informational relation between O and S , where S will be a composite system (c.f. def-inition 3.5) of N ∈ N gbits. Each principle willbe first expressed as an intuitive and colloquialstatement, followed by its mathematical meaningwithin L gbit .We take it as an empirical fact that there ex-ist physical systems about which only a limitedamount of information can be known at any onemoment of time. The standard quantum exampleis a spin- particle about which an experimentermay only know its polarization in a given spatialdirection, but nothing independent of that; thepolarization ‘up’ or ‘down’ corresponds to one bit of information. But there is also a typical classi-cal example, namely a ball which may be locatedin either of two identical boxes, the definite po-sition ‘left’ or ‘right’ corresponding to one bit ofinformation. Informationally, these two examplesincarnate the most elementary of systems, a gbit,supporting maximally just one proposition at atime. But, clearly, there are more complicatedsystems supporting other limited amounts of in-formation. We shall take this simple observationand raise the existence of an information limit tothe level of a principle.More precisely, in analogy to von Weizsäcker’s‘ur-theory’ [78–80], we shall restrict O ’s worldto be a world of elementary alternatives whichthereby consists only of systems which can be de-composed into elementary gbits. All physical However, in contrast to [78–80], we shall be muchless ambitious here and will not attempt to deduce thedimension of space or space-time symmetry groups fromsystems of elementary alternatives. Recent developments[21, 42, 59, 81], on the other hand, unravel a deep rela-
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Quantum2017-11-27, click title to verify uantities in O ’s world are to be finite such thathe can record his information about them on afinite register. We wish to characterize the com-posite systems in O ’s world according to the finitelimit of N bits of information he can maximallyinquire about them. Rule 1. (Limited Information) “The observer O can acquire maximally N ∈ N independent bits of information about the system S at anymoment of time.” There exists a maximal set Q i , i = 1 , . . . , N , of N mutually independent and maximally compat-ible questions in Q N and no subset in Q N cancontain more than N questions with that prop-erty. In other words, O can ask maximally N inde-pendent questions at a time to S . Accordingly,this rule immediately implies that O can distin-guish maximally N states of S in a single shotinterrogation because there will be N possibleanswers to the Q , . . . , Q N . Since Q N , Σ N are in-trinsic to S , also the maximal amount of N bits that each S can carry must be intrinsic to it andthus be observer independent. In fact, this rulecan be regarded as a defining property of all in-ference theories in the gbit landscape L gbit andcan thus clearly not distinguish between classicalbits, rebits, qubits, etc.We take another empirical fact and elevateit to a fundamental principle: despite the lim-ited information accessible to an experimenterat any moment of time, there always exists ad-ditional independent information that she maylearn about the observed system at other times.This is Bohr’s complementarity principle [82].Consider, for instance, the prototypical quantumphysics experiment: Young’s double slit exper-iment. The experimenter can choose whetherto obtain which-way-information or an interfer-ence pattern, but not both; the complete knowl-edge of whether the particle went through theleft or right slit is at the expense of total ig-norance about the information pattern and vice tion between (a) simple conditions on operations with sys-tems carrying finite information (e.g., communication withphysical systems) and (b) the dimension and symmetrygroup of the ambient space or space-time. Obviously, combinations (e.g., ‘correlations’) of thecompatible Q i will define other bits of information which,however, will be dependent once the Q , . . . , Q N are asked(we shall return to this in detail in section 5). versa. (For an informational discussion of theparticle-wave duality in Young’s double slit ex-periment and a Mach-Zehnder interferometer, seealso [32,35].) Similarly, in a Stern-Gerlach exper-iment an experimenter may determine the polar-ization of a spin- particle in x -direction, but willbe entirely oblivious about the polarization in y -and z -direction. A subsequent measurement ofthe spin of the same particle in y -direction willrender her previous information about the polar-ization in x -direction obsolete and keep her ig-norant about the spin in z -direction and so on.That is, systems empirically admit many moreindependent questions than they are able to an-swer at a time – thanks to the information limit.We shall now return to the relation between O and S and accordingly stipulate that complemen-tarity exists in O ’s world, however, we shall saynothing more about how much complementary in-formation may exist. Rule 2. (Complementarity) “The observer O can always get up to N new independent bits ofinformation about the system S . But whenever O asks S a new question, he experiences no netloss in his total information about S .” There exists another maximal set Q ′ i , i =1 , . . . , N , of N mutually independent andmaximally compatible questions in Q N suchthat Q ′ i , Q i are maximally complementary and Q ′ i , Q j = i are maximally compatible and indepen-dent. That is, after asking S a set of N indepen- Alternatively, one could formulate the technical partof this rule as follows:
In the N = 1 case of a singlegbit there exists, for every Q ∈ Q , another Q ′ ∈ Q such that Q , Q ′ are maximally complementary. Defini-tion 3.5 would then immediately imply that a compos-ite system of N gbits features two sets of questions, Q i , i = 1 , . . . , N , and Q ′ j , j = 1 , . . . , N , with the followingproperties: Q i , Q ′ i are questions about the elementary gbitlabeled by i , all Q i are maximally independent and max-imally compatible, all Q ′ j are maximally independent andmaximally compatible and any pair Q i , Q ′ j = i is maximallyindependent and compatible and every pair Q i , Q ′ i corre-sponding to the same gbit is maximally complementary.This is a priori not equivalent to the current formulationof the technical part of rule 2 in the main text as the lat-ter does not necessarily refer to questions about individualsubsystems of a composite system. However, the alterna-tive formulation would also be enough to get the correctstructure of the informationally complete sets for N qubitsand N rebits below. While the alternative formulation issimpler, we keep the other one as it is also published inthis form in the companion article [1]. Accepted in
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Quantum2017-11-27, click title to verify ent elementary propositions Q i , i = 1 , . . . , N ,in a single shot interrogation, O can pose a new ( N + 1) th elementary question Q ′ to S . Since O can only know N independent bits of informa-tion about S at a time and asking a new questiondoes not lead to a net loss of information, thesingle bit of his previous information about Q must have become obsolete upon learning the an-swer to Q ′ , while O ’s total information about S is still N bits . The rule allows O to perform thesame procedure until he replaces his N old by N new bits of information about S . As O ’s infor-mation about S has changed, the state of S rel-ative to O will necessarily experience a ‘collapse’whenever he asks a new complementary question.As a result, it is the observer who decides which information (e.g., spin in x - or y -direction) he willobtain about the system by asking specific ques-tions. But clearly O will have no influence onwhat the answer to these questions will be andany answer will come at the price of total igno-rance about complementary questions.A few further explanations concerning thiscomplementarity rule are in place. First of all,this rule clearly rules out classical bit theory. Sec-ondly, the postulate asserts the existence of max-imally complementary questions in O ’s world,however, makes no statement about whether par-tially independent or partially complementaryquestions may exist too. Thirdly, the peculiarrequirement that every question of rule 2 is com-plementary to exactly one from rule 1 is chosensuch that each subsystem of a composite systemfeatures complementarity. For example, considerthe N = 1 case, corresponding to a single gbit,for which rules 1 and 2 entail a complementarypair Q , Q ′ . This complementarity per gbit gen-eralizes to arbitrary N since every Q i in rule 1and every Q ′ j in rule 2 may correspond to one of N gbits. More complicated complementarity re-lations arising from rules 1 and 2 will be discussedin section 5.Notice that the complementarity rule impliesthe existence of a notion of ‘superposition’ – evenin states of maximal information . For example,take N = 1 and let O know the answer to Q with certainty, y = 1 , such that his informa-tion about S saturates the limit of one bit . Thiswill leave him oblivious about the complementary Q ′ such that he would have to assign y ′ = .The state of information he has about S can be interpreted as being in a ‘superposition’ of the Q ′ alternatives ‘yes’ and ‘no’, but in this casewith ‘equal weight’ because both alternatives areequally likely according to O ’s knowledge. Weemphasize that the necessary presence of super-positions of elementary alternatives, as perceivedby O , is a consequence of the information limitand complementarity.We would like to stress that rules 1 and 2are conceptually motivated by related propos-als which have been put forward first by Rov-elli within the context of relational quantum me-chanics [28] and later, independently, by Bruknerand Zeilinger within attempts to understand thestructure of quantum theory via limited informa-tion [31, 32, 34, 35, 73]. However, in order to com-plete these ideas to a full reconstruction of qubitquantum theory, we have to impose further rules.Specifically, rules 1 and 2 say nothing aboutwhat happens in-between interrogations. We re-quire that O shall not gain or lose informationabout an otherwise non-interacting S without asking questions. Rule 3. (Information Preservation) “The to-tal amount of information O has about (an oth-erwise non-interacting) S is preserved in-betweeninterrogations.” I O → S is constant in time in-between interroga-tions for (an otherwise non-interacting) S . Correspondingly, I O → S is a ‘ conserved charge ’of time evolution; this is a simple observationwhich will become extremely useful later. In fact,notice that rule 3 could also be viewed as defin-ing the notion of non-interacting systems (as per-ceived by O ).Finally, we come to the last rule of thismanuscript which is about time evolution. Em-pirical evidence suggests, at least to a good ap-proximation, that the time evolution of an ex-perimenter’s ‘catalogue of knowledge’ about anobserved system is continuous – in-between mea-surements. More precisely, the specific prob- We shall later see that, even in a state of maximalinformation of N bits , O may possibly have only partialor incomplete knowledge about the outcomes of any ques-tion in an informationally complete set Q M N of pairwiseindependent questions (c.f. assumption 7). In this case, O ’s information about all questions in Q M N will be in ageneral state of superposition because their correspond-ing probabilities cannot all be , otherwise it would bethe state of no information. Accepted in
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Quantum2017-11-27, click title to verify bilistic statements an experimenter can makeabout the outcomes of measurements, i.e. his ac-tual information about the systems, change con-tinuously in time. We promote this to a furtherpostulate for O ’s world, however, with a furtherrequirement.We note that the state space and the time evo-lutions are interdependent as every legal time evo-lution must map a legal state to a legal state, i.e. T ∆ t ( ~y O → S ) ∈ Σ N , ∀ T ∆ t ∈ T N and ∀ ~y O → S ∈ Σ N .Accordingly, what is the set of legal states de-pends on what is the set of legal time evolutions– and vice versa. We would like the pair (Σ N , T N ) to be as ‘big’ as compatibility with the other rulesallows in order to equip the other rules with asgeneral a validity as possible. But there are mul-tiple ways of ‘maximizing’ the pair. Namely, theinterdependence implies that the larger the num-ber of states, the tighter the constraints on the setof time evolutions – and vice versa. We requiredbefore that O ’s world should not be ‘boring’ andtherefore feature a non-trivial time evolution ofstates. We shall now sharpen this requirement: itis more interesting for O to live in a world which‘maximizes’ the number of possible ways in whichany given state of S can change in time, ratherthan the number of states which it can be in rela-tive to O . We shall thus require that any consis-tent, continuous time evolution of O ’s ‘catalogueof knowledge’ about S in-between interrogationsis physically realizable. Since different ways ofevolving the state of a system correspond to dif-ferent interactions (e.g., among the members of acomposite system) we thereby maximize the setof possible interactions among systems. O ’s worldis a maximally interactive place! Rule 4. (Time Evolution) “ O ’s ‘catalogue ofknowledge’ about S evolves continuously in timein-between interrogations and every consistentsuch evolution is physically realizable.” T N is the maximal set of transformations T ∆ t onstates which is continuous in ∆ t and compatiblewith rules 1-3 and the structure of L gbit . Of course, during an interrogation, ~y O → S maychange discontinuously, i.e. ‘collapse’, on accountof complementarity. As innocent as the require-ment of continuity of time evolution in rule 4 ap-pears, it turns out to be absolutely crucial in or-der to single out a unique information measure α i ( ~y O → S ) . Notice also that classical bit theory violates this postulate due to its discrete timeevolution group (see section 3.2.9).It is now our task to verify what the triplesof ( Q N , Σ N , T N ) for each N ∈ N are which obeyrules 1–4. Remarkably, it turns out that thesefour rules cannot distinguish between real andcomplex quantum theory, i.e. between real andcomplex Hilbert spaces. But the state spaces andthe orthogonal and unitary groups of time evo-lutions of rebit and qubit quantum theory are,in fact, the only ‘solutions’ within L gbit to theserules. In the sequel of this manuscript we shallemploy rules 1–4 in order to develop the necessarytools, within L gbit , for eventually proving the fol-lowing more precise claim in [1, 2]. In fact, as asimple example of the newly developed tools, weshall already prove the claim for N = 1 at theend of this article. Claim. L gbit contains only two solutions for thepair (Σ N , T N ) which are compatible with rules 1–4:
1. rebit quantum theory [2], where Σ N co-incides with the space of N × N densitymatrices over ( R ) ⊗ N and T N is PSO(2 N ) .
2. qubit quantum theory [1], where Σ N co-incides with the space of N × N densitymatrices over ( C ) ⊗ N and T N is PSU(2 N ) .Furthermore, states evolve unitarily accord-ing to the von Neumann evolution equation. We remind the reader that the time evolutiongroups T N for density matrices in rebit and qubitquantum theory are projective because they cor-respond to ρ U ρ U † , where for (1) rebits ρ is a N × N real symmetric matrix, U ∈ SO(2 N ) and † denotes matrix transpose; and (2) for qubits ρ is a N × N hermitian matrix, U ∈ SU(2 N ) and † denotes hermitian conjugation.Furthermore, using one additional rule on Q N which allows O to ask S any question that “makes(probabilistic) sense”, we show in [1, 2] that
1. in the rebit case [2], Q N is (isomorphicto) the set of projective measurements ontothe +1 eigenspaces of Pauli operators over R N . This is the set of rank- (2 N − ) -projectors. A real density matrix is a symmetric, positive semidef-inite matrix on ( R ) ⊗ N . The set of Pauli operators over R N is the set of trace-less real symmetric 2 N × N matrices with eigenvalues ± Accepted in
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Quantum2017-11-27, click title to verify . in the qubit case [1], Q N is (isomorphicto) the set of projective measurements ontothe +1 eigenspaces of Pauli operators over C N (i.e., the set of rank- (2 N − ) -projectors)and the outcome probability for any Q ∈ Q N to be answered with ‘yes’ by S in some stateis given by the Born rule for projective mea-surements. Since both real and complex quantum theorycome out of rules 1–4, we shall impose a furtherrule on O ’s information acquisition in [1] in orderto also eliminate rebit theory. However, we notethat rebit theory is mathematically contained inqubit quantum theory and that rebits can ac-tually be produced in laboratories. Hence, onemight also hold rules 1–4 and the rule that O may ask S any question that “makes sense” as physically sufficient.Notice that it is necessary to directly recon-struct the space of density matrices over Hilbertspaces from the rules rather than the underly-ing Hilbert spaces themselves. The reason is thatthe latter contain physically redundant informa-tion (norm and global phase), while the rules referonly to information which is directly accessible to O . Our strategy and procedure for developing toolsand intermediate results before proving the mainclaim is best summarized as a diagram (see figure3).
PSfrag replacementslimited information complementarity information preservation time evolutiontime evolutionindependence, compatibility and cor-relation structure on Q N (in sec. 5) reversible time evolution (in sec. 6)and information measure (in sec. 6.8)Σ is a ball with D = 2 ,
3, and T is either SO(2) or SO(3) (in sec. 7) T is either PSO(4) or PSU(4) ⇒ Σ convex hull of R P or C P (in [1, 2])( Q N , Σ N , T N ) for N > (in [1, 2]) Figure 3: Strategy and steps for the reconstruction.
That is, we shall firstly ascertain, in section 5,the independence, compatibility and correlationstructure on Q N which is induced by rules 1 and2. This section will be particularly instructiveand deliver many elementary technical results The set of Pauli operators over C N is the set of trace-less hermitian 2 N × N matrices with eigenvalues ± which, besides becoming important later, yieldsimple explanations for typical quantum phenom-ena such as, e.g., entanglement, non-locality andmonogamy relations. This is also where we shalldetermine the dimensionality of Σ by a simpleargument and, using this result, derive the di-mensionalities of all other state spaces too. Next,in section 6, we shall use rules 3 and 4 in order Accepted in
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Quantum2017-11-27, click title to verify o conclude that a specific time evolution, is re-versible and, in fact, described by a continuousone-parameter matrix group. These results willthen be used in section 6.8, together with rules3 and 4, to derive an explicit information mea-sure α i ( ~y O → S ) which is unique under elementaryconsistency conditions. In section 7, the conjunc-tion of these outcomes will then be employed toshow that for N = 1 one either obtains a two-or three-dimensional Bloch ball as a state space Σ and that the group of all possible time evo-lutions T is accordingly either SO(2) or SO(3) .The claim for
N > will require substantial addi-tional work and will be proven in the companionpaper [1] for qubits and in [2] for rebits. In this section, we shall solely employ rules 1 and2 in order to deduce the independence, compati-bility and correlation structure for the questionsin an informationally complete set Q M N (c.f. as-sumption 7). To this end it is instructive to lookat the individual cases N = 1 , , in some de-tail before considering general N ∈ N . We shalldenote by D N the dimension of the state space Σ N . Rules 1 and 2 taken together imply that for N = 1 there exists at least one maximally complemen-tary pair Q , Q ′ . Since N = 1 is fixed, it is nowmore convenient to count the independent ques-tions by an index, rather than a prime. Let ustherefore slightly change the notation and write Q ′ henceforth as Q , such that the rules implythe existence of a maximally complementary pair Q , Q . But, applied to a single gbit only, rules1 and 2 tell us nothing more about how manyquestions maximally complementary to Q exist.There may arise another Q maximally comple-mentary to both Q , Q etc. Notice that, for asingle gbit, an informationally complete set ofpairwise independent questions must be given bya maximal set of mutually maximally comple-mentary questions (i.e., no further Q ∈ Q canbe added to this set without destroying mutual Rules 1 and 2 count compatible questions, here weneed to count maximally complementary questions. maximal complementarity) Q M = { Q , Q , . . . , Q D } (11) because rule 1 prohibits further pairwise indepen-dent (partially or maximally) compatible ques-tions. (Recall that maximally complementaryquestions are automatically independent.) Rules1 and 2 imply D ≥ . It turns out that we needto consider two gbits in order to upper bound D .We shall call such questions for a single gbit individual questions . The N = 2 case requires substantially more work.First of all, since this is a composite system (c.f.definition 3.5), the individual questions about thetwo single gbits must be contained in an infor-mationally complete set Q M . We shall againslightly change the notation, as compared to sec-tion 4.1: the maximal complementary set Q M (11) for gbit 1 will be denoted by Q , . . . , Q D ,while the elements of Q M for gbit 2 will be de-noted with a prime Q ′ , . . . , Q ′ D . It will be conve-nient to depict the question structure graphically.Representing individual questions henceforth as vertices , Q M certainly contains the followingPSfrag replacements gbit 1 gbit 2 Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D ... ...... . (12) This structure abides by rules 1 and 2 and any Q i will be independent of and maximally compatiblewith any Q ′ j (cf. definition 3.5). But a composite system may also admit compos-ite questions, built with logical connectives of thesubsystem questions. We must now determinewhich composite questions are allowed and whichmust be added to the individual questions in or-der to form an informationally complete Q M of pairwise independent questions. To this end, wemust firstly unravel which logical connective ∗ O can employ at all in order to construct compos-ite questions which are pairwise independent of Accepted in
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Quantum2017-11-27, click title to verify he individual ones. Recall assumption 5 that O , according to his theoretical model, may onlycompose compatible questions with logical con-nectives ∗ and consider Q i , Q ′ j as an exemplarycompatible pair. In truth tables, we shall hence-forth symbolize ‘yes’ by ‘ ’ and ‘no’ by ‘ ’. Remark. (‘ = ’ in logical expressions.) Toavoid confusion, we emphasize that whenever wewrite an equality sign ‘ = ’ in a logical expressionin the sequel, we do not interpret it as anotherlogical connective ∗ , but as an actual equality ofthe values of the propositions on the left and righthand side. For example, the expression Q i = Q ′ j is not itself a proposition which takes truth val-ues or , but is intended to mean that the truthvalue which Q i takes is identical to that which Q ′ j takes. While permitted binary connectives, it is clearthat, e.g., the AND and OR operations ∧ and ∨ ,respectively, cannot be employed alone to builda new independent question from Q i , Q ′ j because Q i ∧ Q ′ j = 1 implies Q i = Q ′ j = 1 and Q i ∨ Q ′ j = 0 implies Q i = Q ′ j = 0 such that certain answersto these two connectives imply answers to the in-dividuals. But if Q i ∗ Q ′ j is to be pairwise inde-pendent from Q i , Q ′ j , an answer to it must notimply the answer to either of Q i , Q ′ j and, con-versely, the answer to only Q i or only Q ′ j cannotimply Q i ∗ Q ′ j . As can be easily checked, it mustsatisfy the following truth table: Q i Q ′ j Q i ∗ Q ′ j a = b a, b ∈ { , } . (13) The only logical connectives ∗ satisfying thistruth table are the XNOR ↔ for a = 0 and b = 1 and its negation, the XOR ⊕ for a = 1 and b = 0 .As Q i ⊕ Q ′ j = ¬ ( Q i ↔ Q ′ j ) , we may only chooseone of the two binary connectives to define newpairwise independent questions. We henceforth choose to use the XNOR, already introduced in(2), and operationally interpret it as a ‘correlationquestion’; Q ij := Q i ↔ Q ′ j is to be read as ‘arethe answers to Q i and Q ′ j the same?’. But theXOR could equivalently be employed. Since, according to assumption 7, O must beable to build an informationally complete setfor a composite system which, in particular, in-volves pairwise independent composite questions,we would like to conclude that Q i ↔ Q ′ j are notonly implementable, but also contained in a Q M together with the individuals. However, this issubject to a few consistency checks on pairwiseindependence.The XNOR ↔ is a symmetric logical connec-tive, Q ij = Q i ↔ Q ′ j = Q ′ j ↔ Q i (but note that Q ij = Q ji ), and, thanks to its associativity, Q i ↔ Q ij = Q i ↔ ( Q i ↔ Q ′ j )= ( Q i ↔ Q i ) | {z } ↔ Q ′ j = Q ′ j ,Q ij ↔ Q ′ j = Q i (14) such that ↔ defines a closed relation on Q i , Q ′ j , Q ij . This has the following ramifications:(1) For any compatible pair Q i , Q ′ j , the ‘correla-tion’ operation ↔ gives rise to precisely one ad-ditional question Q ij , and (2) the set Q i , Q ′ j , Q ij will indeed be pairwise independent, according tothe definition in section 3.2.4, because neither Q i nor Q ′ j alone can determine Q ij relative to thestate of no information (not even partially), oth-erwise, together with the determined Q ij , theywould (at least partially) determine each othervia (14) – in contradiction with the independenceof Q i , Q ′ j . That is, Q ij is independent of both Q i and Q ′ j and since we assume independence to be asymmetric relation, it must also be true the otherway around.Consequently, the D correlation questions Q ij , i, j = 1 , . . . , D , are candidates for addi-tional questions in Q M besides the D indi-vidual ones of (12). Graphically, we shall repre-sent Q ij as the edge connecting the vertices cor-responding to Q i and Q ′ j . For example, the fol-lowing question graphs As an aside, the XNOR can be expressed in terms ofthe basic Boolean operations as Q i ↔ Q ′ j = ( ¬ Q i ∨ Q ′ j ) ∧ ( Q i ∨ ¬ Q ′ j ). Accepted in
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Quantum2017-11-27, click title to verify Sfrag replacements gbit 1 gbit 2 Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q D D ... ...... PSfrag replacements gbit 1 gbit 2 Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q D D ... ...... PSfrag replacements gbit 1 gbit 2 Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q D D .........represent legal sets of questions. Note that dueto assumption 5, there can only be edges betweengbit 1 and gbit 2 but not between the vertices ofa single gbit, e.g., Q and Q , because they arecomplementary.Our next task is to analyse the independenceand compatibility structure of the Q ij . We begin with a simple, but important observa-tion which we will frequently make use of.
Lemma 5.1.
Let Q , Q , Q ∈ Q N be such that Q is maximally compatible with and pairwise in-dependent of both Q , Q and that Q is maxi-mally complementary to Q ↔ Q . Then Q , Q are maximally complementary.Proof. Suppose Q , Q were at least partiallycompatible. In that case there must exist a stateof S in which O has maximal information α = 1 bit about Q and at least partial information α > bit about Q (or vice versa). Let O now ask Q which is maximally compatible withand pairwise independent of both Q , Q to S in this state. Hence, according to assumption 4,by asking Q , O cannot change his informationabout either Q , Q . Thus, after asking Q , O will have maximal information about Q , Q andat least partial information about Q . But max-imal information about Q , Q implies maximalinformation about Q ↔ Q which is maximallycomplementary to Q . Consequently, Q and Q must be maximally complementary too. This has immediate useful implications.
Lemma 5.2. Q i is maximally compatible with Q ij , ∀ j = 1 , . . . , D and maximally complemen-tary to Q kj , ∀ k = i and ∀ j = 1 , . . . , D . Thatis, graphically, an individual question Q i is max-imally compatible with a correlation question Q ij if and only if its corresponding vertex is a vertexof the edge corresponding to Q ij . By symmetry,the analogous result holds for Q ′ j .Proof. Q i and Q ij are maximally compatible byconstruction. Consider therefore Q i and Q kj for k = i and j = 1 , . . . , D . Clearly, Q i and Q ′ j aremaximally compatible and pairwise independentand so are Q ′ j and Q kj . Maximal complemen-tarity of Q i and Q kj for k = i now follows from Q k = Q kj ↔ Q ′ j (see (14)), which is maximallycomplementary to Q i , and lemma 5.1. For example, Q and Q are maximally com-patible, while Q and Q are maximally com-plementary. The intuitive explanation for theincompatibility of Q and Q is as follows: if O knew the answers to both simultaneously, hewould know more than one bit of informationabout gbit 1 because Q defines a full bit of in-formation about it while Q could be regardedas defining half a bit about each of gbit 1 and2. But in view of rule 1, O should never knowmore than one bit about a single gbit, even in acomposite system. Considering the informationcontained in correlation questions to equally cor-respond to gbit 1 and 2, lemma 5.2 suggests that O ’s information will always be equally distributedover the two gbits for states of maximal knowl-edge, i.e. O will know equally much about gbit 1and 2. We shall make this more precise in [1].We saw before that Q i , Q ′ j , Q ij are pairwise in-dependent. Lemma 5.2 implies that also Q i and Q kj for i = k are independent. We can make useof this result to show the following. Lemma 5.3.
The Q ij , i, j = 1 , . . . , D are pair-wise independent.Proof. Consider Q ij and Q kl and suppose i = k . Then Q i and Q ij are maximally compatible, Clearly, this is not true for non-maximal knowledge.E.g., let O always ask only Q . Accepted in
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Quantum2017-11-27, click title to verify hile Q i and Q kl are maximally complementaryby lemma 5.2. Hence, when knowing Q i and Q ij O cannot know Q kl such that Q ij and Q kl mustbe independent. (Recall from section 3.2.4 thatdependence requires the answer to Q ij to alwaysimply at least partial knowledge about Q kl , i.e., y ij = 0 , y kl = ). The analogous argu-ment holds for when j = l . Recalling from subsection 5.2.1 that the set ofcomposite questions in Q M must be non-empty,lemmas 5.2 and 5.3 have an important corollary. Corollary 5.4. Q i , Q ′ j , Q ij are pairwise indepen-dent for all i, j = 1 , . . . , D and, thanks to as-sumption 7, will thus be part of an information-ally complete set Q M . Next, we consider the compatibility and com-plementarity structure of the correlation ques-tions Q ij . Lemma 5.5. Q ij and Q kl are maximally com-patible if and only if i = k and j = l . That is,graphically, Q ij and Q kl are maximally compat-ible if their corresponding edges do not intersectin a vertex and maximally complementary if theyintersect in one vertex.Proof. Suppose Q ij and Q kj with i = k wereat least partially compatible. Then there mustexist a state of S in which the answer to Q ij is fully known to O and in which also α kj > bit (or vice versa). Let O ask Q ′ j to S in thisstate. Since, by lemma 5.2, this question is max-imally compatible with both Q ij , Q kj (and pair-wise independent of both), according to assump-tion 4, by asking Q ′ j , O cannot change his infor-mation about both Q ij , Q kj . That is, after hav-ing also asked Q ′ j , O must have maximal informa-tion about Q ij , Q ′ j and partial information about Q kj . But maximal information about Q ij , Q ′ j im-plies also maximal information about Q i which,however, is maximally complementary to Q kj bylemma 5.2. Hence, Q ij , Q kj with i = k must bemaximally complementary.Consider now Q ij and Q kl with i = k and j = l .Let O ask S both Q i and Q ′ j the answers to whichimply the answer to Q ij according to (2). Thanksto rule 1, this defines a state of maximal knowl-edge of 2 independent bits and O may not knowany further independent information. Next, let O ask the same S the question Q kl . From lemma 5.2 it follows that Q kl is maximally complemen-tary to both Q i and Q ′ j such that the answer to Q kl will give one new independent bit of infor-mation but renders O ’s information about Q i , Q ′ j obsolete. But by rule 2 O cannot experience a netloss of information by asking a new question andafter asking Q kl he must still know 2 bits about S . Hence, after acquiring the answer to Q kl hemust still know the answer to Q ij such that bothare maximally compatible. To give graphical examples, Q and Q aremaximally complementary due to the intersectionin Q ′ , while Q and Q are maximally compati-ble because their edges do not intersect in verticesPSfrag replacementsgbit 1gbit 2 Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q D D ... PSfrag replacementsgbit 1gbit 2 Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q D D ...This question structure has significant conse-quences: since the correlation questions Q ij and Q kl are both independent and maximally com-patible for i = k and j = l , O can ask both ofthem simultaneously, thereby spending the maxi-mal amount of N = 2 independent bits of infor-mation he may acquire according to rule 1 overcomposite questions. For example, he may ask SQ and Q simultaneouslyPSfrag replacementsgbit 1gbit 2 Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q D D ... (15) upon which he must be entirely oblivious aboutthe individual gbit properties represented by Q i , Q ′ j . (Lemma 5.2 implies that no individualquestion is maximally compatible with two non-intersecting bipartite questions at once.) That is, O has only composite, but no individual infor-mation about the two gbits; they are maximallycorrelated. But this is precisely entanglement .Indeed, the question graph (15) will ultimatelycorrespond to four Bell states in either of the two Accepted in
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Quantum2017-11-27, click title to verify uestion bases { Q , Q ′ } and { Q , Q ′ } , represent-ing the four possible answer configurations ‘yes-yes’, ‘yes-no’, ‘no-yes’ and ‘no-no’ to the corre-lation questions Q , Q (see [31] for a relatedperspective within quantum theory). In fact, if O now ‘marginalized’ over gbit 2, correspondingto discarding any information about questions in-volving gbit 2, he would find gbit 1 in the stateof no information relative to him because thediscarded information also contained everythinghe knew about gbit 1. Other compatible edgeswill correspond to Bell states in different ques-tion bases.Inspired by the compelling proposal withinquantum theory in [33, 34], we shall define states ~y O → S of a bipartite gbit system in L gbit as entangled if the composite information satisfies P D i,j =1 α ij > bit ,where α ij ( ~y O → S ) is O ’s information about thecorrelation question Q ij . While there will bestates with P D i,j =1 α ij ≤ bit which can indeedbe considered as classically composed, in con-trast to [33,34], we emphasize that the above willonly be a sufficient , but not a necessary condi-tion for entanglement, as in the quantum casethere will exist entangled states violating it. It E.g., consider a state with y = 1 and y i = for allother questions in Q M , corresponding to O knowing withcertainty that Q = 1 and nothing else. The discussionsurrounding (1) implies already that P D i,j =1 α ij = 1 bit ,while the individual information is zero. In quantum the-ory, this state would correspond to the separable state ρ = ( + σ x ⊗ σ x ). For the quantum case, states with P D =3 i,j =1 α ij ≤ bit where defined as classically composed in [33, 34]. This is,however, in conflict with the usual definition of entangledstates as those which are non-separable. Namely, considerthe family of quantum states (on the r.h.s. expressed inz-basis) for − ≤ a ≤ ρ a := 14 (cid:16) + 13 ( σ z ⊗ + ⊗ σ z − σ z ⊗ σ z )+2 a ( σ x ⊗ σ x + σ y ⊗ σ y )) = a a
00 0 0 0 . This clearly is a family of non-separable states for a = 0.In particular, for a = this is the reduced state whichone obtains from a W-state upon marginalizing over aqubit. These states yield y z = y z = , y zz = and y xx = y yy = a + and all other y i = , where, e.g., y z and y zz are the probabilities that the spin of qubit 1 isup in z-direction and that the spins of both qubits are would be an interesting question to investigatewhat the precise sufficient and necessary condi-tions for entanglement are in the quantum casein this informational formulation. However, weshall leave this open for now.Since there are only N = 2 independent bits ofinformation to be gained, according to rule 1, theabove sufficient condition for entanglement thusmeans, in particular, that O has more compos-ite than individual information. In general, twogbits will be referred to as maximally entangledrelative to O if he has only composite informa-tion about them which is incompatible with anyindividual information. By contrast, two gbitswould be in a ‘product state’ if O has maximalindividual knowledge of bits about them. Forinstance, if he knows that Q = Q ′ = 1 , S wouldbe in such a product state relative to him. Butnotice that even in this case, O would have onedependent bit of information about the correla-tions because clearly Q = 1 too.Note also that O can ‘collapse’ two gbits intoan entangled state: as the most extreme exam-ple, consider the case that O receives a ‘productensemble state’ of two gbits in a multiple shotinterrogation. O may then ask the next pair ofgbits the questions Q , Q . Upon receiving theanswers, the two gbits will have ‘collapsed’ intoa maximally entangled posterior state relative to O , despite having been in a ‘product state’ priorto interrogation. Entanglement is thus a propertyof O ’s information about S .We emphasize that, for systems with limited in-formation content, entanglement is a direct con-sequence of complementarity – at least for statesof maximal information. To illustrate this obser-vation, consider two classical bits (cbits). Since the same in z-direction, respectively. Using the quadraticinformation measure α ij = (2 y ij − which we derive insubsection 6.8 and which was also proposed for quantumtheory by the authors of [33, 34], these states give I comp := X i,j =1 α ij = (2 y xx − +(2 y yy − +(2 y zz − +0 · · · + 0 = (cid:16)
19 + 8 a (cid:17) bit ≤ bit , and for the individual information I indiv := (2 y z − +(2 y z − + 0 · · · + 0 = 2 / bit . I comp can be arbitrar-ily close to 1 / bit for a sufficiently small a = 0 so thateven I comp < I indiv , and yet the corresponding state is en-tangled in violation of the proposed condition of classicalcomposition in [33, 34]. Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify here is no complementarity in this case, there areonly three pairwise independent questions: theindividuals Q , Q ′ about cbit 1 and cbit 2, re-spectively, and the correlation Q would form aninformationally complete Q M N , graphically rep-resented asPSfrag replacements cbit 1 cbit 2 Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q D D ... Q , Q ′ , Q are mutually maximally compatibleand this composite system satisfies rule 1 too suchthat O may only acquire maximally N = 2 inde-pendent bits of information about this pair ofcbits. Clearly, also in this classical case it is pos-sible for O to acquire only composite informationabout the pair of cbits, namely by only asking Q and not bothering about the individuals. O would be able to find out whether identically pre-pared pairs of cbits in an ensemble are correlatedby always asking Q in a multiple shot interro-gation. There will indeed exist states such that y = 1 and y = y ′ = , however, the pair ofcbits will not be in a state of maximal informa-tion relative to O because he has only spent one ofhis two available bits . A classical state of max-imal information corresponds to O knowing theanswers to two questions of the three Q , Q ′ , Q ,but then by (2, 14) O will also know the answerto the third. That is, in a state of maximal in-formation about two cbits, O will always havemaximal individual information about the pair.For cbits, O cannot spend the second bit also incomposite information. By contrast, it is a conse-quence of the complementarity rule 2 that in ourcase here, O can actually exhaust the entire infor-mation available to him by composite questions,thereby giving rise to entanglement.Thus far, we have only considered individualand correlation questions. All are part of Q M .But can there be more pairwise independent ques-tions in Q M ? These would have to be obtainedfrom the individuals and correlations via anothercomposition with an XNOR. Composing the indi-viduals Q i , Q ′ j with the correlation questions viaXNOR, e.g. Q i ↔ Q il , will yield nothing newbecause questions need to be maximally compat-ible in order to be logically connected by O andfor compatible pairs it already follows from (14)that the individual and correlation questions arelogically closed under ↔ . However, what about correlations of correlation questions, e.g., whatabout Q ↔ Q ? The answer to the last question, in fact, is inter-twined with the dimensionality D of the statespace Σ of a single gbit and thus, ultimately,with the dimensionality of the Bloch-sphere. De-riving the dimension of the Bloch sphere has alsobeen a crucial step in the successful reconstruc-tions of quantum theory via the GPT framework.While Hardy’s pioneering reconstruction [14] didnot fully settle this issue (it required an opera-tionally somewhat obscure ‘simplicity axiom’ toobtain D = 3 ), it was later explicitly solvedby impressive group-theoretic arguments in [59]which show that, under certain information the-oretic constraints, qubit quantum theory is theonly theory with non-trivial entangling dynam-ics. This result was then exploited to relate thedimension of the Bloch-sphere to the dimensionof space via a communication thought experi-ment [21]. However, unfortunately, the ingredi-ents of these arguments neither fit mathemati-cally nor conceptually fully into our frameworkwhich relies on distinct structures than GPTs.On the other hand, although not yielding fullquantum theory reconstructions, approaches as-serting an epistemic restriction (i.e. a restrictionof knowledge) over ontic states admit very sim-ple and elegant arguments for a 1- bit system tohave a state space spanned by three independentepistemic states [40, 41, 54]. These argumentscan be carried out by considering a single systemover a - bit ontic state which at the epistemiclevel, however, is effectively a - bit system dueto epistemic restrictions. The arguments involvebinary connectives (at the ontic level) of comple-mentary questions which, while non-problematicwhen dealing with ‘hidden variables’, are, how-ever, deemed illegal in O ’s model of the world ac-cording to our assumption 5. We do not wish tomake any ontological commitments in our purelyoperational approach. Consequently, we mustreason differently and without ontic states to de-duce the dimensionality of Σ . In our case, thisrequires to consider two gbits rather than justone and involves entanglement, in analogy to thegroup-theoretic GPT derivation in [21, 59] whichlikewise requires two gbits and entanglement. Accepted in
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Quantum2017-11-27, click title to verify heorem 5.6. D = 2 or .Proof. By lemma 5.5, Q ij and Q kl are indepen-dent and maximally compatible if i = k and j = l . But any maximal set of pairwise maxi-mally compatible correlation questions then con-tains precisely D questions. Graphically, thisis easy to see: one can always find D non-intersecting edges between the D vertices of gbit1 and the D vertices of gbit 2. For example, the D ‘diagonal correlations’ Q ii , i = 1 , . . . , D PSfrag replacements ... Q Q Q D D Q are pairwise maximally compatible and pairwiseindependent (lemma 5.3) such that, by theorem3.2 (Specker’s principle), they must also be mu-tually maximally compatible. Hence, O may ac-quire the answers to all D Q ii at the same time.However, rule 1 forbids O ’s information about S to exceed the limit of N = 2 independent bits . Accordingly, the D correlation questions Q ii cannot be mutually independent if D > O has asked Q and Q . The pair must de-termine the answers to Q , . . . , Q D D such thatthe latter must be Boolean functions of Q , Q .Hence, Q jj = Q ∗ Q , j = 1 ,
2, for some logicalconnective ∗ which preserves that Q , Q , Q jj are pairwise independent. Table (13) implies that ∗ must either be the XNOR ↔ or the XOR ⊕ such that either Q jj = Q ↔ Q , or Q jj = ¬ ( Q ↔ Q ) , ∀ j = 3 , . . . , D . But then clearly Q jj , j = 3 , . . . , D could not be pairwise independent if D >
3. The sameargument can be carried out for any set of D pairwise independent and maximally compatiblecorrelations Q ij , i.e. for any question graph with D non-intersecting edges, and any choice of apair of maximally compatible correlations which O may ask first. We must therefore conclude that D ≤
3, while rule 2 implies that D ≥ We shall refer to D = 2 as the ‘rebit case’ andto D = 3 as the ‘qubit case’; for the momentthis is just suggestive terminology, however, weshall see later and in [2] that the D = 2 casewill, indeed, result in rebit quantum theory (two-level systems over real Hilbert spaces), while the D = 3 case will give rise to standard quantumtheory of qubit systems [1]. N = 2 qubits The two cases have distinct properties as regardsquestions which ask for the correlations of bi-partite questions and it is necessary to considerthem separately. We begin with the simpler qubitcase D = 3 for which correlations of correlationquestions do not define new information for O about S such that six individual and nine cor-relation questions constitute an informationallycomplete set Q M . These will ultimately cor-respond to the propositions ‘the spin is up in x − , y − , z − direction’ for the individual qubits 1and 2 and ‘the spins of qubit 1 in i − and qubit 2in j − direction are the same’ where i, j = x, y, z .Note that the density matrix for two qubits has15 parameters. Theorem 5.7. (Qubits) If D = 3 then the correlation questions Q ij are logically closed under ↔ and Q M = { Q i , Q ′ j , Q ij } i,j =1 , , constitutes an informationally complete set for N = 2 with D = 15 .Furthermore, for any two permutations σ, σ ′ of { , , } either Q σ (3) σ ′ (3) = Q σ (1) σ ′ (1) ↔ Q σ (2) σ ′ (2) , or Q σ (3) σ ′ (3) = ¬ ( Q σ (1) σ ′ (1) ↔ Q σ (2) σ ′ (2) ) (16) and either Q σ (3) σ ′ (3) = Q σ (1) σ ′ (2) ↔ Q σ (2) σ ′ (1) , or Q σ (3) σ ′ (3) = ¬ ( Q σ (1) σ ′ (2) ↔ Q σ (2) σ ′ (1) ) (17) Accepted in
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Quantum2017-11-27, click title to verify uch that either Q σ (1) σ ′ (1) ↔ Q σ (2) σ ′ (2) = Q σ (1) σ ′ (2) ↔ Q σ (2) σ ′ (1) , or Q σ (1) σ ′ (1) ↔ Q σ (2) σ ′ (2) = ¬ ( Q σ (1) σ ′ (2) ↔ Q σ (2) σ ′ (1) ) . (18) Proof.
Statements (16, 17) are an immediateconsequence of the argument in the proof of the-orem 5.6 which can be applied to any correla-tion question graph with D = 3 non-intersectingedges and thus to any two permutations σ, σ ′ of { , , } . From this it directly follows that thecorrelation questions Q ij are logically closed un-der ↔ such that in the case D = 3 correlationsof correlations such as Q ij ↔ Q kl do not defineany new independent questions. Accordingly, Q M = { Q i , Q ′ j , Q ij } i,j =1 , , is an informationally complete set of questionsfor N = 2 and D = 3 such that D = 15. For example, if σ, σ ′ are both trivial then thetheorem applied to the following two questiongraphsPSfrag replacements... Q Q Q Q Q PSfrag replacements... Q Q Q Q Q implies for D = 3 either Q = Q ↔ Q , or Q = ¬ ( Q ↔ Q ) (19) and either Q = Q ↔ Q , or Q = ¬ ( Q ↔ Q ) (20) such that either Q ↔ Q = Q ↔ Q , or Q ↔ Q = ¬ ( Q ↔ Q ) (21) In sections 5.2.7 and 5.4 we shall determinewhether negations ¬ occur in the expressions (16–18). For Q, Q ′ , Q ′′ maximally compatible and re-lated by an XNOR, we shall henceforth distin-guish between even correlation: if Q = Q ′ ↔ Q ′′ , and odd correlation: if Q = ¬ ( Q ′ ↔ Q ′′ ) . N = 2 rebits Next, let us consider the rebit case D = 2 . O canask the four individual questions Q , Q , Q ′ , Q ′ and the four correlations Q , Q , Q , Q . But O can also define the two new correlation of cor-relations questions Q := Q ↔ Q , ˜ Q := Q ↔ Q (22) We note that this also implies Q ′ = ¬ ( Q ↔ Q ′′ ) and Q ′′ = ¬ ( Q ↔ Q ′ ). corresponding to the two correlation questionsgraphsPSfrag replacements Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q ˜ Q ... PSfrag replacements Q Q Q Q D Q ′ Q ′ Q ′ Q ′ D Q Q Q Q Q ˜ Q ...(we add the correlation of correlations as a newedge without vertices to the graph). The dif-ference to the qubit case is that Q , ˜ Q cannow not be written as correlations of individ-ual questions Q , Q ′ since the latter do not ex-ist. Clearly, Q , Q , Q and Q , Q , ˜ Q aretwo pairwise independent sets. The question iswhether Q , ˜ Q are independent of each otherand of the individuals. The following lemma evenasserts complementarity to the latter. (Note thatthis lemma holds trivially in the case D = 3 .) Accepted in
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Quantum2017-11-27, click title to verify emma 5.8. The questions asking for the cor-relation of the bipartite correlations, Q ↔ Q and Q ↔ Q , are maximally complementaryto and independent of any Q , Q , Q ′ , Q ′ .Proof. We firstly demonstrate independence of Q and Q ↔ Q . This follows from not-ing that Q is maximally complementary to Q (lemma 5.2) and maximally compatible with Q ↔ Q and arguments completely analogousto those in the proof of lemma 5.3. Next, we es-tablish complementarity of Q ↔ Q and Q .Suppose the contrary, namely, that Q ↔ Q and Q were at least partially compatible suchthat there must exist a state with α Q ↔ Q = 1 bit and α > bit . Clearly, both are maxi-mally compatible with and independent of Q .According to assumption 4, asking Q to thisstate does not change O ’s information about Q ↔ Q and Q . However, maximal infor-mation about Q ↔ Q and Q also impliesmaximal information about Q which is max-imally complementary to Q . Hence, Q and Q ↔ Q must be maximally complementary.The arguments works analogously for the othercases. Finally, we show that (19–21) hold analogouslyfor the rebit case D = 2 . From this it followsthat the four individual questions Q i , Q ′ j , the fourcorrelations Q ij , i, j = 1 , , and the correlationof correlations Q form an informationally com-plete set Q M for rebits. That is, in contrastto the qubit case, there is now one non-trivialcorrelation of correlations question which is pair-wise independent from the individuals and corre-lations. Theorem 5.9. (Rebits) If D = 2 then Q M = { Q i , Q ′ j , Q ij , Q } i,j =1 , constitutes an informa-tionally complete set for N = 2 with D = 9 andeither Q ↔ Q = Q ↔ Q , or Q ↔ Q = ¬ ( Q ↔ Q ) . Proof. O can begin by asking S the maximallycompatible Q and Q upon which he alsoknows the answer to ˜ Q . O then possesses themaximal amount of N = 2 independent bits of information about S . Next, O can ask Q (or Q ) which, according to lemma 5.5, is max-imally complementary to Q , Q . But, by rule 2, O is not allowed to experience a net loss ofinformation. Hence, Q (or Q ) must be max-imally compatible with ˜ Q (they are also in-dependent). That is, Q , Q , ˜ Q are mutu-ally maximally compatible according to theorem3.2 (Specker’s principle) and O may also ask allthree at the same time. But then the same ar-gument as in the proof of theorem 5.6 appliessuch that either ˜ Q = Q ↔ Q or ˜ Q = ¬ ( Q ↔ Q ). This implies either Q = ˜ Q or Q = ¬ ˜ Q . Accordingly, there is only one in-dependent correlation of correlations Q . Since Q , Q , Q , Q and Q are then, by construc-tion, logically closed under the XNOR ↔ and Q is maximally complementary to all individuals Q , Q , Q ′ , Q ′ , no further pairwise independentquestion can be built from this set such that it isinformationally complete. Hence, D = 9. The rebit case D = 2 thus has a very spe-cial question and correlation structure: Q = Q ↔ Q is the only composite question whichis maximally complementary to all individuals,but maximally compatible with all correlations Q ij . By contrast, e.g., Q is maximally comple-mentary to Q , Q ′ , Q , Q and maximally com-patible with Q , Q ′ , Q and maximally compati-ble with the correlation of correlations Q . Con-sequently, Q assumes a special role in the en-tanglement structure: once O knows the answerto this question he may no longer have any fur-ther information about the outcomes of individ-ual questions, but may have information aboutcorrelation questions. That is, two rebits can bein a state of non-maximal information of bit relative to O , corresponding to the latter onlyhaving maximal knowledge about the answer to Q and no information otherwise, and still be maximally entangled because any individual in-formation is forbidden in that situation. Noticethat this is not true for pairs of qubits becauseeven if everything O knew about the pair was theanswer to Q , he could still acquire informationabout the individuals Q , Q ′ such that one couldnot consider such a state as maximally entangled.Even stronger, O will always know the answerto Q if the two rebits are maximally entangledand he has maximal information about S . Thisfollows from (6): for a maximally entangled state Accepted in
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Quantum2017-11-27, click title to verify f maximal information (no information about in-dividuals), the following must hold I O → S = X i =1 ( α i + α ′ i ) | {z } =0 + X i,j =1 α ij + α = α + α + α + α + α = 3 bits . The last equality follows from the fact that once O knows the answers to two maximally compat-ible questions, he will also know the answer totheir correlation and since this correlation is inthe pairwise independent question set the totalinformation defined in (6) will always yield 3 bits for N = 2 independent bit systems in states ofmaximal knowledge. If we now require that O cannot know more than a single bit about max-imally complementary questions then, in a stateof maximal information of N = 2 independent bits , we must have α + α = α + α = 1 bit ⇒ α = 1 bit such that O must have maximal informationabout Q . Therefore, the correlation of corre-lations Q can be viewed as the litmus test forentanglement of two rebits.We note that the individuals Q , Q , Q ′ , Q ′ will ultimately correspond to projections onto the +1 eigenspaces of σ x , σ z , while the Q ij corre-spond to projections onto the +1 eigenspaces of σ i ⊗ σ j , i, j = x, z , and Q corresponds to theprojection onto the +1 eigenspace of σ y ⊗ σ y ,where σ x , σ y , σ z are the Pauli matrices [1, 2]. Ob-servables and density matrices on a real Hilbertspace correspond to real symmetric matrices.This is the reason why σ y is not an observableon R (it corresponds to the ‘missing’ Q ), but σ y ⊗ σ y is a real symmetric matrix and thus anobservable on R ⊗ R .This gives a novel and simple explanation forthe discovery that σ y ⊗ σ y determines the entan-glement of rebits [83]: a two-rebit density matrix ρ is separable if and only if Tr( ρ σ y ⊗ σ y ) = 0 ,i.e. if the state has no σ y ⊗ σ y component. Thisstatement means in our language that a state isseparable if and only if α = 0 and is consistentwith our observation above because any informa-tion about the individuals is incompatible withinformation about Q due to complementarity. We are implicitly using here also rules 3 and 4.
This question structure also has severe reper-cussions for rebit state tomography: it must ul-timately be non-local . For rebits, D = 2 , theprobability that Q = 1 could be written as y = p ( Q = 1 , Q = 1) + p ( Q = 0 , Q = 0) where p ( Q i , Q j ) denotes here the joint probabil-ity distribution over Q i , Q j . That is, in a multipleshot interrogation, O could ask both Q , Q tothe identically prepared rebit couples and fromthe statistics over the answers, he could also de-termine y . But this probability cannot be de-composed into joint probabilities over the individ-ual questions according to Q = “( Q ↔ Q ′ ) ↔ ( Q ↔ Q ′ )” because Q , Q and Q ′ , Q ′ are max-imally complementary. Therefore, O would notbe able to determine y by only asking individ-ual questions to the two rebits and the statis-tics over these answers. That is, for rebits,state tomography would always require correla-tion questions and in this sense be non-local.Note that this stands in stark contrast to qubitpairs where Q = Q ↔ Q ′ can be writtenin terms of individual questions such that also y = p ( Q = 1 , Q ′ = 1) + p ( Q = 0 , Q ′ = 0) canbe determined by the statistics over the answersto Q , Q ′ only. Qubit systems (and quantum the-ory in general) are thus tomographically local.The requirement of tomographic locality , ac-cording to which the state of a composite systemcan be determined by doing statistics over mea-surements on its subsystems, is a standard con-dition in the GPT framework [15–18, 20, 55, 59]and thus directly rules out rebit theory. How-ever, in contrast to derivations within the GPTlandscape, we shall not implement local tomog-raphy here because it will be interesting to seethe differences between real and complex quan-tum theory from the perspective of informationinference and we shall thus carry out the recon-struction of both here and in [1, 2]. For example, in a multiple shot interrogation O couldfirst ask Q , Q ′ on a set of identically prepared rebit cou-ples to find out whether the answers are correlated. On asecond identically prepared set, O could then ask Q , Q ′ which are maximally complementary to the first questionshe asked. From the statistics over the answers, O would beable to determine also the probabilities for Q and Q .But since he needed two separate interrogation runs todetermine the statistics for Q and Q , he would not beable to infer from this any information whatsoever aboutthe statistics of answers to Q . Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify ocal tomography is usually taken as the ori-gin of the tensor product structure for compositesystems in quantum theory [16, 17, 55, 59]. How-ever, one has to be careful with this statementbecause there exist two distinct tensor products:there is (a) the tensor product of Hilbert spaces,e.g., ( C ) ⊗ N for qubits and ( R ) ⊗ N for rebits, and(b) the tensor product of unnormalized probabil-ity vectors or density matrices. A tensor prod-uct of type (b) defines a sufficient support onlyfor composite qubit systems, but not for compos-ite rebit systems; the space of hermitian matricesover ( C ) ⊗ N is the N -fold tensor product of her-mitian matrices over C , but the space of sym-metric matrices over ( R ) ⊗ N is not the N -foldtensor product of symmetric matrices over R (seealso [83]). What local tomography implies is thetensor product (b), but as the example of rebitsshows, the tensor product of type (a) also existswithout it. local hidden variables We shall now settle the issue of the relative nega-tion ¬ , i.e. whether Q ↔ Q = Q ↔ Q , or Q ↔ Q = ¬ ( Q ↔ Q ) , for both rebits and qubits; one of these equationsmust be true (see theorems 5.7 and 5.9). It is instructive to illustrate these equations by meansof a question configuration analogous to a Bellscenario. (A) Suppose Q ↔ Q = Q ↔ Q wastrue. Before we determine whether it is consis-tent with our background assumptions and pos-tulates, we note that this is the case of classi-cal (or realist) logic: O can consistently interpretany such configuration by means of a local ‘hid-den variable’ model. For example, consider thecase Q = Q = 1 (which is compatible withthe rules) such that Q ↔ Q = 1 and, conse-quently, also Q ↔ Q = 1 . We represent thelast two equations by the following two graphsPSfrag replacements Q Q Q Q PSfrag replacements Q Q Q Q where both edges solid and of equal colour meansthat the corresponding questions are correlated. O could consistently read this configuration asfollows: “Since “ Q = Q ′ ” and “ Q = Q ′ ” , Iwould precisely have to conclude that “ Q = Q ” , i.e. Q ↔ Q = 1 , if the individuals Q , Q , Q ′ , Q ′ had definite values which, how-ever, I do not know.” For instance, Q = Q = Q ↔ Q = 1 would be consistent with the fourontic states:PSfrag replacements
111 10
PSfrag replacements
110 0
PSfrag replacements
10 00 0
PSfrag replacements . Therefore, O could join the two graphs consis-tently (without knowing the truth values of theindividuals) (23) i.e., draw all four edges simultaneously which cor-responds to all four edges having definite (al-beit unknown) values at the same time. Sim- ilarly, O can interpret any other configurationwith Q ↔ Q = Q ↔ Q in terms of alocal ‘hidden variable’ model which assigns defi-nite values to the individuals and, conversely, ascan be easily checked, any assignment of definite(or ontic) values to the individuals Q , Q , Q ′ , Q ′ leads necessarily to Q ↔ Q = Q ↔ Q .We now preclude this possibility. We note that Q ↔ Q = Q ↔ Q can be rewritten interms of the individuals as ( Q ↔ Q ′ ) ↔ ( Q ↔ Q ′ ) = ( Q ↔ Q ′ ) ↔ ( Q ↔ Q ′ ) (24) Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify hich is a classical logical identity that is al-ways true for classical Boolean logic. It statesthat the brackets can be equivalently regrouped,i.e. that the order in which the XNOR ↔ is executed in the logical expression can be switched. In classical Boolean logic, this iden-tity follows from the associativity and symme-try of the XNOR. Namely, suppose for a momentthat Q , Q , Q ′ , Q ′ had simultaneous truth val-ues. Then Boolean logic would tell us that ( Q ↔ Q ′ ) ↔ ( Q ↔ Q ′ ) = (cid:0) ( Q ↔ Q ′ ) ↔ Q (cid:1) ↔ Q ′ = (cid:0) Q ↔ ( Q ↔ Q ′ ) (cid:1) ↔ Q ′ = Q ′ ↔ (cid:0) Q ↔ ( Q ↔ Q ′ ) (cid:1) = ( Q ↔ Q ′ ) ↔ ( Q ↔ Q ′ ) . (25) Of course, relative to O , Q , Q , Q ′ , Q ′ do nothave simultaneous truth values. Instead, assump-tion 6 implies that, for a classical logical iden-tity to hold in O ’s model, either (a) all ques-tions in the involved logical expressions are mu-tually maximally compatible or (b) the identityfollows from applying classical rules of inferenceto (sub-)expressions which are entirely written interms of mutually maximally compatible ques-tions and subsequently decomposing the resultlogically into other questions (with possibly dis-tinct compatibility relations). However, since Q , Q and Q ′ , Q ′ form maximally complemen-tary pairs, neither is the case here. In particu-lar, the right hand side in (24) does not followfrom applying any rules of Boolean logic to thetotal expression Q ↔ Q or those subexpres-sions Q ↔ Q ′ and Q ↔ Q ′ on the left handside which are fully composed of maximally com-patible questions. Hence, (24) violates assump-tion 6 such that we must rule out the possibility Q ↔ Q = Q ↔ Q . (B) We are thus already forced to the conclu-sion that Q ↔ Q = ¬ ( Q ↔ Q ) . (26) must hold. Indeed, its equivalent representation ( Q ↔ Q ′ ) ↔ ( Q ↔ Q ′ )= ¬ ( Q ↔ Q ′ ) ↔ ( Q ↔ Q ′ ) does not constitute a classical logical identityand is thus consistent with assumption 6. It The only relevant rules in this case, as in (25), followfrom the properties of the XNOR, namely its symmetryand associativity in Boolean logic. However, it is clearthat, thanks to the occurring complementarity, the argu-ments of (25) no longer apply. For instance, the first stepin (25) is illegal, according to assumption 5, because Q and Q ↔ Q ′ are complementary such that they may notbe directly connected. is also consistent with assumption 5 since onlymaximally compatible questions are directly con-nected.It is impossible for O to interpret this situationin terms of local ‘hidden variables’ which assigndefinite values to the individuals simultaneously.Namely, consider, again, the case Q = Q =1 such that Q ↔ Q = 1 and, consequently,now Q ↔ Q = 0 . This configuration may begraphically represented asPSfrag replacements Q Q Q Q PSfrag replacements Q Q Q Q where one edge solid the other dashed, but samecolour, means that the corresponding answers areanti-correlated. One can easily check that, in con-trast to (23), it is impossible to consistently jointhe two diagrams by drawing all four edges si-multaneously (which would correspond to all in-dividuals and thus all edges having definite truthvalues), as one would obtain a frustrated graphPSfrag replacements < PSfrag replacements . (27) O must conclude that neither the four individu-als nor the four edges can have definite (but un-known) values all at the same time. The sameverdict holds for any other configuration with Q ↔ Q = ¬ ( Q ↔ Q ) .Since either (A) or (B) must be true by theo-rems 5.7 and 5.9, and (A) violates assumption 6,while (B) is consistent with both assumptions 5and 6, we conclude that (26) is the correct rela-tion. Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify ote that this argument holds for both rebitsand qubits and, for qubits, also for any other pairs Q i , Q j = i and Q ′ k , Q ′ l = k of pairs of maximally com-plementary individuals (see theorem 5.7). In fact,even more generally, the same argument can be made for any four questions (i.e., not necessar-ily individuals) Q, Q ′ , Q ′′ , Q ′′′ which are such that Q, Q ′ and Q ′′ , Q ′′′ are maximally complementarypairs, while Q and Q ′ are each maximally com-patible with both Q ′′ , Q ′′′ ; also in this case onewould have to conclude that ( Q ↔ Q ′′ ) ↔ ( Q ′ ↔ Q ′′′ ) = ¬ (cid:0) ( Q ↔ Q ′′′ ) ↔ ( Q ′ ↔ Q ′′ ) (cid:1) . (28) This will become relevant later on. For now weobserve that exchanging the positions of the com-plementary Q ′′ , Q ′′′ from the left to the right handside introduces a negation ¬ .This relative negation ¬ in (28) precludes aclassical reasoning for the distribution of truthvalues over O ’s questions. In fact, as just seen,it rules out local ‘hidden variable models’, analo-gously to the Bell arguments. We also recall thatassumption 6 was a statement about which rulesof inference and logical identities are not appli-cable to logical compositions involving mutuallycomplementary questions. By contrast, (28) isnow a first non-classical logical identity showingone possible non-classical way of reasoning in thepresence of complementarity.The correlation structure for rebits is now clearfrom these results; (26) implies Q = ¬ ˜ Q forthe correlations of correlations defined in (22).This settles the fate of all possible relative nega-tions ¬ for rebits. However, these results do not fully determine the odd and even correla-tion structure for qubits: we still have to clarifywhether there is an overall negation ¬ relative to Q in (19, 20) and more generally in theorem 5.7for other permutations of non-intersecting edges.This difference between rebits and qubits resultsagain from the fact that Q is defined as thecorrelation of the individuals Q , Q ′ for qubits,while it is a correlation of correlations for rebits.This has a remarkable consequence: rebit theoryis its own ‘logical mirror image’, while qubit the-ory’s ‘logical mirror image’ is distinct from qubittheory. This topic will be deferred to section 5.4because we firstly need to understand the ques-tion structure for the N = 3 case in order todiscuss the odd and even correlation structure oftwo qubits further. It will be both useful and instructive to explicitlyconsider the N = 3 case for rebits and qubits.As a composite system, we can view three gbits,labeled by A, B, C , either as three individual sys-tems, as three combinations of one individual anda bipartite composite system or as a tripartitesystem:PSfrag replacements
A BC
PSfrag replacements
A BC
PSfrag replacements
A BC
PSfrag replacements
A BC
PSfrag replacements
A BC (29)
According to definition 3.5 of a composite system, Q must then contain the individual questions ofall three gbits, any bipartite correlation questions and any permissible logical connectives thereof.This results in different structures for rebits andqubits. Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify
We shall begin with the qubit case D = 3 .Clearly, according to definition 3.5, all × individual questions, henceforth denoted as Q i A , Q j B , Q k C , and all × bipartitecorrelation questions, from now on written as Q i A j B , Q i A ,k C , Q j B k C , i, j, k = 1 , , , will be partof an informationally complete set Q M . As be-fore, we represent individuals and correlationsgraphically as vertices and edges, respectively,e.g.PSfrag replacements A B CQ A Q A Q A Q B Q B Q B Q C Q C Q C Q A C Q A B Q B C Q A B Q B C Q Q Q PSfrag replacements
A B CQ A Q A Q A Q B Q B Q B Q C Q C Q C Q A C Q A B Q B C Q A B Q B C Q Q Q depict valid question graphs.But in order to complete the individuals andbipartite correlations to Q M we now have toconsider logical connectives of these questionswhich are pairwise independent. This will nec-essarily involve tripartite questions because thebipartite structure is already exhausted with in-dividuals and bipartite correlations. Clearly, wecannot add a question ˜ Q , representing theproposition “the answers to Q A , Q B , Q C are thesame" to the individuals and bipartite correla-tions because, e.g., ˜ Q = 1 would always imply Q A B = Q A C = Q B C = 1 . Also, Q A B = 0 would imply ˜ Q = 0 such that the bipartite correlations and ˜ Q would not be pairwise in-dependent.From (13) we already know that the logi-cal connective yielding new pairwise independentquestions must either be the XNOR or the XOR.For consistency with the bipartite structure, wecontinue to employ the XNOR. There is an obvi-ous candidate for an independent tripartite ques-tion, namely Q ijk = Q i A ↔ Q j B ↔ Q k C (30) which thanks to the associativity and symmetryof ↔ can also equivalently be written as Q ijk = Q i A ↔ Q j B k C = Q i A j B ↔ Q k C = Q i A k C ↔ Q j B . (31) (Since the notation for this tripartite ques-tion is unambiguous from the ordering of i, j, k we drop the subscripts A, B, C in Q ijk .)This structure is also natural from the differ-ent compositions in (29). Q ijk thus definedis by construction maximally compatible with Q i A , Q j B , Q k C , Q i A j B , Q i A k C , Q j B k C and, for sim-ilar reasons to the independence of Q i A j B from Q i A , Q j B (see the discussion below (14)), alsopairwise independence of the latter. Note thatthis question does not stand in one-to-one cor-respondence with the proposition “the answersto Q i A , Q j B , Q k C are the same"; e.g., Q i A = 1 and Q j B = Q k C = 0 also gives Q ijk = 1 . It iseasier to interpret this question via (31) as ei-ther of the three questions “are the answers to Q i A , Q j B k C / Q i A j B , Q k C / Q i A k C , Q j B the same?".There are × × such tripartitequestions Q ijk , i, j, k = 1 , , . We shall repre-sent them graphically as triangles. For example, Q , Q , Q are depicted as follows:PSfrag replacements A B CQ A Q A Q A Q B Q B Q B Q C Q C Q C Q A C Q A B Q B C Q A B Q B C Q Q Q Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify
We have to delve into some technical details onthe independence and compatibility structure toexplain and understand monogamy and the en-tanglement structure of three qubits. These re-sults will also be used to prove the analogous re-sults by induction in section 5.5.1 for N qubits.Individual questions from qubit A are maxi-mally compatible with the individual questionsfrom qubits B and C , etc. But what about thecompatibility of bipartite and tripartite correla-tions? The compatibility structure of bipartitecorrelations of a fixed qubit pair is clear fromlemma 5.5, but we have to investigate compat-ibility of bipartite correlation questions involvingall three qubits. Lemma 5.10. Q i A j B and Q l B k C are maximallycomplementary if j = l . On the other hand, Q i A j B and Q j B k C are maximally compatible andit holds Q i A j B ↔ Q j B k C = Q i A k C . The analogous statements hold for any permuta-tion of
A, B, C . That is, graphically, two bipar-tite correlations involving three qubits are max-imally compatible if the corresponding edges in-tersect in a vertex and maximally complementaryotherwise.Proof.
Maximal complementarity of Q i A j B and Q l B k C for j = l is proven by noting that bothare maximally compatible with and independentof Q i A , the relation Q j B = Q i A ↔ Q i A j B andlemma 5.1. Q i A j B and Q j B k C are evidently maximally com-patible since Q i A , Q j B , Q k C are maximally com-patible. Moreover, Q i A j B ↔ Q j B k C = Q i A ↔ ( Q j B ↔ Q j B ) | {z } =1 ↔ Q k C = Q i A k C thanks to the associativity of ↔ . For example, Q A B and Q B C intersect in Q B and are thus maximally compatible, while Q A B and Q B C do not share a vertex and are therefore maximally complementary:PSfrag replacements ABCQ A Q A Q A Q B Q B Q B Q C Q C Q C Q B C Q A B Q B C Q A B Q B C Q Q Q PSfrag replacements
ABCQ A Q A Q A Q B Q B Q B Q C Q C Q C Q B C Q A B Q B C Q A B Q B C Q Q Q We continue with the tripartite questions.
Lemma 5.11. Q ijk is maximally compatible with Q i A , Q j B , Q k C and maximally complementary to Q l A = i A , Q m B = j B , Q n C = k C . That is, graphically, Q ijk is maximally compatible with an individ-ual Q i A,B,C if the corresponding vertex is one ofthe vertices of the triangle representing Q ijk andmaximally complementary otherwise.Proof. Q ijk is by construction maximally com-patible with Q i A , Q j B , Q k C . On the other hand,complementarity of Q ijk and Q l A = i A is shown bynoting that both are maximally compatible withand independent of Q j B k C and lemma 5.1. Oneargues analogously for the individuals of qubits B and C . For instance, Q is maximally compatiblewith Q C and maximally complementary to Q C :PSfrag replacements ABCQ A Q A Q A Q B Q B Q B Q C Q C Q C Q B C Q A B Q B C Q A B Q B C Q Q Q This lemma also directly implies that any in-dividual and any tripartite correlation questionare pairwise independent because (1) maximallycomplementary questions are in particular inde-pendent, and (2) for maximally compatible ques-tion pairs such as Q i A , Q ijk independence argu-ments analogous to those surrounding (14) apply. Accepted in
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Quantum2017-11-27, click title to verify ext we consider bipartite and tripartite cor-relation questions. Lemma 5.12. Q ijk is maximally compatiblewith Q i A j B , Q i A k C , Q j B k C and, furthermore, with Q m B n C , Q l A n C and Q l A m B for l = i , m = j and k = n . On the other hand, Q ijk is max-imally complementary to Q i A m B , Q i A n C , Q l A j B , Q j B n C , Q l A k C , Q m B k C for l = i , m = j and k = n . That is, graphically, Q ijk is maximallycompatible with a bipartite correlation if the edgeof the latter is either an edge of the triangle cor-responding to Q ijk or if the edge and triangle donot intersect. Q ijk is maximally complementaryto a bipartite correlation question if the edge ofthe latter and the triangle corresponding to Q ijk share one common vertex.Proof. Q ijk is by construction maximally com-patible with Q i A j B , Q i A k C , Q j B k C . Q ijk = Q i A ↔ Q j B k C and Q m B n C are also maximally compat-ible for j = m and k = n because Q m B n C ismaximally compatible with Q i A and thanks tolemma 5.5 also with Q j B k C . Complementarity of Q ijk and Q i A m B for j = m follows from notingthat both are maximally compatible with and in-dependent of Q k C and lemma 5.1. The reasoningfor all other cases is analogous. To give a graphical example, Q is maximallycompatible with Q B C and Q A C and maxi-mally complementary to Q A B :PSfrag replacements ABCQ A Q A Q A Q B Q B Q B Q C Q C Q C Q B C Q A B Q A B Q A C Q B C Q Q Q PSfrag replacements
ABCQ A Q A Q A Q B Q B Q B Q C Q C Q C Q B C Q A B Q A B Q A C Q B C Q Q Q We still have to check pairwise independence ofthe bipartite and tripartite correlation questions.
Lemma 5.13.
Any bipartite Q m B n C , Q l A n C , Q l A m B and any tripartite corre-lation question Q ijk are independent from oneanother.Proof. Lemma 5.12 implies that we onlyhave to check pairwise independence of Q m B n C , Q l A n C , Q l A m B from Q ijk for l = i , m = j and k = n because maximally comple-mentary questions are by definition independentand Q ijk and Q i A j B , Q i A k C , Q j B k C are pairwiseindependent. Consider therefore Q ijk and Q m B n C for j = m and k = n . By lemma 5.11, Q k C is maximally compatible with Q ijk and bylemma 5.2 maximally complementary to Q m B n C .This implies, using the arguments from the proofof lemma 5.3, independence of Q ijk , Q m B n C .The other cases follow similarly. Lemma 5.14.
The tripartite correlation ques-tions Q ijk , i, j, k = 1 , , are pairwise indepen-dent.Proof. Consider Q ijk and Q lmn for i = l . Bylemma 5.11, Q i A is maximally compatible with Q ijk and maximally complementary to Q lmn . Us-ing the analogous arguments from the proof oflemma 5.3, this implies that Q ijk , Q lmn are inde-pendent. The same reasoning holds when j = m and k = n . This has an immediate consequence:
Corollary 5.15.
The individuals Q i A , Q j B , Q k C ,the bipartite Q i A j B , Q i A k C , Q j B k C and the tripar-tite Q ijk , i, j, k = 1 , , are pairwise independentand thus, thanks to assumption 7, contained inan informationally complete set Q M . Lastly, we consider the complementarity andcompatibility structure of the tripartite correla-tions.
Lemma 5.16. Q ijk and Q lmn are maximallycompatible if { i, j, k } and { l, m, n } overlap in oneor three indices and maximally complementary if { i, j, k } and { l, m, n } overlap in zero or two in-dices. That is, graphically, Q ijk and Q lmn aremaximally compatible if their corresponding tri-angles intersect in one vertex (or coincide) andmaximally complementary if the triangles sharean edge or do not intersect.Proof. Compatibility for an overlap in all threeindices is trivial. But also Q ijk = Q i A ↔ Q j B k C and Q imn = Q i A ↔ Q m B n C are clearly maxi-mally compatible for j = m and k = n becauseby lemma 5.5 Q j B k C , Q m B n C are maximally com-patible in this case. Compatibility for the othercases of an overlap of Q ijk and Q lmn in one indexfollows by permutation. Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify he proof of the complementarity of Q ijk and Q lmn for i = l , j = m and k = n follows fromlemma 5.1. One may use the fact that, by lemma5.12, both questions are maximally compatiblewith and independent of Q j B k C and that, bylemma 5.11, Q i A = l A is maximally complementaryto Q lmn .Similarly, one proves complementarity of Q ijk = Q i A ↔ Q j B k C and Q ljk = Q l A ↔ Q j B k C for i = l by using that both are maximally com-patible with and independent of Q j B k C . Com-plementarity of tripartite correlations for otheroverlaps in precisely two indices follows by per-mutation. For example, Q and Q intersect in thevertex Q B and are thus maximally compatible.By contrast, Q shares the edge Q A B with Q and does not intersect at all with Q suchthat Q is maximally complementary to both. Q and Q intersect in the vertex Q C and aretherefore maximally compatible:PSfrag replacements Q Q Q Q PSfrag replacements Q Q Q Q We now show that the individuals, the bi-partite and the tripartite correlation questionsform an informationally complete set Q M for N = 3 qubits. Theorem 5.17. (Qubits)
The individuals Q i A , Q j B , Q k C , the bipartite Q i A j B , Q i A k C , Q j B k C and the tripartite Q ijk , i, j, k = 1 , , are logi-cally closed under ↔ such that they form an in-formationally complete set Q M with D = 63 for D = 3 .Proof. For individuals and bipartite correlationsof any pair of qubits the logical closure under theXNOR was already shown in section 5.2. Cor-relation of individuals with a bipartite correla-tion question from another pair of qubits yieldsthe tripartite correlation questions. Lemma 5.10shows that also the bipartite correlation ques-tions involving distinct pairs of qubits are logi-cally closed under ↔ . We thus only have to checklogical closure of XNOR combinations involvingtripartite correlation questions.Lemma 5.11 shows that tripartite correlationsand individuals are only maximally compatible ifthe individual is a vertex of the tripartite trian-gle. But then a combination such as Q ijk ↔ Q i A = ( Q i A ↔ Q i A ) ↔ Q j B k C = Q j B k C produces another bipartite correlation, thanks tothe associativity of ↔ . Similarly, lemma 5.12 as-serts that tripartite and bipartite correlations areonly maximally compatible if the edge of the bi-partite correlation is either contained in the tri-partite triangle or if the edge and triangle do notintersect. However, combinations of such maxi-mally compatible pairs also do not yield any newquestions because, e.g., Q ijk ↔ Q i A j B = Q k C and, using again the associativity of XNOR andtheorem 5.7, Q σ A (1) σ B (1) k ↔ Q σ A (2) σ B (2) = ( Q σ A (1) σ B (1) ↔ Q σ A (2) σ B (2) ) | {z } = Q σA (3) σB (3) or ¬ Q σA (3) σB (3) ↔ Q k C = Q σ A (3) σ B (3) k or ¬ Q σ A (3) σ B (3) k , where σ A , σ B are permutations of { , , } . Fi-nally, lemma 5.16 entails that tripartite correla-tions are only maximally compatible if they inter- sect in one vertex. But, using theorem 5.7 oncemore, Accepted in
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Quantum2017-11-27, click title to verify σ A (1) σ B (1) k ↔ Q σ A (2) σ B (2) k = Q σ A (1) σ B (1) ↔ Q σ A (2) σ B (2) = Q σ A (3) σ B (3) or ¬ Q σ A (3) σ B (3) . Permuting these examples implies that all in-dividuals, bipartite and tripartite correlationsare logically closed under ↔ which by (13) isthe only independent logical connective possiblyyielding new independent questions. These ques-tions therefore form an informationally completeset Q M with D = 63. Let us now put all these detailed results to gooduse and reward ourselves with the observationthat they naturally explain monogamy of entan-glement for the qubit case. This is best illus-trated with an example. Let O have asked thequestions Q A B and Q A B to qubits A, B suchthat the two are in a state of maximal informa-tion of N = 2 independent bits and maximalentanglement relative to O .PSfrag replacements Q A B Q A B Intuitively, this already suggests monogamy ofentanglement because O has spent the maximallyattainable amount of information over the pair A, B such that even if now a third qubit C entersthe game he should no longer be able to ask any question to the triple which gives him any fur-ther simultaneous independent information aboutthe pair A, B . But any correlation of A or B with C would constitute additional informationabout A or B which would violate the informa-tion bound for the subsystem of the compositesystem of three qubits.We shall now make this more precise. Thequestion is, which bipartite or tripartite correla-tions involving qubit C are maximally compatiblewith Q A B , Q A B . Firstly, lemma 5.10 states that bipartite correlations of two distinct qubitpairs are only maximally compatible if their cor-responding edges intersect in a vertex. Clearly,there can then not exist any edge which is max-imally compatible with both Q A B , Q A B . Sec-ondly, lemma 5.12 asserts that a bipartite correla-tion is only maximally compatible with a tripar-tite correlation if the corresponding edge is eithercontained in the tripartite triangle or does notintersect the triangle. This means that the onlytripartite correlations maximally compatible with Q A B , Q A B are (a) the correlations of the lat-ter with the individuals Q C , Q C , Q C of qubit C and (b) Q k , k = 1 , , .For example, in the first of the following twoquestion graphs Q B C is maximally compati-ble with Q A B but maximally complementary to Q A B , while Q B C is maximally complementaryto both. But O could ask Q together with Q A B , Q A B , as depicted in the second graph:PSfrag replacements Q A B Q A B Q Q C Q B C Q B C PSfrag replacements Q A B Q A B Q Q C Q B C Q B C However, asking Q A B , Q A B and Q simul-taneously is equivalent to asking Q A B , Q A B and Q C because Q A B ↔ Q = Q C .The analogous conclusion holds for any othertripartite question maximally compatible with Q A B , Q A B . Therefore, once O has askedthe last to bipartite correlations about qubits A, B , he can only acquire individual informa-tion corresponding to Q C , Q C , Q C about qubit C . Clearly, the same state of affairs is true forany permutation of the three qubits. This is monogamy of entanglement in its most extremeform: two qubits which are maximally entangledcan not be correlated whatsoever with any othersystem. Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify he non-extremal form of monogamy, namelythe case that a qubit pair is not maximally en-tangled and can thus share a bit of entanglementwith a third qubit, can also be explained. To thisend, we recall from quantum information theorythat monogamy is generally described with so-called monogamy inequalities , e.g., the Coffman-Kundu-Wootters inequality [84] τ A | BC ≥ τ AB + τ AC , (32) where ≤ τ A | BC = 2(1 − Tr ρ A ) ≤ measuresthe entanglement between qubit A and the qubitpair B, C and ρ A is the marginal state of qubit A obtained after tracing qubit B and C out of the(not necessarily pure) tripartite state. τ AB , τ AC ,on the other hand, measure the bipartite entan-glement of the pairs A, B and
A, C and are simi-larly obtained by tracing either C or B out of thetripartite qubit state. The inequality (32) formal-izes the intuition that the correlation between A and the pair B, C is at least as strong as the cor-relation of A with B and C individually. Thisis the general form of monogamy. One usuallyalso defines the so-called three-tangle [84] as thedifference between left and right hand side τ ABC = τ A | BC − τ AB − τ AC ∈ [0 , to measure the genuine tripartite entanglementshared among all three qubits. This three-tangleturns out to be permutation invariant τ ABC = τ BCA = τ CAB .In our current case, a measure of entanglementsharing must be informational. At this stage,in line with our previous definition of entangle-ment, we simply define the analogous entangle-ment measures to be the sum of O ’s informationabout the various independent bipartite or tripar-tite correlation questions ˜ τ A | BC := X i,j,k =1 α ijk + X i,j =1 α i A j B + X i,k =1 α i A k C ˜ τ AB := X i,j =1 α i A j B , ˜ τ AC := X i,k =1 α i A k C . (33) With this definition, an informational inequalityanalogous to the inequality (32) is trivial ˜ τ A | BC ≥ ˜ τ AB + ˜ τ AC as is the fact that the informational three-tangle ˜ τ ABC := ˜ τ A | BC − ˜ τ AB − ˜ τ AC = X i,j,k =1 α ijk is permutation invariant and measures only theinformation contained in the tripartite questionsand thus genuine tripartite entanglement. Thisis how one can describe the general form ofmonogamy of entanglement in our language. Ofcourse, at this stage the information measure α i about the various questions Q i is only implicit,but we shall derive its explicit form in section6.8, upon which the above statements becometruly quantitative. These informational versionsof monogamy inequalities and tangles naturallysuggest themselves for simple generalizations toarbitrarily many qubits, thereby complementingcurrent efforts in the quantum information liter-ature (e.g., see [85]).Before we move on, we briefly emphasize thatmonogamy is a consequence of complementarity– as is entanglement. For example, three classi-cal bits also satisfy the limited information rule1, but due to the absence of complementarity, O could ask all correlations Q AB , Q AC , Q BC and Q ABC at oncePSfrag replacements
A BC which, however, is equivalent to asking the threeindividuals Q A , Q B , Q C . Let us elucidate maximal tripartite entanglementfor three qubits, corresponding to O asking threemutually maximally compatible and independenttripartite correlation questions such that he ex-hausts the information bound of N = 3 indepen-dent bits with tripartite information. Lemma5.16 asserts that tripartite questions are maxi-mally compatible if and only if they intersect inexactly one vertex. For example, O could ask Q , Q , Q simultaneously. These are mu-tually independent because by (19, 20, 26) Q ↔ Q = Q A B or ¬ Q A B Q ↔ Q = Q A C or ¬ Q A C Q ↔ Q = Q B C or ¬ Q B C , (34) i.e., their binary connectives with an XNOR donot imply each other and the bipartite correla-tions are pairwise independent from the tripar-tite ones. Accordingly, asking Q , Q , Q Accepted in
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Quantum2017-11-27, click title to verify ill provide O with three independent bits ofinformation about the qubit triple. We note thatlemma 5.11 implies that once O has posed thequestions Q , Q , Q , every individual ques-tion Q i A , Q j B , Q k C is maximally complementaryto at least one of the three tripartite correlations.That is, O cannot acquire any individual infor-mation about the three qubits and will only havecomposite information. This is a necessary con-dition for maximal entanglement.In addition to the three implied binary corre-lations (34), O would clearly also know the tri-partite correlation of all three Q , Q , Q ,which amounts to Q ↔ Q ↔ Q = Q or ¬ Q . This exhausts the list of questions aboutwhich O would have information by asking Q , Q , Q . The list can be represented bythe following question graph: (35) This graph will ultimately correspond tothe eight possible Greenberger-Horne-Zeilinger(GHZ) states in either of the three questionbases { Q A , Q B , Q C } , { Q A , Q B , Q C } and { Q A , Q A , Q C } , corresponding to the eightanswer configurations ‘yes-yes-yes’, ‘yes-yes-no’,‘yes-no-no’, ‘yes-no-yes’, ‘no-yes-yes’, ‘no-yes-no’,‘no-no-yes’ and ‘no-no-no’ to the three tripar-tite correlations Q , Q , Q . Similarly, anyother three maximally compatible tripartite cor-relations will correspond to GHZ states in otherquestion bases. The correlation information in the graph (35) isclearly democratically distributed over the threequbits
A, B, C . (Intuitively, one could even As an aside, we note that these propositions solve thelittle riddle in Zeilinger’s festschrift for D. Greenberger[31]. view the bit stemming from, say, the answer to Q A B C as being carried to one-third by eachof A, B, C .) Furthermore, if O now wanted to‘marginalize’ over qubit C , i.e. discard any infor-mation involving qubit C , all that would be left ofhis knowledge about the qubit triple would be theanswer to Q A B . But, as argued in section 5.2.6,this resulting ‘marginal state’ of qubits A, B rel-ative to O cannot be considered entangled. Theanalogous results hold for ‘marginalization’ overeither A or B . As a result, the question graph(35) corresponds to genuine maximal tripartiteentanglement. We shall now repeat the same exercise for threerebits but can benefit from the results of the N =3 qubit case. We shall thus be briefer here. Thereader exclusively interested in qubits may jumpto section 5.4.According to definition 3.5, O ’s questions tothe three rebit system must contain the indi-viduals Q i A , Q j B , Q k C , bipartite correlations Q i A j B , Q i A k C , Q j B k C , i, j, k = 1 , , and bipartitecorrelations of correlations Q A B , Q A C , Q B C .To render this set informationally complete, wehave to top it up with tripartite questions. Thereare now tripartite correlations Q ijk , i, j, k =1 , , of the kind (30, 31) and, moreover, tri-partite correlations of a rebit individual questionwith the question Q asking for the correlationof bipartite correlations of the other rebit pair Q i := Q i A ↔ Q B C ,Q j := Q j B ↔ Q A C ,Q k := Q A B ↔ Q k C , i, j, k = 1 , . Given that the individuals Q A , Q B , Q C do notexist for rebits there is now also no tripartite cor-relation Q . The independence, compatibility and comple-mentarity structure for the questions not involv-ing an index i, j, k = 3 directly follows from thequbit discussion as lemmas 5.10–5.16 also hold inthe present case for i, j, k = 1 , . But we nowhave to clarify the question structure once twoindices are equal to (an odd number of indices Accepted in
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Quantum2017-11-27, click title to verify annot be ). The status of any such purely bi-partite relations was clarified in section 5.2 suchthat here we have to consider the case that allthree rebits are involved. Lemma 5.18. Q B C is maximally complemen-tary to Q i A j B , i, j = 1 , , and maximally compat-ible with Q A B . Furthermore, Q A B ↔ Q B C = Q A C . (36) The same holds for any permutations of
A, B, C . Proof.
One proves complementarity of Q i A j B and Q B C = Q B C ↔ Q B C by employing thatboth are maximally compatible with and inde-pendent of Q i A and lemma 5.1. This also requiresinvoking that Q j B and Q B C are maximally com-plementary by lemma 5.8.Compatibility of Q A B and Q B C fol-lows indirectly. Namely, Q A B , Q B C and Q A B , Q B C are two maximally compatiblepairs by lemma 5.10. We can then apply thereasoning around (28) to find Q A B ↔ Q B C = ( Q A B ↔ Q A B ) ↔ ( Q B C ↔ Q B C )= ¬ (( Q A B ↔ Q B C ) | {z } = Q A C ↔ ( Q B C ↔ Q A B ) | {z } = Q A C )= ¬ ( Q A C ↔ Q A C ) = Q A C . The last equality holds thanks to theorem 5.9.The same reasoning applies to any permuta-tion of
A, B, C . Next, we discuss the tripartite correlations ofindividuals with the questions Q asking for thecorrelation of bipartite correlations of rebit pairs. Lemma 5.19. Q i is maximally compatible with Q i A and maximally complementary to Q j B , Q k C .The same holds for any permutation of A, B, C .Proof.
Compatibility of Q i with Q i A is trueby construction. Complementarity of Q j B and Q i = Q i A ↔ ( Q B C ↔ Q B C ) follows fromthe observation that Q i and Q j B are both max-imally compatible with and independent of Q i A and a similar reasoning to the previous proof. Lemma 5.20. Q i is maximally compatible with Q B C , Q j B k C , j, k = 1 , , and Q l A k C , Q l A j B for i = l . On the other hand, Q i is maximally com-plementary to Q i A j B , Q i A k C and Q A B , Q A C .Furthermore, Q ijk is maximally compatible with Q A B . The same holds for any permutation of A, B, C .Proof.
Compatibility of Q i with Q j B k C and Q B C , as well as compatibility of Q ijk with Q A B is obvious. Compatibility of Q i with Q l A k C for i = l follows indirectly by noting that Q i A , Q l A and Q B C , Q k C are two pairs of max-imally complementary questions, but that the questions in one pair are maximally compatiblewith both questions of the other. In this casethe reasoning of (28) applies and entails the cor-relation of Q i A with Q B C must be maximallycompatible with the correlation of Q l A with Q k C .Compatibility of Q i with Q l A j B follows simi-larly.Complementarity of Q i and Q i A j B followsfrom the fact that both are maximally compati-ble with and independent of Q i A and using argu-ments as in previous proofs. Likewise, Q i and Q A B follows similarly by noting that both aremaximally compatible with Q B C . Lemma 5.21.
Any of Q i , Q j , Q k isindependent from any bipartite correla-tion question Q i A j B , Q i A k C , Q j B k C andany bipartite correlation of correlationsquestion Q A B , Q A C , Q B C . Q ijk is alsopairwise independent from the latter. Further-more, the Q ijk and Q i , Q j , Q k are pairwiseindependent.Proof. The proof is completely analogous to theproofs of lemmas 5.3, 5.13 and 5.14.
As before this has an important consequence.
Corollary 5.22.
The individuals Q i A , Q j B , Q k C ,the bipartite correlations Q i A j B , Q i A k C , Q j B k C ,the bipartite correlations of correlations Q A B , Q A C , Q B C , the tripartite correla-tions Q ijk and the tripartite Q i , Q j , Q k , Accepted in
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Quantum2017-11-27, click title to verify , j, k = 1 , , are pairwise independent andthus, thanks to assumption 7, part of aninformationally complete set Q M . The compatibility and complementarity struc-ture of questions involving the ‘correlation of cor-relations’ Q is analogous to lemma 5.16. Lemma 5.23. Q ijk is maximally compatible with Q i , Q j , Q k and maximally complementaryto Q l , Q m , Q n for i = l , j = m and k = n .Furthermore, Q i is maximally compatible with Q j , Q k , but Q and Q are maximallycomplementary. The analogous result holds forall permutations of A, B, C .Proof.
Compatibility of Q ijk with Q i , Q j , Q k follows from the fact thatthe constituents of the latter Q i A , Q B C , . . . are maximally compatible with Q ijk . Comple-mentarity of, e.g., Q ijk and Q l for i = l canbe shown by noting that both are maximallycompatible with and independent of Q B C andlemma 5.1. Compatibility of, say, Q i and Q j can be demonstrated by using (28) andnoting that Q i A , Q A C and Q j B , Q B C are twopairs of maximally complementary questionswhich are such that each question in one pairis maximally compatible with both questionsof the other pair. Finally, Q and Q aremaximally complementary because both aremaximally compatible with Q B C and Q A , Q A are maximally complementary. This finishes our considerations of the inde- pendence and complementarity structure of threerebits.
The rebit questions considered thus far comprisean informationally complete set of elements. Theorem 5.24. (Rebits)
The individu-als Q i A , Q j B , Q k C , the bipartite correlations Q i A j B , Q i A k C , Q j B k C , the bipartite correlationsof correlations Q A B , Q A C , Q B C , the tri-partite correlations Q ijk and the tripartite Q i , Q j , Q k , i, j, k = 1 , , are logically closedunder ↔ and thus form an informationallycomplete set Q M with D = 35 for D = 2 .Proof. Logical closure under the XNOR for anypair of rebits follows from section 5.2. Combin-ing individuals with bipartite questions of an-other rebit pair produces the tripartite questions.Lemma 5.10 and 5.18 assert logical closure ofbipartite correlation questions and the bipartitequestions Q asking for the correlation of bipar-tite correlations among all three rebits. We musttherefore only check logical closure of combina-tions involving tripartite questions which involveindices taking the value 3 (the other cases arecovered by theorem 5.17 for i, j, k = 1 , Q and Q A B aremaximally compatible. The conjunction with ↔ yields Q ↔ Q A B = ( , ) ( Q A ↔ Q B C ) ↔ ¬ ( Q A B ↔ Q A B )= ( , ) Q B ↔ Q A C = Q . Similarly, by lemma 5.23, Q i , Q j are max- imally compatible. Their XNOR conjunctiongives Q i ↔ Q j = ( Q i A ↔ Q B C ) ↔ ( Q A C ↔ Q j B ) = ( ) ¬ ( Q A B ↔ Q i A j B )which again coincides with some Q l A m B with i = l and j = m (or the negation thereof). In Accepted in
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Quantum2017-11-27, click title to verify omplete analogy to these explicit examples andto the proof of theorem 5.17, one shows the log-ical closure under the XNOR of all other cases.This gives the desired result. Rebits are considered as non-monogamous in theliterature [86, 87]. However, this conclusion de-pends somewhat on one’s notion of monogamyand can be clarified within our language. Firstly,rebits, just as qubits, are clearly monogamous inthe following sense: if two rebits
A, B are maxi-mally entangled in a state of maximal informationrelative to O , then they cannot share any entan-glement whatsoever with a third rebit C . Thiscan be seen by repeating the argument of section5.3.4 which holds analogously for three rebits.The situation changes slightly for entangledstates of non-maximal information involving ei-ther of Q A B , Q A C , Q B C and for tripartitemaximally entangled states. As an example forthe former case, O could ask Q A B , Q B C simul-taneously which gives him two independent bits of information about the rebit triple and implies Q A C by (36) as well such that his informationcould be summarized asPSfrag replacements Q A B Q A C Q B C (we depict the Q A B , Q A C , Q B C without ver-tices to emphasize the absence of the individu-als Q A , Q B , Q C for rebits). This graph cor-responds to non-monogamously entangled rebitstates: by lemma 5.8 all individuals are maxi-mally complementary to two of the three knownquestions such that O cannot acquire any indi-vidual information about the three rebits at thesame time. The three rebits are pairwise maxi-mally entangled, albeit not in a state of maximalinformation (see also section 5.2.6). However, the three rebits can be similarlynon-monogamous in a tripartite maximally en-tangled state of maximal information. As inthe qubit case (35) in section 5.3.5, one canwrite down a question graph corresponding to therebit analogues of GHZ states, representing theeight possible answers to the tripartite questions Q , Q , Q PSfrag replacements Q A B Q A C Q B C (to justify this graph one has to employ the-orem 5.24). Thanks to the information aboutthe answers to Q A B , Q A C , Q B C , such a statewould contain a non-monogamous distribution ofentanglement over A, B, C . In particular, if O ‘marginalized’ over rebit C , by discarding all in-formation involving C , he would be left with theanswer to Q A B which defines a maximally en-tangled two-rebit state of non-maximal informa-tion (see section 5.2.6) – in contrast to the qubitcase of section 5.3.5. N = 2 gbits While we were able to settle the relative negation ¬ between the correlations of bipartite correla-tion questions for both qubits and rebits in (28)(e.g., Q ↔ Q = ¬ ( Q ↔ Q ) ), we still haveto clarify the odd and even correlation structurefor qubits in (19, 20) and more generally in theo-rem 5.7. For example, we have to clarify whether Q = Q ↔ Q or Q = ¬ ( Q ↔ Q ) , etc.This will involve the notion of the ‘logical mir-ror’ of an inference theory and require the toolsfrom the previous section, covering the N = 3 case. We recall that the odd and even correla-tion structure for rebits has already been settledin section 5.2.7 through (28). This, as we shallsee shortly, is a consequence of rebit theory beingits own mirror image. Accepted in
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Consider a single qubit, described by O via an in-formationally complete set Q , Q , Q which canbe viewed as a question basis on Q . As such, itdefines a ‘logical handedness’ in terms of whichoutcomes to Q , Q , Q O calls ‘yes’ and which‘no’. Clearly, this is a convention made by O and he can easily change the handedness bysimply swapping the assignment ‘yes’ ↔ ’no’ ofone question, say, Q , which is tantamount to Q
7→ ¬ Q . This corresponds to taking the log-ical mirror image of a single qubit. For a singlequbit this will not have any severe consequencesand O is free to choose whichever handedness hedesires.However, the handedness of a single qubit ques-tion basis does become important when consider-ing composite systems. Consider, e.g., two ob-servers O, O ′ each having one qubit and describ-ing it with a certain handed basis. They can alsoconsider correlations Q ij of their qubit pair. If O now decided to change the handedness of his localquestion basis by Q
7→ ¬ Q this would result in Q , Q , Q
7→ ¬ Q , ¬ Q , ¬ Q ,Q ij Q ij , i = 1 and therefore Q ↔ Q
7→ ¬ ( Q ↔ Q ) , and Q ↔ Q
7→ ¬ ( Q ↔ Q ) . Since Q would remain unaffected by the changeof local handedness ( Q , Q ′ remain invariant), O ’s change of local handedness would have re-sulted in a switch between even and odd correla-tion structure for (19, 20). Since the handednessof the local question basis is just a conventionby O or O ′ , we shall allow either. Consequently,whether three correlation questions are related byeven or odd correlation (see theorem 5.7) dependson the local conventions by O, O ′ and both are, infact, consistent. However, (28) will always hold.Usually, of course, one would favour the situationthat both O, O ′ make the same conventions suchthat their local question bases are equally handed(or that one observer O describes all qubits withthe same handedness). Nevertheless, physicallyall local conventions are fully equivalent, yet willlead to different representations of composite sys-tems in the inference theory.But there are important consistency condi-tions on the distribution of odd and even cor-relations. This becomes obvious when examiningthree qubits A, B, C . To this end, consider thetrivial conjunction Q A B ↔ Q A C ↔ Q B C = lemma 5.10 . (37) From (19) we know that the correlation is eithereven or odd, respectively, Q A B = Q A B ↔ Q A B , or Q A B = ¬ ( Q A B ↔ Q A B ) (38) and analogously for A, C and
B, C . Suppose nowthat all three bipartite correlations in the con-junction (37) were even. Then we immediatelyget a contradiction because, using lemma 5.10, Q A B ↔ Q A C ↔ Q B C = ( Q A B ↔ Q A B ) ↔ ( Q A C ↔ Q A C ) ↔ ( Q B C ↔ Q B C )= ( ) ¬ ( Q A B ↔ Q A C ) | {z } = Q B C ↔ ( Q A B ↔ Q A C ) | {z } = Q B C ↔ ( Q B C ↔ Q B C )= 0 . But this violates the identity (37) and resultsfrom the relative negation between the left andright hand side in (28). One would get the same Ultimately, these three questions will define an or-thonormal Bloch vector basis in the Bloch sphere (see sec-tion 7). The handedness or orientation of this basis willdepend on the labeling of question outcomes. contradiction if one of the correlations in (37)was even and two were odd because then the twonegations from the odd correlation would canceleach other and one would still be left with thenegation coming from (28).On the other hand, everything is consistent ifeither all bipartite correlations in (38) are odd orone is odd and two are even because then the odd
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Quantum2017-11-27, click title to verify umber of negations from the odd correlations cancels the negation coming from (28): Q A B ↔ Q A C ↔ Q B C = ¬ ( Q A B ↔ Q A B ) ↔ ¬ ( Q A C ↔ Q A C ) ↔ ¬ ( Q B C ↔ Q B C )= ( ) ¬ ( Q A B ↔ Q A C ) | {z } = Q B C ↔ ( Q A B ↔ Q A C ) | {z } = Q B C ↔ ¬ ( Q B C ↔ Q B C )= 1 . We can also quickly check that this is consistentwith (20) Q A B = Q A B ↔ Q A B , or Q A B = ¬ ( Q A B ↔ Q A B ) (39) and (26) (and analogously for A, C and
B, C ) Q A B ↔ Q A B = ¬ ( Q A B ↔ Q A B ) . That is, if all correlations in (38) are odd, thenall correlations in (39) must be even. Indeed, Q A B ↔ Q A C ↔ Q B C = ( Q A B ↔ Q A B ) ↔ ( Q A C ↔ Q A C ) ↔ ( Q B C ↔ Q B C )= ( ) ¬ ( Q A B ↔ Q A C ) | {z } = Q B C ↔ ( Q A B ↔ Q A C ) | {z } = Q B C ↔ ( Q B C ↔ Q B C )= Q B C ↔ Q B C = 1 is consistent.In conclusion, if O wants to treat the bipar-tite relations among all three A, B, C identically ,then the following distribution of odd and evencorrelations Q A B = Q A B ↔ Q A B = ¬ ( Q A B ↔ Q A B ) , (40) and analogously for A, C and
B, C , is the only consistent solution.
We shall henceforth make theconvention that the bipartite correlation structureamong any pair of qubits be the same such that(40) holds . This turns out to be the case of qubitquantum theory. The tacit assumption, underlying the stan-dard representation of qubit quantum theory, isthat the handedness of each local qubit ques-tion basis for
A, B, C is the same, e.g., all ‘left’or ‘right’ handed. But we emphasize, that itwould be equally consistent to choose one basis as‘left’ (‘right’) and the other two as ‘right’ (‘left’)handed. A qubit pair with equally handed baseswill be described by the odd and even correla-tion distribution as in (40), while a qubit pairwith oppositely handed bases will be describedby the opposite distribution of odd and even cor-relations. In terms of whether Q = Q ↔ Q is even or odd Q = ¬ ( Q ↔ Q ) , the threequbit relations yield only four consistent graphsfor the distribution of ‘left’ and ‘right’ handednessPSfrag replacements odd oddoddeven ‘left’‘left’ ‘left’‘right’ PSfrag replacements oddodd oddeven‘left’ ‘right’ ‘right’‘right’
PSfrag replacements oddeveneven‘left’ ‘left’‘right’
PSfrag replacements oddeveneven‘left’ ‘right’‘right’ (41)
This gives a simple graphical explanation forthe consistency observations above. The firsttwo graphs correspond to quantum theory. The framework with the opposite correlation structureof quantum theory, corresponding to the last twographs, is sometimes referred to as mirror quan-
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Quantum2017-11-27, click title to verify um theory [15]. While mirror quantum theorywas considered inconsistent in [15], we see herethat inconsistencies would only arise if the bi-partite correlation structure of mirror quantumtheory (and in particular the even correlation Q = Q ↔ Q ) was used for all three pairs ofqubits. However, the proper formulation of mir-ror quantum theory for three qubits correspondsto the last two graphs in (41) and gives a per-fectly consistent framework. It could be easilygeneralized in an obvious manner to arbitrarilymany qubits. Obviously, the same argument can be carriedout for any other triple of maximally compatiblebipartite correlations appearing in theorem 5.7.In conclusion, the two distinct consistent distri-butions of odd and even correlations, correspond-ing to quantum theory and mirror quantum the-ory,(a) result from different conventions of local ba-sis handedness, and(b) are in one-to-one correspondence through alocal relabeling ‘yes’ ↔ ‘no’ of one individualquestion.As such, the two distinct correlation structuresultimately give rise to two distinct representa-tions of the same physics and are thus fully equiv-alent. The transformation (b) between the tworepresentations – being a translation between twoconventions/descriptions – is a passive one and In fact, one could even produce the correlation struc-ture of mirror quantum theory in the lab by using oppo-sitely handed bases for two qubits in an entangled pair.The resulting state would be represented by the partialtranspose of an entangled qubit state. This state wouldnot be positive if represented in terms of the standardPauli matrices and thus not correspond to a legal quan-tum state [88]. However, the point is that mirror quantumtheory would not be represented in the standard Paulimatrix basis but in the partially transposed basis whichcorresponds to replacing σ y = (cid:18) − ii (cid:19) by σ Ty = (cid:18) i − i (cid:19) , for one of the two qubits (this is a switch of the y -axisorientation). In this basis, the state would be positive. Notice, however, that for more than three qubits onewould get more than two different consistent distributionsof odd and even correlations. For example, for four qubitsthere will be three cases: (1) all four have equally handedbases, (2) three have equally handed bases, (3) two haveequally handed bases. can be carried out on a piece of paper; it is always allowed. However, clearly, there cannot exist anyactual physical transformation in the laboratorywhich maps states from one convention into theother.Within the formalism of quantum theory thetransformation (b) Q
7→ ¬ Q corresponds tothe partial transpose (e.g., see [15]). It is wellknown that the partial transpose defines a sepa-rability criterion for quantum states which is bothnecessary and sufficient for a pair of qubits [88]:a two-qubit density matrix ρ is separable if andonly if its partial transpose is positive relative toa basis of standard Pauli matrix products (i.e.,represents a legal quantum state). This criterionholds analogously in our language here: as seen atthe beginning of this section, the transformation Q
7→ ¬ Q changes between odd and even corre-lations of bipartite correlations Q ij . This would,in fact, be unproblematic if O only had individ-ual information about the two qubits; it wouldmap a classically composed state even of maxi-mal information, say, Q = 1 and Q ′ = 1 andthus Q = 1 , to another legal classically com-posed state Q = 0 , Q ′ = 1 and thus Q = 0 .Both states exist within both conventions. How-ever, applying this transformation to a maxi-mally entangled state with odd correlation, say, Q = Q = 1 and thus, by (40), Q = 0 yieldsan even correlation Q = 0 = Q and Q = 1 .The former state only exists in the quantum the-ory representation, while the latter exists only inthe mirror image. For other entangled states onewould similarly find that Q
7→ ¬ Q necessar-ily maps from one representation into the other.The same conclusion also holds for a transfor-mation { Q , Q , Q } 7→ {¬ Q , ¬ Q , ¬ Q } , whichone might call total inversion , because the oddnumber of negations involved in the transforma-tion would likewise lead to a swap of odd andeven correlations.Lastly, we note that the situation is very differ-ent for rebit theory because it is its own mirrorimage, i.e. rebit theory and mirror rebit theory are identical representations. O will describe asingle rebit by a question basis Q , Q . Suppose O decided to swap the ‘yes’ and ‘no’ assignmentsto the outcomes of Q , such that equivalently Q
7→ ¬ Q . For a pair of rebits, this would have Accepted in
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Quantum2017-11-27, click title to verify he following ramification Q , Q
7→ ¬ Q , ¬ Q ,Q , Q Q , Q , and therefore Q ↔ Q
7→ ¬ ( Q ↔ Q ) ⇒ Q
7→ ¬ Q . In contrast to the qubit case, Q is defined asa correlation of correlations Q := Q ↔ Q (22) and can not be written in terms of localquestions Q , Q ′ . Hence, Q also changes un-der this transformation by construction. Accord-ingly, Q
7→ ¬ Q does not lead to a swap of oddand even correlations for the rebit case. This‘partial transpose’ therefore always maps statesto other states within the same representation.For example, even a maximally entangled stateof maximal information and even correlation, say, Q = Q = 1 and Q = 1 , would be mapped toanother evenly correlated state Q = 0 , Q = 1 and Q = 0 . As a consequence, the Peres sep-arability criterion [88] which is valid for qubits does not hold in analogous fashion for rebits. Forcompletely equivalent reasons, the total inver-sion , corresponding to { Q , Q } 7→ {¬ Q , ¬ Q } ,is also a transformation which preserves the rep-resentation. N = 2 After the many technical details it is useful tocollect all the results concerning the compatibil-ity, complementarity and correlation structure fortwo qubits, derived in lemmas 5.2 and 5.5, theo-rem 5.7, equation (28) and in the previous section5.4.1 in a graph to facilitate a visualization. Weshall henceforth abide by the convention that allbipartite relations for arbitrarily many qubits betreated equally such that (40) must hold. Forthe other relations of theorem 5.7 one finds theanalogous results. As can be easily verified, theensuing question structure has the lattice patternof figure 4, where the trianglesPSfrag replacements − + Q Q ′ Q ′′ ⇔ Q = ¬ ( Q ′ ↔ Q ′′ ) , PSfrag replacements − + Q Q ′ Q ′′ ⇔ Q = Q ′ ↔ Q ′′ (42) denote odd and even correlation, respectively.We recall that (26, 28) imply alternating oddand even correlation triangles for the bipartitecorrelation questions. However, we emphasizethat the Bell scenario argument of section 5.2.7does not require the correlation triangles involv-ing individual questions to also admit such analternating odd and even pattern. For instance,the following relations of Q Q = Q ↔ Q ′ = Q ↔ Q = ¬ ( Q ↔ Q ) are consistent with assumptions 5 and 6, despitethe absence of a negation in Q ↔ Q ′ = Q ↔ Q because the latter is not a classical logicalidentity. The graph corresponding to the last re-lation,PSfrag replacements... Q Q Q Q Q Q Q ′ is also different from the graphs (23) as it in-volves all six individual questions and the argu-ment leading to (28) does not apply. Clearly,given the definition Q ij := Q i ↔ Q ′ j , all suchtriangles must be even.The lattice structure in figure 4 contains 15 tri-angles and × distinct edges correspond-ing to compatibility relations. There are 60 ‘miss-ing edges’ corresponding to complementarity re-lations. Every question resides in three compat-ibility triangles and is therefore maximally com-patible with the six other questions in these threetriangles and maximally complementary to theeight remaining questions not contained in thosethree triangles. This means that once one ques-tion is fully known, all other information availableto O must be distributed over the three adjacent Accepted in
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Quantum2017-11-27, click title to verify Sfrag replacements ++ + − −− Q Q Q Q ′ Q ′ Q ′ Q Q Q Q Q Q Q Q Q Q Q identify identify PSfrag replacements +++ + +++ + + − Q Q Q Q ′ Q ′ Q ′ Q ′ Q ′ Q ′ Q Q Q Q Q Q Q Q Q identify identifyidentify Figure 4: A lattice representation of the complete compatibility, complementarity and correlation structure of theinformationally complete set Q M for two qubits . Every vertex corresponds to one of the 15 pairwise independentquestions. If two questions are connected by an edge, they are maximally compatible. If two questions are not connected by an edge, they are maximally complementary. Thanks to the logical structure of the XNOR ↔ , defininga question as a correlation of two other questions, the compatibility structure results in a lattice of triangles. Asclarified in (42), red triangles denote odd, while green triangles denote even correlation. Note that the two latticesrepresented here are connected through the nine correlation questions Q ij and form a single closed lattice (which,however, is easier to represent in this disconnected manner). Every question resides in exactly three triangles and isthereby maximally compatible with six and maximally complementary to eight other questions. triangles. In particular, if O knows the answersto two of the questions in the lattice, he will alsoknow the answer to the third sharing the sametriangle such that every triangle corresponds toa specific set of states of maximal information.(Although, as we shall see shortly in section 6.8and, in more detail, in [1], the time evolution rule4 implies that states of maximal information willlikewise exist such that two independent bits canbe distributed differently over the lattice.)Notice that every triangle is connected by anedge (adjacent to one of its three vertices) to ev-ery other vertex in the lattice. This embodiesthe statement ‘whenever O asks S a new ques- tion, he experience no net loss of information’ ofthe complementarity rule 2. For instance, if O has maximal knowledge about two questions inthe lattice, corresponding to maximal informa-tion about one triangle, it is impossible for himto loose information by asking another questionfrom the lattice because it will be connected toone of the questions from the previous triangle.As a concrete example, suppose O knew the an-swers to Q , Q , Q . He could ask Q next.Since Q is connected by an edge to Q , hewould know the answers to both upon asking Q and thereby then also the answer to Q – no netloss of information occurs. Accepted in
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Quantum2017-11-27, click title to verify t is straightforward to check, e.g., using theansatz | ψ i = α | z − z − i + β | z + z + i + γ | z − z + i + δ | z + z − i for a two qubit pure state and translating it intothe various basis combinations xx, xy, yx, yy, . . . ,that the lattice structure of figure 4 is pre-cisely the compatibility and correlation struc-ture of qubit quantum theory. For instance, Q , Q , Q correspond to projectors onto the +1 -eigenspaces of σ x ⊗ σ x , σ y ⊗ σ y and σ z ⊗ σ z .The three questions sharing a red triangle means,e.g., that Q = Q = 1 and Q = 0 is an al-lowed state while Q = Q = Q = 1 is illegal.Indeed, ignoring normalization, in quantum the-ory one finds | x + x + i − | x − x − i = − i | y + y + i + i | y − y − i = | z + z − i + | z − z + i for α = β = 0 and γ = δ = 1 , corresponding tothe propositions Q = 1 : “the spins are corre-lated in x -direction"; Q = 1 : “the spins arecorrelated in y -direction"; and Q = 0 : “thespins are anti-correlated in z -direction". Butno quantum state exists such that the spins arealso correlated in z -direction if they are correlatedin x - and y -direction (this would be mirror quan-tum theory). Every other triangle in the latticecorresponds similarly to four pure quantum states(representing the answer configurations ‘yes-yes’,‘yes-no’, ‘no-yes’, ‘no-no’ to the two independentquestions per triangle).Finally, we also collect the results on the com-patibility, complementarity and correlation struc-ture of two rebits, derived in lemmas 5.2, 5.5 and5.8, theorem 5.9 and equation (26), in a latticestructure in figure 5. There are six triangles and × edges representing compatibility rela-tions. Every question resides in exactly two tri-angles and is thereby maximally compatible withfour and maximally complementary to four otherquestions. As in the qubit case in figure 4, everytriangle is connected by an edge to every othervertex in the lattice, in conformity with rule 2 Similarly, | ψ i = α | z − z − i + β | z + z + i = α + β ( | x + x + i + | x − x − i ) + β − α ( | x + x − i + | x − x + i ) = α + β ( | y + y − i + | y − y + i ) + β − α ( | y + y + i + | y − y − i ). But | x + x + i + | x − x − i = | y + y − i + | y − y + i and | y + y + i + | y − y − i = | x + x − i + | x − x + i .Hence, once Q = 1, one indeed gets Q ↔ Q = 0even though α, β are unspecified such that Q , Q maybe unknown. PSfrag replacements ++ ++ + − Q Q Q Q ′ Q ′ Q ′ Q Q Q Q Q ′ Q ′ Q Q ′ Q ′ Q Q identify identify Figure 5: A lattice representation of the complete com-patibility, complementarity and correlation structure ofthe informationally complete set Q M for two rebits .The explanations of the caption of figure 4 also applyhere. asserting that O shall not experience a net loss ofinformation by asking further questions. N > gbits We are now prepared to investigate the indepen-dence, compatibility and correlation structure en-suing from rules 1 and 2 on a general Q N , i.e. of O ’s possible questions to a system S composed of N > gbits. In particular, we shall exhibit an in-formationally complete set Q M N for both qubitsand rebits. O can view the N gbits as a compositesystem in many different ways: as N individualgbits, as one individual gbit and a composite sys-tem of N − gbits, as a system composed of gbits and a system composed of N − gibts, andso on. All these different compositions yield, ofcourse, the same question structure. It is simplestto interpret the N gbits recursively as being com-posed of a composite system of N − gbits and anew individual gbit. Definition 3.5 of a compositesystem then implies that Q N must contain (1) thequestions of Q N − for N − gbits, (2) the set Q of the new gbit, and (3) all logical connectives ofthe maximally compatible questions of those twosets. This entails slightly different repercussionsfor qubits and rebits. N > qubits A natural candidate for an informationally com-plete question set is given by the set of all possibleXNOR conjunctions of the individual questions of
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Quantum2017-11-27, click title to verify he N gbits Q µ µ ··· µ N := Q µ ↔ Q µ ↔ · · · ↔ Q µ N (43) (we recall from (13) that the logical connectiveyielding independent questions is either ↔ or ⊕ ).Here we have introduced a new index notation: µ a is the question index for gbit a ∈ { , . . . , N } and can take the values , , , . As before theindex values i = 1 , , correspond to the indi-viduals Q a , Q a , Q a . On the other hand, theindex value µ a = 0 implies that none of the threeindividual questions of gbit a appears in the con-junction (43), i.e. always Q a ≡ . For instance, Q ··· := Q , Q ··· := Q ↔ Q , etc. Note that Q ··· corresponds to no ques-tion. The set (43) thus contains all individual,bipartite, tripartite, and up to N -partite cor-relation questions for N gbits. We emphasizethat, due to the special multipartite structure(and associativity) of the XNOR, a question suchas Q ··· does not incarnate the question ‘arethe answers to Q , Q , · · · , Q N all the same?’.For example, for N = 4 , Q = Q = 0 and Q = Q = 1 also yield Q = 1 . This isimportant for the entanglement structure.We begin with an important result. Lemma 5.25.
The N − questions Q µ ··· µ N , µ = 0 , , , , are pairwise independent.Proof. Consider Q µ ··· µ N and Q ν ··· ν N . The twoquestions must disagree in at least one index oth-erwise they would coincide. Let the questionsdiffer on the index of gbit a , i.e. µ a = ν a . Sup-pose µ a , ν a = 0. Then Q µ a is maximally compat-ible with Q µ ··· µ a ··· µ N = Q µ ↔ · · · ↔ Q µ a ↔· · · ↔ Q µ N and maximally complementary to Q ν ··· ν a ··· ν N = Q µ ↔ · · · ↔ Q ν a ↔ · · · ↔ Q ν N since Q µ a , Q ν a are maximally complemen-tary. Using the same argument as in the proof of lemma 5.3 this implies independence of Q µ ··· µ N and Q ν ··· ν N . Lastly, suppose now µ a = 0 and ν a = 0 (as we have µ a = ν a not both can be 0).Then Q i a = ν a with i = 0 is maximally compatiblewith Q µ ··· µ N and maximally complementary to Q ν ··· ν N . By the same argument, this again im-plies independence of Q µ ··· µ N and Q ν ··· ν N . Consequently, the set (43) will be part of aninformationally complete set. We note that ahermitian matrix of trace equal to on a N -dimensional complex Hilbert space (i.e. qubitdensity matrix) is described by N − param-eters.Next, we must elucidate the compatibility andcomplementarity structure of this set. Lemma 5.26. Q µ ··· µ N and Q ν ··· ν N are maximally compatible if the index sets { µ , . . . , µ N } and { ν , . . . , ν N } differ by an even number (incl. ) of non-zero indices, maximally complementary if the index sets { µ , . . . , µ N } and { ν , . . . , ν N } differ by an odd number of non-zero indices. For example, for N = 2 , Q and Q differby two non-zero indices and are thus maximallycompatible. By contrast, Q = Q and Q dif-fer by an odd number of non-zero indices and arethereby maximally complementary. Proof.
Let Q µ ··· µ N and Q ν ··· ν N disagree in an odd number, call it 2 n + 1, of non-zero indices.We can always reshuffle the index labeling of the N qubits such that now a = 1 , . . . , n + 1 cor-responds to the qubits on which Q µ ··· µ N and Q ν ··· ν N differ by non-zero indices, i.e. µ a = ν a and µ a , ν a = 0. The remaining qubits labeledby b = 2 n + 2 , . . . , N are then such that Q µ ··· µ N and Q ν ··· ν N either agree on the non-zero index, µ b = ν b = 0 or at least one of µ b , ν b is 0. Thatis, after the reshuffling the index labeling, we canwrite the questions as Q µ ··· µ N = ( Q µ ↔ · · · ↔ Q µ n +1 ) | {z } disagreement ↔ ( Q µ n +2 ↔ · · · ↔ Q µ N ) | {z } maximally compatible Q ν ··· ν N = z }| { ( Q ν ↔ · · · ↔ Q ν n +1 ) ↔ z }| { ( Q ν n +2 ↔ · · · ↔ Q ν N ) . (44) We deduct the trivial question Q ··· . Obviously,one arrives at the same number by counting the distri-bution of individuals, bipartite,...and N -partite correla-tions over N qubits as a binomial series P Nk =1 (cid:0) Nk (cid:1) k =(3 + 1) N − Accepted in
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Quantum2017-11-27, click title to verify he parts of the questions where the indexsets either agree or feature zeros coincide with Q µ n +2 ··· µ N and Q ν n +1 ··· ν N and are clearly maxi-mally compatible (dropping here zero indices). We can now proceed by induction. Lemmas5.2, 5.5, 5.10–5.16 imply that the statement ofthis lemma is correct for n = 0 ,
1. Let the state-ment therefore be true for n and consider n + 1.Then, (44) reads Q µ ··· µ N =( Q µ ··· µ n +3 ) | {z } disagreement ↔ Q µ n +4 ··· µ N | {z } maximally compatible = ( Q µ ↔ · · · ↔ Q µ n +1 ) | {z } maximally complementary ↔ ( Q µ n +2 µ n +3 ) | {z } disagreement ↔ Q µ n +4 ··· µ N | {z } max. compat. Q ν ··· ν N = z }| { ( Q ν ··· ν n +3 ) ↔ z }| { Q ν n +4 ··· ν N = z }| { ( Q ν ↔ · · · ↔ Q ν n +1 ) ↔ z }| { ( Q ν n +2 ν n +3 ) ↔ z }| { Q ν n +4 ··· ν N . But, by lemma 5.5, Q µ n +2 µ n +3 and Q ν n +2 ν n +3 are maximally compatible with each other andtherefore also with Q µ ··· µ N and Q ν ··· ν N . Thisimplies that both Q µ ··· µ N and Q ν ··· ν N are max-imally compatible with and, thanks to lemma5.25, independent of, e.g., Q µ n +2 ··· µ N . Com-plementarity of Q µ ··· µ N and Q ν ··· ν N now fol-lows from lemma 5.1 and noting that Q µ ··· µ n +1 and Q ν ··· ν N are maximally complementary be- cause they disagree in 2 n + 1 non-zero indices(for which, by assumption, the statement of thelemma holds).Finally, let Q α ··· α N and Q β ··· β N disagree inan even number 2 n of non-zero indices. Usingan analogous reshuffling of the index labeling asin the odd case above, one can rewrite the twoquestions as Q α ··· α N = Q α α | {z } differ ↔ Q α α | {z } differ ↔ · · · ↔ Q α n − α n | {z } differ ↔ Q α n +1 ··· α N | {z } maximally compatible Q β ··· β N = z }| { Q β β ↔ z }| { Q β β ↔ · · · ↔ z }| { Q β n − β n ↔ z }| { Q β n +1 ··· β N . (45)That is, one can decompose the disagreeing partsof the questions into bipartite correlations. Butthanks to lemma 5.5 two bipartite correlations ofthe same qubit pair are maximally compatible ifand only if they differ in both indices. Conse-quently, all the pairings of question componentsof the upper and lower line, as written in (45),are maximally compatible and, hence, so must be Q α ··· α N and Q β ··· β N . Fortunately, it turns out that the N − ques-tions (43) are logically closed under the XNOR. Theorem 5.27. (Qubits)
The N − questions Q µ ··· µ N , µ = 0 , , , , are logically closed under ↔ and thus form an informationally complete set Q M N with D N = 4 N − for the case D = 3 .Proof. We shall prove the statement by induc-tion. The statement is trivially true for N = 1and, by theorems 5.7 and 5.17, holds also for N = 2 ,
3. Let the statement therefore be truefor N − N qubits. Since O can treat the N qubit systemin many different ways as a composite systemand the statement holds for N −
1, all XNORconjunctions of questions involving at least onezero index will be contained in (43). We thusonly need to show that all ↔ conjunctions in-volving at least one N -partite correlation Q i ··· i N , i a = 0 ∀ a = 1 , . . . , N , produce questions alreadyincluded in (43).Consider Q i ··· i N and Q ν ··· ν N . Lemma 5.26 im-plies that these two questions are maximally com-patible – and can thus be connected by ↔ – if andonly if { i , . . . , i N } and { ν , . . . , ν N } disagree inan even number of non-zero indices. There arenow two cases that we must consider:(a) Suppose Q i ··· i N and Q ν ··· ν N disagree inan even number of non-zero indices and, further-more, agree on at least one index i a = ν a . Thanksto Q i a ↔ Q i a = 1, the conjunction then yields Accepted in
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Quantum2017-11-27, click title to verify i ··· i a ··· i N ↔ Q ν ··· i a ··· ν N = Q i ··· position a ··· i N ↔ Q ν ··· position a ··· ν N . Hence, the two questions on the right hand sideboth contain less than N non-zero indices suchthat the result must lie in the set (43) becausethe statement is true up to N − Q i ··· i N and Q ν ··· ν N disagree in aneven number 2 n of non-zero indices and do notagree on any non-zero index. Reshuffling the in-dex labelings as in the proof of lemma 5.26, onecan then write Q i ··· i N = Q i i | {z } differ ↔ Q i i | {z } differ ↔ · · · ↔ Q i n − i n | {z } differ ↔ Q i n +1 ··· i N Q ν ··· ν N = z }| { Q ν ν ↔ z }| { Q ν ν ↔ · · · ↔ z }| { Q ν n − ν n ↔ Q n +1 ··· N . (We obtain here Q n +1 ··· N because Q i ··· i N and Q ν ··· ν N do not agree on any common index and Q i ··· i N does not feature zero-indices.) By lemma5.5, the pairs of bipartite correlations differing intwo indices, e.g., Q i i and Q ν ν , are maximally compatible and by theorem 5.7 their XNOR con-junction will yield another bipartite correlation ofthe same qubit pair. For example, Q i i ↔ Q ν ν equals either Q j j or ¬ Q j j for j = i , ν and j = i , ν . Accordingly, up to negation, one finds( j a = i a , ν a , a = 1 , . . . , n ) Q i ··· i N ↔ Q ν ··· ν N = Q j j ↔ Q j j ↔ · · · ↔ Q j n − j n ↔ Q i n +1 ··· i N = Q j ··· j n i n +1 ··· i N which is another N -partite correlation containedin the set (43). As in the cases N ≤ , we could representthe compatibility and complementarity relationsfor N > qubits geometrically by a simplicialquestion graph where a D -partite question corre-sponds to a ( D − -dimensional simplex withinthe graph ( D ≤ N ). The compatibility and com-plementarity relations of lemma 5.26 then trans-late into abstract geometric relations accordingto whether a D -simplex and a D ′ -simplex shareor disagree on subsimplices. In particular, thecriterion that two distinct questions in Q M N aremaximally compatible if and only if they disagreeon an even number of non-zero indices means ge-ometrically that the two questions are maximallycompatible if and only if the two subsimplices inthe two question simplices which correspond tothe qubits they share in common either(a) do not overlap and are odd -dimensional, or(b) coincide. For example, for N = 2 , Q , Q correspond totwo non-intersecting edges, i.e. -simplices; theyare maximally compatible because they disagreeon their -simplices, both involving qubit and .On the other hand, for N = 3 , Q A B , Q B C alsocorrespond to two non-intersecting -simplicesbut are maximally complementary because thetwo edges involve different qubits – A, B for thefirst and
B, C for the second edge.This also helps us to understand entanglementfor arbitrarily many qubits. Specifically, maxi-mal entanglement will correspond to O spendingthe N independent bits he is allowed to acquireabout the system S of N qubits on N -partite cor-relation questions. Lemma 5.26 guarantees thatfor every N there will exist N maximally com-patible N -partite questions. For instance, thereare (cid:0) N (cid:1) ways of having N − of the indices takevalue and indices take the value . Any twoof the (cid:0) N (cid:1) corresponding questions will disagreein two or four non-zero indices (and agree on therest) and will thus be maximally compatible (evenmutually according to theorem 3.2). These will Accepted in
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Quantum2017-11-27, click title to verify orrespond to a set of (cid:0) N (cid:1) maximally compati-ble ( N − -simplices in the question graph suchthat any two of them either disagree on an edgeor a tetrahedron. (There will exist even moremaximally compatible N -partite questions.) For N ≥ it also holds that (cid:0) N (cid:1) ≥ N .It is definitely possible to choose N such maxi-mally compatible N -partite correlation questionsout of the (cid:0) N (cid:1) many such that these N questionsdo not all agree on a single index. For similar rea-sons to the N = 2 , cases, this choice will con-stitute a mutually independent set such that ev-ery individual question Q i , . . . , Q i N will be max-imally complementary to at least one of these NN -partite questions. (As a consequence of rule 1,once the answers to these N N -partite questionsare known, they will also imply the answers to the (cid:0) N (cid:1) − N remaining ones by the same reasoningas in section 5.2.3.) Accordingly, O can exhaustthe information limit with these N -partite corre-lation questions, while not being able to have anyinformation whatsoever about the individuals – anecessary condition for maximal entanglement.There will exist many different ways of havingsuch multipartite entanglement for arbitrary N .One could describe such different ways of entan-glement by generalizing the correlation measures(33) and informational monogamy inequalities re-sulting therefrom. These monogamy inequalitiescould also be considered as simplicial relations:they restrict the way in which the available (inde-pendent and dependent) information can be dis- tributed over the various simplices and subsim-plices in the question graph. However, we abstainfrom analyzing such relations here further. N > rebits We briefly repeat the same procedure for N rebits. In analogy to (43), the natural candidateset for an informationally complete Q M N will con-tain Q µ µ ··· µ N := Q µ ↔ Q µ ↔ · · · ↔ Q µ N ,µ a = 0 , , , a = 1 , . . . , N, (46) where the notation should be clear from sec-tion 5.5.1. However, being a composite sys-tem, by definition 3.5 we must permit the cor-relation of correlations Q a b (22) for all a, b ∈{ , . . . , N } because clearly O is allowed to ask Q a , Q a , Q b , Q b . Furthermore, thanks to (36)we have, e.g., Q = Q ↔ Q = Q ↔ Q = Q ↔ Q such that no confusion can arise about the mean-ing of Q although there are no individu-als Q a into which the question could be decom-posed. The same holds similarly for any othereven number of indices taking the value . Con-sequently, the candidate set for Q M N can be writ-ten as ˜ Q M N := (cid:8) Q µ ··· µ N , µ a = 0 , , , , a = 1 , . . . , N (cid:12)(cid:12) only even number of indices taking value (cid:9) with an evident meaning of each such question.Let us count the number of elements within ˜ Q M N . Lemma 5.28. ˜ Q M N contains N − (2 N + 1) − non-trivial questions.Proof. If arbitrary distributions of the values µ a = 0 , , , a = 1 , . . . , N were per-mitted, we would obtain 4 N − Q ··· N as in the qubitcase. In order to obtain the number of questionswithin ˜ Q M N we thus still have to subtract all thepossible ways of distributing an odd number of 3’s over the N indices. There are precisely O N := N ! N − + N ! N − + · · · + N n + 1 ! N − (2 n +1) such ways, where 2 n +1 is the largest odd numbersmaller or equal to N . Similarly, the number ofways an even number of 3’s can be distributedover N indices is given by E N := 3 N + N ! N − + · · · + N m ! N − m , Accepted in
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Quantum2017-11-27, click title to verify here 2 m is the largest even number smaller orequal to N . We then have E N + O N = (3 + 1) N = 4 N ,E N − O N = (3 − N = 2 N and thus O N = 12 (4 N − N )which yields4 N − − O N = 2 N − (2 N + 1) − Q M N . We note that the number of parameters in asymmetric matrix with trace equal to on a N -dimensional real Hilbert space (i.e. rebit densitymatrix) is precisely N (2 N + 1) − .Next, we assert pairwise independence as re-quired for an informationally complete set. Lemma 5.29.
The N − (2 N + 1) − non-trivialquestions in ˜ Q M N are pairwise independent.Proof. The proof is entirely analogous to theproofs of lemmas 5.3, 5.13, 5.14 and 5.25.
Likewise, the complementarity and compatibil-ity structure of ˜ Q M N is analogous to the qubitcase. Lemma 5.30. Q µ ··· µ N , Q ν ··· ν N ∈ ˜ Q M N are maximally compatible if the index sets { µ , . . . , µ N } and { ν , . . . , ν N } differ by an even number (incl. ) of non-zero indices,and maximally complementary if the index sets { µ , . . . , µ N } and { ν , . . . , ν N } differ by an odd number of non-zero indices.Proof. Thanks to lemmas 5.8, 5.18–5.20 and5.23, the proof of lemma 5.26 also applies tothe rebit case with the sole difference that onlyan even number of indices in the questions cantake the value 3 and that correlations of correla-tions Q a b cannot be decomposed into individu-als Q a , Q b . Finally, ˜ Q M N is indeed logically closed. Theorem 5.31. (Rebits) ˜ Q M N is logicallyclosed under ↔ and is thus an informationallycomplete set ˜ Q M N = Q M N with D N = 2 N − (2 N +1) − for the case D = 2 . Proof. Thanks to theorems 5.9, 5.24 and lemma5.30, the proof of theorem 5.27 also applies here,except that only an even number of indices cantake the value 3 and Q a b cannot be decomposedinto individuals Q a , Q b . We close with the observation that simi-larly to the N qubit case in section 5.5.1, onecould represent the compatibility and comple-mentarity structure via a simplicial questiongraph. Maximally entangled states (of maximalinformation) will correspond to O spendingall N available bits over a mutually inde-pendent set of N N -partite questions whichis maximally complementary to every indi-vidual question. In fact, the prescription forconstructing a maximally N -partite entangledqubit state provided at the end of section 5.5.1also applies to N rebits since only indices withvalues , were employed. Furthermore, forrebits one can similarly generate maximallyentangled states of non-maximal information;e.g., O could ask only the N − questions Q ··· , Q ··· , Q ··· , . . . , Q ··· N − N .If O has maximal information about each suchquestion, every rebit pair will be maximallyentangled, while O has still not reached theinformation limit (see also section 5.2.6). Thus far we have only applied rules 1 and 2 toderive the basic question structure on Q N . Weshall now slightly switch topic and return to theproblems of time evolution, begun in subsection3.2.8, and of explicitly quantifying the informa-tion which O has acquired about S by means ofinterrogations with questions. The informationmeasure and time evolution of S ’s state in be-tween interrogations are intertwined through rule3 of information preservation and rule 4 of max-imality of time evolution such that we have todiscuss these topics together.In (6) we have implicitly defined the totalamount of O ’s information about S – once thelatter is in a state ~y O → S – as I O → S ( ~y O → S ) = D N X i =1 α i ( ~y O → S ) , Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify here α i quantifies O ’s information about theoutcome of question Q i ∈ Q M N and satisfies thebounds (1). We shall begin in subsection 6.1 byclarifying that the information measure we seekto derive here is conceptually distinct from thestandard Shannon entropy, before imposing el-ementary consistency conditions on the relation α i ( ~y O → S ) in subsection 6.2. Subsequently, we usethese conditions and implement rules 3 and 4 toestablish that the set of possible time evolutionsdefines a group and, finally, to derive the explicitfunctional form of I O → S in subsection 6.8. Thisdiscussion will also unravel properties of statespaces and lead to the notion of pure states. Consider a random variable experiment with dis-tinct outcomes each of which is described by aparticular probability. Outcomes with a smallerlikelihood are more informative (relative to theprior information) than those with a larger like-lihood. The Shannon information quantifies theinformation an observer gains, on average , via aspecific outcome if she repeated the experimentmany times. (Equivalently, it quantifies the un-certainty of the observer before an outcome ofthe experiment occurs.) The probability distri-bution over the different outcomes is assumed tobe known. In particular, none of the outcomes inany run of the experiment will lead to an updateof the probability distribution.Here, by contrast, we are not interested in howinformative one specific question outcome is rel-ative to another and how much information O would gain, on average, with a specific answer ifhe repeated the interrogation with a fixed ques-tion many times on identically prepared systems.Instead, we wish to quantify O ’s prior informa-tion (e.g., gained from previous interrogations)about the possible question outcomes on only the next system to be interrogated. The role ofthe probability distribution over the various out-comes (to all questions) is here assumed by thestate which O ascribes to S . This state is not as-sumed to be ‘known’ absolutely, instead, as dis-cussed in section 3.2, it is defined only relativeto the observer and represents O ’s ‘catalogue ofknowledge’ or ‘degree of belief’ about S . In par-ticular, this state is updated after any interro-gation if the outcome has led to an information gain. This update takes the form of a ‘collapse’ ofthe prior into a posterior state of s single systemin a single shot interrogation, while it yields anupdate of the ensemble state in a multiple shotinterrogation (see subsection 3.2.3). The proba-bility distribution represented by the state is thusa prior distribution for the next interrogation andonly valid until the next answer – unless the lat-ter yields no information gain. We thus seek toquantify the information content in the state of S relative to O . This is not equivalent to the sumof the average information gains, relative to thatsame state used as a fixed prior probability dis-tribution, which are provided by specific answersto the questions in an informationally completeset over repeated interrogations.The reader should thus not be surprised to findthat the end result of the below derivation willnot yield the Shannon entropy. We emphasize,however, that this clearly does not invalidate theuse of the Shannon entropy (in the more generalform of the von Neumann entropy) in quantumtheory as long as one employs it what it is de-signed for: to describe average information gainsin repeated experiments on identically preparedsystems – relative to a prior state which is as-signed before the repeated experiments are car-ried out and which represents the ‘known’ prob-ability distribution. There are a few natural requirements on the re-lation between α i and ~y O → S :(i) The Q i ∈ Q M N are pairwise independent.Accordingly, α i should not depend on the‘yes’-probabilities y j = i of other questions Q j = i such that α i = α i ( y i ) .(ii) All Q i ∈ Q M N are informationally of equiv-alent status. The functional relation be-tween y i and α i should be the same for all i : α i = α ( y i ) , i = 1 , . . . , D N .(iii) If O has no information about the outcomeof Q i , i.e. y i = 1 / , then α i = 0 bit .(iv) If O has maximal information about the out-come of Q i , i.e. y i = 1 or y i = 0 , then α i = 1 bit ; both possible answers give bit of in-formation. Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify v) The assignment of which answer to Q i is‘yes’ and which is ‘no’ is arbitrary and thefunctional relation between α i and y i (or n i )should not depend on this choice. Hence, α ( y i ) = α ( n i ) must be symmetric around y i = 1 / (see also (iv)); α i quantifies theamount of information about Q i , but doesnot encode what the answer to Q i is.(vi) On the interval y i ∈ (1 / , , the relationbetween α and y i should be monotonically increasing such that O ’s information aboutthe answer to Q i is quantified as higher, thehigher the assigned probability for a ‘yes’outcome. Likewise, on [0 , / , α ( y i ) mustbe monotonically decreasing . In particular, α shall be continuous and strictly convex;i.e., for all y i , y i ∈ [0 , and every λ ∈ [0 , α ( λ y i + (1 − λ ) y i ) ≤ λ α ( y i ) + (1 − λ ) α ( y i ) and the inequality shall be strict whenever y i = y i and < λ < . Hence, y i = is theglobal minimum of α on [0 , .Since now I O → S ( ~y O → S ) = D N X i =1 α ( y i ) (47) and each summand is strictly convex, we alsohave that I O → S is a strictly convex function on Σ N . Accordingly, in the coin flip scenario of sub-section 3.2.8, O ’s information about the mixedstate ~y O → S = λ ~y O → S +(1 − λ ) ~y O → S is smaller than his maximal information about any of itsconstituents ~y O → S , ~y O → S – unless the outcomeof the coin flip was certain, or S , S are in thesame state. This is consistent with the fact that O ’s information about the outcome of any ques-tion (asked to either S or S , depending on theoutcome of the coin flip) is clouded by the randomcoin flip outcome. In order to discuss the action of time evolutionon the state space Σ N , we have to derive a fewfurther properties of the latter, some of whichdepend on the properties of I O → S .Firstly, since D N is finite for finite N , Σ N is finite dimensional in this case. In fact, Σ N contains a basis of R D N so that it is a D N -dimensional closed convex subset of R D N . Fur-thermore, since y i ∈ [0 , for all y i in ~y O → S , Σ N is clearly bounded and thus, by the Heine-Borel-theorem, compact. As a compact convex set withnon-empty interior, Σ N is thus a convex bodyand, in particular, has volume [89].Next, we focus on the state of no information ~y O → S = ~ . Given strict convexity of I O → S on Σ N , the conditions of the previous subsection en-tail that the state of no information is the globalminimum of I O → S with I O → S ( ~
1) = 0 bits –in agreement with its uniqueness. It is also aninterior state.
Lemma 6.1.
The state of no information lies inthe interior of Σ N .Proof. The Q i ∈ Q M N are pairwise indepen-dent so for each such Q i there exist two states ~y y i O → S = ( , · · · , , , , · · · , ) and ~y n i O → S =( , · · · , , , , · · · , ) corresponding to O having only asked Q i to S in the state of no informationand having received the answers Q i = ‘yes’ and Q i = ‘no’, respectively. Since Σ N is convex, italso contains all convex mixtures of these specialstates. The D N ~y y i O → S , as well as the D N ~y n i O → S each define a basis of R D N such that the set of alltheir convex mixtures define a D N -dimensionalconvex polytope contained in Σ N ⊂ R D N . Inparticular, given that ~y y i O → S + ~y n i O → S = ~ i , itis clear that it lies in the interior of the convexpolytope. Given that time evolution preserves the totalinformation by rule 3, we shall also be interestedin the level sets of I O → S . Lemma 6.2.
Let I O → S fulfill conditions (i)–(vi)of section 6.2. For sufficiently small ε > , thelevel set L ε := { ~y O → S ∈ Σ N | I O → S ( ~y O → S ) = ε } (48) lies in the interior of Σ N and is homeomorphicto S D N − . In this case, L ε constitutes the fullboundary of the fat level set L I ≤ ε := { ~y O → S ∈ Σ N | I O → S ( ~y O → S ) ≤ ε } which contains the state Each of the following D N -dimensional vectors(1 , , , · · · , ) , ( , , , · · · , ) , · · · , ( , · · · , ,
1) rep-resents a legal state ~y O → S , corresponding to O onlyknowing the answer to precisely one question from Q M N with certainty and ‘nothing else’. Accepted in
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Quantum2017-11-27, click title to verify f no information and is homeomorphic to theclosed unit ball in R D N .Proof. The state of no information is the globalminimum of I O → S with I O → S ( ~
1) = 0 and,thanks to lemma 6.1, lies in the interior of Σ N .Furthermore, condition (vi) of subsection 6.2 re-quires that α ( y i ) increases monotonically awayfrom y i = for each i so that I O → S in (47) mustlikewise increase monotonically in each directionaway from ~y O → S = ~ R D N . Owing to thecontinuity of I O → S , we can therefore find a suffi-ciently small ε > any direction until reaching I O → S = ε still liesin the interior of Σ N . Accordingly, for such ε > L ε must lie in the interior of Σ N and so must L I ≤ ε with the state of no information in its own inte-rior. Given that I O → S : Σ N → R is continuousand Σ N is closed, I O → S is a closed convex func-tion. Closed convex functions have closed convexfat level sets [89] so that L I ≤ ε is a closed convexsubset in the interior of Σ N . Clearly, L I ≤ ε is D N -dimensional since we moved away in all directionsfrom ~y O → S = ~ L ε and the states with I O → S = ε define the limit points of L I ≤ ε in eachdirection so that L ε constitutes its full bound-ary. Any D N -dimensional closed convex subsetof R D N with non-empty interior is homeomorphic to the closed unit ball and its boundary is home-omorphic to S D N − . These results will become important for estab-lishing that the set of time evolutions is a group.
In subsection 3.2.8, we have only considered thetime evolution of the redundantly parametrized D N -dimensional state ~P O → S . According to (10),this time evolution is linear, described by a ma-trix A (∆ t ) , and only depends on the interval ∆ t = t − t , where t , t are arbitrary instants oftime in between O ’s interrogations on a given sys-tem. However, because of (4), it clearly sufficesto consider the yes-vector ~y O → S (or no-vector ~n O → S ) alone to describe the state of S relativeto O . Let us therefore determine, how ~y O → S and ~n O → S evolve under time evolution. To this end,we decompose A (∆ t ) and ~P O → S in (10), ~y O → S (∆ t ) ~n O → S (∆ t ) ! = a (∆ t ) b (∆ t ) c (∆ t ) d (∆ t ) ! ~y O → S (0) ~n O → S (0) ! , where a (∆ t ) , b (∆ t ) , c (∆ t ) , d (∆ t ) are nonnegative(real) D N × D N matrices. Using the normal-ization (4) (with p = 1 ), one finds that both ~y O → S , ~n O → S evolve affinely , ~y O → S (∆ t ) = [ a (∆ t ) − b (∆ t )] ~y O → S (0) + b (∆ t ) ~ ,~n O → S (∆ t ) = [ d (∆ t ) − c (∆ t )] ~n O → S (0) + c (∆ t ) ~ . (49) We shall now determine relations among the fourmatrices a, b, c, d . Firstly, the normalization (4) must hold for all initial states and all times.Hence, for all ∆ t ∈ R , ~ ~y O → S (∆ t ) + ~n O → S (∆ t )= [ a (∆ t ) − b (∆ t )] ~y O → S (0) + [ d (∆ t ) − c (∆ t )] ~n O → S (0) + [ b (∆ t ) + c (∆ t )] ~ ( ) [ a (∆ t ) − b (∆ t ) + c (∆ t ) − d (∆ t )] ~y O → S (0) + [ b (∆ t ) + d (∆ t )] ~ . Since the set of all possible initial states ~y O → S (0) contains a basis of R D N , we must conclude a (∆ t ) − b (∆ t ) = d (∆ t ) − c (∆ t ) , [ b (∆ t ) + d (∆ t )] ~ ~ . (50) Next, we note that, by assumption 3, the state ofno information is unique and, by rule 3, clearlypreserved under time evolution. Inserting thestate of no information, e.g., as ~n O → S (∆ t ) = ~ Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify nto (49), this entails [ c (∆ t ) + d (∆ t )] ~ ~ . (51) In conjunction, (50, 51) therefore imply ~y O → S (∆ t ) = [ a (∆ t ) − b (∆ t )] ~y O → S (0) + b (∆ t ) ~ ,~n O → S (∆ t ) = [ a (∆ t ) − b (∆ t )] ~n O → S (0) + b (∆ t ) ~ . Consequently, defining the D N × D N time evolu-tion matrix T (∆ t ) := a (∆ t ) − b (∆ t ) , (52) yields a linear time evolution of what we shallhenceforth call the generalized Bloch vector ~y O → S − ~ : ~y O → S (∆ t ) − ~ ~y O → S (∆ t ) − ~n O → S (∆ t )= T (∆ t ) (cid:16) ~y O → S (0) − ~ (cid:17) . (53) Notice that T (∆ t ) need neither be nonnegativenor stochastic in any pair of its components (incontrast to A (∆ t ) ). Henceforth, we shall onlyconsider the T (∆ t ) governing the time evolutionof the Bloch vector ~r O → S := 2 ~y O → S − ~ and often employ the latter to parametrize states. We continue by demonstrating that time evolu-tion of S ’s states must be injective on Σ N . Insubsection 6.6, we will use this result to provethat time evolution is also surjective on Σ N andthus, in fact, reversible . Lemma 6.3.
Let T (∆ t ) : Σ N → Σ N be a timeevolution as given in (52) for any ∆ t ∈ R . If I O → S is strictly convex, rule 3 entails that T (∆ t ) is injective.Proof. Assume T (∆ t ) was not injective. Thenthere would exist states ~y ′ O → S ( t ) = ~y ′′ O → S ( t )such that ~y O → S ( t ) := T (∆ t ) ~y ′ O → S ( t )= T (∆ t ) ~y ′′ O → S ( t ) . (55)Now consider the coin flip scenario of section3.2.8. O can prepare S in the state ~y O → S ( t ) := ~y ′ O → S ( t ) and S in the state ~y O → S ( t ) := ~y ′′ O → S ( t ) at time t before tossing the coin. By(8), ~y O → S ( t ) = λ ~y O → S ( t ) + (1 − λ ) ~y O → S ( t ) , and, on account of (55) and rule 3, I O → S ( ~y O → S ( t )) = I O → S ( ~y O → S ( t ))= I O → S ( ~y O → S ( t ))= I O → S ( ~y O → S ( t )) . (56)Thus, using (56) and strict convexity of I O → S ,this yields for a coin flip with 0 < λ < I O → S ( ~y O → S ( t )) < I O → S ( ~y O → S ( t )) = I O → S ( ~y O → S ( t )) . (57)On the other hand, (55) implies that, at time t , O would find ~y O → S ( t ) = ~y O → S ( t ) = ~y O → S ( t ) As an aside, note that A (∆ t ) can thus be replaced by A (∆ t ) = (cid:18) a (∆ t ) b (∆ t ) b (∆ t ) a (∆ t ) (cid:19) , so that time evolution is symmetric under a swap of the‘yes/no’-labeling for all Q ∈ Q M N at once, i.e. under ~y O → S ↔ ~n O → S . such that I O → S ( ~y O → S ( t )) = I O → S ( ~y O → S ( t ))= I O → S ( ~y O → S ( t )) . (58)Hence, (57, 58) entail that O ’s information about S has increased between t and t , despite nothaving tossed the coin and asked any questions.This is in contradiction with rule 3. We concludethat T (∆ t ) must be injective. Since Σ N contains a basis of R D N (see subsec-tion 6.3) it is clear that T (∆ t ) is also injectiveon R D N . For finite dimensional square matrices, T (∆ t ) being injective is equivalent to it being Accepted in
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Quantum2017-11-27, click title to verify ijective on R D N . Hence, to every T (∆ t ) theremust exist an inverse matrix T − (∆ t ) such that T (∆ t ) T − (∆ t ) = T − (∆ t ) T (∆ t ) = .However, is this T − (∆ t ) also a legal time evo-lution? This would be the case if T (∆ t ) wasalso surjective and thereby reversible on Σ N ⊂ R D N . Indeed, we shall demonstrate this propertynext. To establish reversibility of time evolution weshall resort to tools from topology and convexsets.
Lemma 6.4.
Restricting any time evolution T (∆ t ) : Σ N → Σ N to an interior level set L ε (48) defines a continuous, open and closed mapfrom L ε to itself.Proof. As an injective square matrix T (∆ t ) de-fines a continuous invertible map from R D N to it-self and so the pre-image of any open set in R D N is another open set. Rule 3 implies [ T (∆ t )]( L ε ) ⊆ L ε and that no state from outside L ε can evolveinto L ε . The open sets of L ε in the induced topol-ogy are the intersections of L ε with open sets of R D N . Consider any such open set V ⊂ L ε andany open set U ⊂ R D N such that V = U ∩ L ε .The pre-image [ T − (∆ t )]( U ) is again open as istherefore [ T − (∆ t )]( U ) ∩ L ε which, thanks to rule3, must be non-empty if V is non-empty. Hence, T (∆ t ) defines a continuous map from L ε to it-self. For similar reasons, this map is also open.Finally, the closed map lemma states that a con-tinuous map from a compact to a Hausdorff spaceis also closed. Thanks to lemma 6.2, we have L ε ≃ S D N − which is both compact and Haus-dorff and so the map from L ε to itself defined by T (∆ t ) is also closed. Next, we employ this result to show that timeevolution is surjective on interior level sets.
Lemma 6.5.
Any allowed time evolutionis surjective on the interior level sets, i.e. [ T (∆ t )]( L ε ) = L ε and thus also [ T (∆ t )]( L I ≤ ε ) = L I ≤ ε .Proof. Lemma 6.4 entails that T (∆ t ) defines acontinuous, open and closed map from L ε to it-self. It therefore maps all open sets to open setsand all closed sets to closed sets in L ε . In par-ticular, it must map all sets that are both openand closed in L ε to other sets which are bothopen and closed. Since L ε ≃ S D N − is con-nected, the empty set and the full L ε are theonly sets in L ε which are both closed and open.Now, by lemma 6.3, T (∆ t ) is injective so that[ T (∆ t )]( L ε ) = L ε . The last result is sufficient to finally establishthat time evolution is surjective also on the statespace.
Theorem 6.6.
Any allowed time evolution issurjective on Σ N , i.e. [ T (∆ t )](Σ N ) = Σ N , andhas | det T (∆ t ) | = 1 .Proof. Given that an interior fat level set L I ≤ ε is, by lemma 6.2, homeomorphic to the closedunit ball in R D N it is a convex body and hasvolume [89]. Lemma 6.5 yields [ T (∆ t )]( L I ≤ ε ) = L I ≤ ε and so its volume is left invariant by T (∆ t ) which, according to (53), acts linearly onstates ~r O → S in the Bloch vector parametriza-tion. Theorem 6.2.14 in [89] then implies that | det T (∆ t ) | = 1 and therefore also that the vol-ume of [ T (∆ t )](Σ N ) is equal to the volume of Σ N .Consider now the volume difference δ (cid:16) [ T (∆ t )](Σ N ) , Σ N (cid:17) := vol (cid:16) [T(∆t)](Σ N ) ∪ Σ N (cid:17) − vol (cid:16) [T(∆t)](Σ N ) ∩ Σ N (cid:17) . Clearly, [ T (∆ t )](Σ N ) ⊆ Σ N , andso δ (cid:0) [ T (∆ t )](Σ N ) , Σ N (cid:1) = vol (Σ N ) − vol (cid:0) [T(∆t)](Σ N ) (cid:1) = 0. Exercise 6.2.2 in [89]then shows that δ (cid:0) [ T (∆ t )](Σ N ) , Σ N (cid:1) = 0 is onlypossible if [ T (∆ t )](Σ N ) = Σ N . Hence, T − (∆ t ) maps all legal states to legalstates and must, by rule 4, be legal also. Corollary 6.7.
Any time evolution permitted bythe rules is reversible . I thank a referee for pointing out that addressing thisquestion was overlooked in a previous draft version.
Accepted in
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Quantum2017-11-27, click title to verify .7 Time evolution defines a group Given that any time interval can be decomposed into two time intervals, ∆ t = ∆ t + ∆ t , and theevolution of ~r O → S = 2 ~y O → S − ~ is continuous by rule 4, O must find T (∆ t ) T (∆ t ) ~r O → S (0) = T (∆ t ) ~r O → S (∆ t ) = ~r O → S (∆ t ) = T (∆ t ) ~r O → S (0) , and thus that multiplication is abelian , T (∆ t + ∆ t ) = T (∆ t ) T (∆ t ) = T (∆ t ) T (∆ t ) . (The last equality follows from time translationinvariance.) From the last equation and T (0) = , using ∆ t − ∆ t = 0 , we can also infer that T − (∆ t ) = T ( − ∆ t ) . If we permit O to considerthe time evolution of S for any duration ∆ t ∈ R ,it follows that the product of any two time evolu-tion matrices is again a time evolution matrix. Insummary, we therefore gather that a given timeevolution, as perceived by O and under the cir-cumstances to which he has subjected S , is suchthat:(i) T (0) = ,(ii) to every T (∆ t ) there exists an inverse T − (∆ t ) = T ( − ∆ t ) ,(iii) the multiplication of any two time evolutionmatrices is again a time evolution matrix,and(iv) matrix multiplication is obviously associa-tive.In conclusion, under the assumptions of section3.2 and rules 3 and 4, a given time evolution de-fines therefore an abelian, one-parameter matrixgroup. Hence, a given time evolution is described by a single evolution generator and single param-eter ∆ t parametrizing the duration. We note,however, that a multiplicity of time evolutions of S is possible, depending on the physical circum-stances (interactions) to which O may subject S .Different time evolutions will be generated by dif-ferent generators, but each time evolution willform a one-parameter group as discussed above.In fact, the full set of time evolutions T N which O is able to implement must likewise be a group.Namely, if T (∆ t ) , T ′ (∆ t ′ ) correspond to two dis-tinct interactions, then rule 4 implies that also T (∆ t ) · T ′ (∆ t ′ ) is a legal time evolution for anystate and since both T, T ′ are invertible their fullset must be a group. This implies, in particular,that T N is a group. We shall return to this furtherbelow and in [1]. We now have sufficient structure in our handto determine the functional relation between α i and y i . Given that the generalized Bloch vector ~y O → S − ~ transforms nicely under time evolution(53), it is useful to parametrize α i by y i − , i.e. α i = α (2 y i − . Rule 3 entails that O ’s total in-formation about (an otherwise non-interacting) S is a ‘conserved charge’ of time evolution I O → S ( ~y O → S (∆ t )) = I O → S ( ~y O → S (0)) which translates into the condition I O → S (cid:16) T (∆ t ) (cid:16) ~y O → S (0) − ~ (cid:17)(cid:17) = D N X i =1 α D N X j =1 T ij (∆ t ) (2 y j (0) − = D N X i =1 α (2 y i (0) −
1) = I O → S (2 ~y O → S (0) − ~ . (59) If T (∆ t ) was a permutation matrix, (59) wouldhold for any function α (2 y i − . For exam- ple, N classical bits are governed by the evolu-tion group Z × · · · × Z . However, permutations Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify orm a discrete group, while in our present case { T (∆ t ) , ∆ t ∈ R } constitutes a continuous one-parameter group. This is where continuity of timeevolution, as asserted by rule 4, becomes crucial.Under a reasonable assumption on the informa-tion measure, we shall now show that continuityof time evolution, together with conditions (i)– (vi) of subsection 6.2, enforces the quadratic re-lation α i = (2 y i − . To this end, we once moreinvoke the coin flip scenario. Given our parametrization in terms of theBloch vector ~y O → S − ~ , O ’s information aboutthe outcomes of his questions in the coin flip sce-nario can be written as follows I O → S (cid:16) λ ~y O → S + (1 − λ ) ~y O → S ) − ~ (cid:17) = I O → S (cid:16) λ (2 ~y O → S − ~
1) + (1 − λ )(2 ~y O → S − ~ (cid:17) = D N X i =1 α (cid:16) λ (2 y i −
1) + (1 − λ )(2 y i − (cid:17) . It is instructive to consider the case in which O is entirely oblivious about S such that the lat-ter is in the state of no information ~y O → S = ~ relative to him, but that O has some informa-tion about S . In this case, ~y O → S − ~ λ (2 ~y O → S − ~ and (assuming the outcome of thecoin flip is not certain) strict convexity of I O → S (see subsection 6.2) implies I O → S ( λ (2 ~y O → S − ~ < λ I O → S (2 ~y O → S − ~ or, equivalently, I O → S ( λ (2 ~y O → S − ~ f · I O → S (2 ~y O → S − ~ , where f < is a factor parametrizing O ’s in-formation loss relative to the case in which hedoes not toss a coin and, instead, directly asks S . The reason O experiences such a relativeinformation loss about the outcome of his inter-rogation is, of course, entirely due to the random-ness of the coin flip. But the coin flip is indepen-dent of the systems S , and, in particular, of thestates in which these are relative to O ; the fac-tor λ by which the probabilities ~y O → S become We suspect that this result may be derivable frompurely group theoretic arguments without an operationalsetup by employing the mathematical fact that to everycontinuous matrix group acting linearly on some spacethere corresponds a conserved inner product which isquadratic in the components of the vectors. In fact, one can equivalently interpret the situation asfollows: O only considers a system S which would be inthe state ~y O → S if it was present with certainty. However, λ in the state λ (2 ~y O → S − ~
1) represents the probabilitythat S ‘is there’ at all. Indeed, in this case, (4) can bewritten as λ ( ~y O → S + ~n O → S ) = λ · ~ λ ( ~y O → S − ~n O → S ) = λ (2 ~y O → S − ~ rescaled is state independent. For that reason,the relative information loss should likewise de-pend only on the coin flip, quantified by λ , and not on the state ~y O → S . For instance, if we alsoconsidered the case that the coin flip was certain,i.e. λ = 0 , , then clearly for λ = 1 we must have f = 1 ∀ ~y O → S ∈ Σ N and for λ = 0 it must hold f = 0 ∀ ~y O → S ∈ Σ N . We shall make this intoa requirement on the information measure for allvalues of λ : Requirement 1.
The relative information lossfactor f in (61) is a state independent (contin-uous) function of the coin flip probability λ with f ( λ ) < for λ ∈ (0 , . The binary Shannon entropy H ( y ) = − y log y − (1 − y ) log(1 − y ) in the role of α wouldfail this requirement. Namely, the measure α thusfactorizes, α ( λ (2 y i − f ( λ ) α (2 y i − , while H ( y ) would not. Setting λ = λ · λ yields f ( λ · λ ) α (2 y i −
1) = α ( λ · λ (2 y i − f ( λ ) α ( λ (2 y i − f ( λ ) f ( λ ) α (2 y i − and therefore f ( λ ) · f ( λ ) = f ( λ · λ ) , whichimplies f ( λ ) = λ p , for some power p ∈ R . Butthen, α must be a homogeneous function α (2 y i − Such a factorization of coin flip probabilities could beachieved, e.g., if O decided to use one coin, with ‘heads’probability λ , to firstly decide which of two possible con-vex mixtures to prepare where both possible mixtures aregenerated with a second coin with ‘heads’ probability λ .If three of the four states within the two mixtures arechosen as the state of no information, one would obtainprecisely such an equation. Accepted in
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Quantum2017-11-27, click title to verify ) = k (2 y i − p with some constant k ∈ R . Inconsequence, the information I O → S (2 ~y O → S − ~ is(up to k ) the p -norm of the Bloch vector ~y O → S − ~ . We can rule out that p ∈ ( −∞ , because inthis case, as one can easily check, it is impossi-ble to satisfy all the consistency conditions (i)–(vi) of subsection 6.2. Hence, p > . At thisstage we can make use of (59) and a result byAaronson [90] which implies that the only vector p -norm with p > which is preserved by a contin-uous matrix group is the -norm. Since any giventime evolution of the Bloch vector ~y O → S − ~ isgoverned by a continuous, one-parameter matrixgroup, we conclude that α (2 y i −
1) = k (2 y i − .Imposing condition (iv) yields k = 1 and there-fore ultimately I O → S ( ~y O → S ) = D N X i =1 (2 y i − . (61) It is straightforward to convince oneself that allof (i)–(vi) of subsection 6.2 are satisfied by thisquadratic information measure. O ’s total amountof information about S is thus the squared lengthof the generalized Bloch vector, thereby assuminga geometric flavour. It is important to emphasizethat, had we not imposed continuity of time evo-lution in rule 4, we would not have been able toarrive at (61); if time evolution was not continu-ous, many solutions to α in terms of y i would bepossible.The quadratic information measure (61) hasbeen proposed earlier by Brukner and Zeilingerin [32, 34, 91, 92] from a different perspective, em-phasizing that this is the most natural measuretaking into account an observer’s uncertainty –due to statistical fluctuations – about the out-come of the next trial of measurements on asystem in a multiple shot experiment. Further-more, taking the formalism of quantum theoryas given, Brukner and Zeilinger [73] later singledout the quadratic measure from the set of Tsal-lis entropies by imposing an ‘information invari-ance principle’, according to which a continuoustransformation among any two complete sets ofmutually complementary measurements in quan-tum theory should leave an observer’s informa-tion about the system invariant. While [73] iscertainly compatible with the present framework,here we come from farther away to the same re-sult: we do not pre-suppose quantum theory and derive the quadratic measure more generally bystarting from the landscape of information infer-ence theories and imposing rule 3 of informationpreservation and rule 4 of maximality of time evo-lution thereon. In this regard, the present deriva-tion may similarly be taken as a strong justifica-tion for the original Brukner-Zeilinger proposal. SO(D N ) Thus far, we only gathered that a given time evo-lution is described by a one-parameter group and,by theorem 6.6, must have | det T (∆ t ) | = 1 . Now,given (61), we are in the position to say quite a bitmore. The full matrix group leaving (61) invari-ant is O( D N ) . However, a given time evolutionis a continuous one-parameter group and musttherefore be connected to the identity. Hence, T (∆ t ) ∈ SO( D N ) , ∀ T (∆ t ) . Consequently, thegroup corresponding to any fixed time evolutionis a one-parameter subgroup of SO( D N ) . Wealso noted before that the set of all possible timeevolutions T N is a group thanks to rule 4. Itmust therefore likewise satisfy T N ⊂ SO( D N ) .This topic is thoroughly discussed in the com-panion articles, where it is shown that the rulesimply T N = PSU(2 N ) for the D = 3 [1] and T N = PSO(2 N ) for the D = 2 case [2]. Notethat PSU(2 N ) is a proper subgroup of SO(D N =4 N − for N > . The generators of T N are theset of possible time evolution generators of S ’sstates. The explicit quantification of O ’s informationnow permits us to render the distinction betweenthree informational classes of S ’s states – whichwe already loosely referred to as ‘states of max-imal knowledge’ or ‘states of non-maximal infor-mation’ in previous sections – precise.Firstly, we determine the maximally attain-able (independent and dependent) informationcontent within a state ~y O → S of a system of N gbits. This can be easily counted: once O knowsthe answers to N mutually independent questions(these do not need to be individuals), he will alsoknow the answers to all their bipartite, tripar-tite,... and N -partite correlation questions – allof which are contained in Q M N too by theorems Accepted in
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Quantum2017-11-27, click title to verify .27 and 5.31. But these are then N ! + N ! + · · · NN ! = N X i =1 Ni ! = 2 N − answered questions from Q M N , while all remain-ing questions in Q M N will be maximally comple-mentary to at least one of the known ones. O ’s to-tal information, as quantified by I O → S (47), thuscontains plenty of dependent bits of information– a result of the fact that the questions in Q M N are pairwise but not necessarily mutually inde-pendent.Using this observation, we shall characterize S ’s states according to their information content,i.e. squared length of the Bloch vector. By rules 3and 4, this distinction applies to all states whichare connected via some time evolution to thestates above, including those for which the infor-mation may be distributed partially over manyelements of a fixed Q M N . Specifically, we shallrefer to a state ~y O → S as a pure state: if it is a state of maximal information content, i.e. maximal length I O → S ( ~y O → S ) = D N X i =1 (2 y i − = (2 N − bits , mixed state: if it is a state of non-extremal information content, i.e. non-extremal length bit < I O → S ( ~y O → S ) = D N X i =1 (2 y i − < (2 N − bits , totally mixed state: if it is the state of no information ~y O → S = ~ with zero length I O → S (cid:18) ~y O → S = 12 ~ (cid:19) = 0 bit . We note that these characterizations of states interms of their length are indeed true in quantumtheory. In particular, N qubit pure states actu-ally have a Bloch vector squared length equal to N − . Our reconstruction gives this peculiarfact a clear informational interpretation.One may wonder whether the above definitionof a pure state is directly equivalent to being ex-tremal in Σ N and thereby to the usual defini-tion of pure states in GPTs or quantum theory.This is not obvious from what we have estab-lished thus far. However, since clearly we as-sume I O → S = (2 N − bits to be the maxi-mally attainable information, we can already con-clude that pure states so defined must lie on theboundary of Σ N because a non-constant convexfunction cannot have its maximum in the inte-rior of its convex domain [89]. Showing that thepure states, as defined above, of the theory sur-viving the imposition of all rules – quantum the- ory – are, indeed, the extremal states, requiresmore work. For N = 1 we shall demonstrate thisshortly, while the discussion for N > is deferredto [1]. N = 1 case and the Bloch ball Before closing this toolkit for now, we quickly givea flavor of the capabilities of the newly developedconcepts and tools by applying them to show thatin the simplest case of a single gbit ( N = 1 ) rules1–4 indeed only have two solutions within L gbit ,namely the qubit and the rebit state space includ-ing their respective time evolution groups. Thisproves the claim of section 4.1 for N = 1 .To this end, recall theorem 5.6 which assertsthat the dimension of the N = 1 state space Σ is either D = 2 or D = 3 which thus far wesuggestively referred to as the ‘rebit’ and ‘qubitcase’, respectively. Accepted in
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Quantum2017-11-27, click title to verify .1 A single qubit and the Bloch ball We begin with the D = 3 case. Σ will be parametrized by a three-dimensional vector ~y O → S = y y y , where y , y , y are the ‘yes’-probabilities of a mutually maximally complementary question set Q , Q , Q constituting an informationally complete Q (11).The informational distinction of states introduced in section 6.10 reads in this case pure states: ~y O → S such that I O → S = (2 y − + (2 y − + (2 y − = 1 bit , mixed states: ~y O → S such that bit < I O → S = (2 y − + (2 y − + (2 y − < bit , totally mixed state: ~y O → S = ~ such that I O → S = (2 y − + (2 y − + (2 y − = 0 bit . Recall from section 6.9 that the set of possibletime evolutions T must be contained in SO(3) and that the time evolution rule 4 requires theset of states into which any state ~y O → S can evolveto be maximal, while being compatible with therules. Thus, in particular, the image of the cer-tainly legal pure state (1 , , (rules 1 and 2 im-ply its existence in Σ ) under T must be maxi-mal. Applying all of SO(3) to this state generates all states with | ~y − ~ | = 1 bit and these cer-tainly abide by the rules. Consequently, the setof all possible time evolutions, compatible withrules 1–4, is the rotation group T ≃ SO(3) ≃ PSU(2) . This is the component of the isometry group ofthe Bloch ball which is connected to the identityand it is required in full in order to maximize thenumber of states into which (1 , , can evolve.However, since time evolution is state indepen-dent (c.f. assumption 8), this is the time evolu-tion group for all states. PSU(2) is precisely theadjoint action of
SU(2) on density matrices ρ × over C , ρ × U ρ × U † , U ∈ SU(2) and thuscoincides with the set of all possible unitary timeevolutions of a single qubit in standard quantum theory.The set of allowed states populates the entireunit ball in the three-dimensional Bloch ball, i.e. Σ ≃ B . This follows from the fact that apply-ing the full group T = SO(3) to (1 , , gener-ates the entire Bloch sphere as a closed set of ex-tremal states and the fact that Σ must be closedconvex according to assumption 2. Hence, werecover the well-known three-dimensional Blochball state space of a single qubit of standard quan-tum theory with the set of all pure states definingthe boundary sphere S , the totally mixed state(as the state of no information) constituting thecenter and the set of mixed states filling the inte-rior in between, as illustrated in figure 6a. Thisis precisely the geometry of the set of all normal-ized density matrices on C . Notice that the purestate space S ≃ C P indeed coincides with theset of all unit vectors in C (modulo phase).Given the complete symmetry of the Bloch ballas the state space for N = 1 , there should not ex-ist a distinguished informationally complete ques-tion set Q M , corresponding to a distinguished or-thonormal Bloch vector basis, by means of which O can interrogate S . While it is an additionalassumption, it is thus natural to stipulate that Accepted in
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Quantum2017-11-27, click title to verify Sfrag replacements totally mixed statemixed states pure states ~r ~r = 2 ~y O → S − ~ (a) 3D qubit Bloch ball PSfrag replacements totally mixed statemixed statespure states ~r ~r = 2 ~y O → S − ~ (b) 2D rebit Bloch discFigure 6: The three-dimensional Bloch ball (a) and thetwo-dimensional Bloch disc (b) are the correct statespaces Σ of a single qubit in standard quantum theoryand a single rebit in real quantum theory, respectively.The vector ~r parametrizing the states is the Bloch vector ~y O → S − ~ . to every Q ∈ Q there exists a unique pure statein Σ which represents the truth value Q = ‘yes’and, conversely, that every pure state of this sys-tem corresponds to the definite answer to onequestion in Q . But then Q ≃ S which alsocoincides with the set of all possible projectivemeasurements onto the +1 (or, equivalently, the − ) eigenspaces of the Pauli operators ~n · ~σ over C which is parametrized by ~n ∈ R , | ~n | = 1 ,and where ~σ = ( σ x , σ y , σ z ) are the usual Paulimatrices. This set of permissible questions Q N for all N will be discussed more thoroughly in [1]together with a derivation of the Born rule forprojective measurements.The ‘ballness’ and three-dimensionality of the state space of a single qubit can also be derivedfrom various operational axioms within GPTs[15, 16, 18, 21, 23, 59, 93] and constitutes a cru-cial step in most GPT based reconstructions ofquantum theory [14–16, 20, 21]. The principle of continuous reversibility , according to which everypure state of the convex set can be mapped intoany other by means of a continuous and reversibletransformation, usually assumes a crucial role insuch derivations. Here we offer a novel perspec-tive on the origin of the Bloch ball by deriving itfrom elementary rules for the informational rela-tion between O and S ; in particular, we recovercontinuous reversibility as a by-product. The analogous result holds for the D = 2 case: Σ will be parametrized by a two-dimensionalvector ~y O → S = y y ! , where y , y are the ‘yes’-probabilities of twomaximally complementary questions Q , Q con-stituting an informationally complete Q (11).We then have pure states: ~y O → S such that I O → S = (2 y − + (2 y − = 1 bit , mixed states: ~y O → S such that bit < I O → S = (2 y − + (2 y − < bit , and the totally mixed state: ~y O → S = ~ such that I O → S = (2 y − + (2 y − = 0 bit . In analogy to the qubit case, the time evolutionrule 4 implies that(a) the set of all possible time evolutions is the(projective) rotation group T ≃ SO(2) ≃ PSO(2) ≃ SO(2) / Z because SO(2) is the full (connected com-ponent of the orientation preserving) isom-etry group of the Bloch disc and all time
Accepted in
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Quantum2017-11-27, click title to verify volutions are permitted which preserve theBloch vector length, representing O ’s totalinformation about S , and the orientation ofthe Bloch vector basis. Orientation preser-vation means that O ’s convention about the‘yes’-‘no’-labelling of the question outcomesis preserved: The isomorphism denoted witha SO(2) is actually a double cover of
PSO(2) as indi-cated on the right. Indeed, the real densitymatrix of a single rebit, ρ × = ( + r x σ x + r z σ z ) on R where r i = (2 y i − , evolvesunder the adjoint action of SO(2) , ρ × O ρ × O T for O = e iσ y t ∈ SO(2) . This isequivalent to an action of
PSO(2) becausethe non-trivial center element − ∈ SO(2) acts trivially on ρ × and thus factors out.However, as a subcase of the single qubitcase this adjoint action of SO(2) on ρ × isalso equivalent to O ′ ~r for O ′ ∈ SO(2) in theBloch vector representation.(b) the state space Σ coincides with the two-dimensional Bloch disc, as depicted in figure6b, with the totally mixed state in the center,the pure states on the boundary circle S andthe mixed states in the interior in between.This is precisely the geometry of the set ofunit trace, positive-semidefinite, symmetricmatrices over R – the space of density ma-trices of a single rebit. Similarly, the purestate space S ≃ R P coincides with the setof all unit vectors in the Hilbert space R modulo Z .Again, given the symmetry of the Bloch disc,there is no reason for a distinguished informa-tionally complete question basis Q M on Q , in-carnated as a distinguished Bloch vector basis,to exist. Accordingly, we assume that any purestate on S corresponds to the definite answerof some question in Q which O may ask therebit. In this case, Q ≃ S , which coincideswith the set of projective measurements onto the +1 eigenspaces of the Pauli operators ~n · ~σ over R which are parametrized by ~n ∈ R , | ~n | = 1 and where ~σ = ( σ x , σ z ) are the two real and sym-metric Pauli matrices. Based on the premise to only speak about in-formation which an observer (or more generally system) has access to by interaction with othersystems, we have developed a novel frameworkfor characterizing and (re-)constructing the quan-tum theory of qubit systems. We have laid down,without ontological statements, the mathemati-cal and conceptual foundations for a landscapeof theories describing how an observer acquiresinformation about physical systems through in-terrogation with yes-no-questions.Within this landscape four elementary ruleshave been given which constrain the observer’sacquisition of information for qubit (and rebit)systems. The rule of limited information and thecomplementarity rule imply an independence andcompatibility structure of the binary questionswhich, in fact, reproduces that of projective mea-surements onto the +1 eigenspaces of Pauli op-erators in quantum theory. In particular, theserules entail in a constructive and simple way1. a novel argument for the dimensionality ofthe Bloch ball,2. a new method for determining thecorrect number of independent ques-tions/measurements necessary to describe asystem of N qubits (or rebits),3. a natural explanation for entanglement,monogamy of entanglement and quantumnon-locality,4. the explicit correlation structure of twoqubits and rebits, and5. more generally the correlation structure forarbitrarily many qubits and rebits.Furthermore, the rules of information preserva-tion and maximality of time evolution are shownto result6. in a reversible time evolution, and7. under elementary consistency conditions, ina quadratic information measure, quantify-ing the observer’s prior information aboutthe answers to the various questions he mayask the system.This measure has been earlier proposed byBrukner and Zeilinger from a different perspec-tive [32, 34, 73, 91, 92], complementing our presentderivation.Combining these results, we then show, as thesimplest example, how the rules entail that Accepted in
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Quantum2017-11-27, click title to verify . the Bloch ball and Bloch disc are recoveredas state spaces for a single qubit and rebit,respectively, together with the correct timeevolution groups and question sets.The full reconstruction of qubit quantum the-ory, following from the present four rules (and twoadditional ones), is completed in the companionpaper [1] (the rebit case which violates one of theadditional rules is considered separately [2]). Inconjunction, this derivation highlights the partial interpretation of quantum theory as a law book,describing and governing an observer’s acquisi-tion of information about physical systems. Inparticular, it highlights the quantum state as theobserver’s ‘catalogue of knowledge’.Certainly, there are some limitations to thepresent approach: First of all, we employ a cleardistinction between observer and system whichcannot be considered as fundamental. Secondly,the construction is specifically engineered for thequantum theory of qubit systems. While arbi-trary finite dimensional quantum systems could,in principle, be immediately encompassed by im-posing the so-called subspace axiom of GPTs[14, 16], the latter does not naturally fit into ourset of principles, mostly because rules 1 and 2quantify the information content of a system interms of N ∈ N bits which is only suitable forcomposite gbit systems. More generally, the lim-itation is that the current approach only encom-passes finite dimensional quantum theory, but notquantum mechanics . As it stands, the mechani-cal phase space language does not naturally fitinto the present framework and more sophisti-cated tools are required in order to cover mechan-ical systems, let alone anything beyond that.While this informational construction recoversthe state spaces, the set of possible time evolu-tions and projective measurements of qubit quan-tum theory, in other words, the architecture ofthe theory, it clearly does not tell us much aboutthe concrete physics (other than demanding it tofit into this general construction). This purelyinformational framework is rather universal andinformation carrier independent. But qubits asinformation carriers can be physically incarnatedin many different ways: as electron or muon spins,photon polarization, quantum dots, etc. Theframework cannot distinguish among the differ-ent physical incarnations, underlining the obser-vation that not everything can be reduced to in- formation and additional inputs are necessary inorder to do proper physics.Nevertheless, despite its current limitations,this informational approach teaches us somethingnon-trivial about the structure and physical con-tent of quantum theory.While this work is more generally motivated bythe effort to understand physics from an infor-mational and operational perspective and high-lights a partial interpretation of quantum the-ory, it clearly does not single out ‘the right one’(see also the discussion in [94]). For example, byspeaking exclusively about the information acces-sible to an observer, we are by default silent onthe fate of hidden variables and on whether theycould give rise to the assumptions and rules whichwe impose. The status of hidden variables is sim-ply not relevant here (other than that local hiddenvariables are ruled out).Let us, nonetheless, make a few possible(but not inescapable) interpretational state-ments. Given the absence of ontological commit-ments, this informational approach is generallycompatible with (but does not rely on) the re-lational [28, 29], Brukner-Zeilinger informational[31–35], or QBist [36–39] interpretations of quan-tum mechanics. They depart from the traditionalidea that systems necessarily have, absolute, i.e.observer independent properties (or, more gen-erally, properties independent of their relationswith other systems). Instead, many physicalproperties are interpreted as relational; the inter-action between systems establishes a relation be-tween them, permitting an information exchangewhich reveals certain physical properties relativeto one another and in the absence of hidden vari-ables this would be all there is. Certainly, in or-der not to render such a view hopelessly solip-sistic, systems ought to have certain intrinsic at-tributes, e.g. a corresponding state space or setof permissible interactions/measurements, suchthat different observers have a basis for agree-ing or disagreeing on the description of physicalobjects. However, a state of a system or measure-ment/question outcome is taken relative to who-ever performs the measurement or asks the ques-tion. Different observers may agree on states ormeasurement outcomes by communication (i.e.,physical interaction) but if one rejects the idea ofan absolute and omniscient observer it is naturalto also abandon the idea of an absolute and ex- Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify ernal standard by means of which properties ofsystems could be defined.From such a perspective, one could say thatthe relational character of the qubits’ proper-ties is a consequence of a universal limit on theamount of information accessible to the observerand the mere existence of complementary infor-mation such that the observer can not know theanswers to all his questions simultaneously. It isthe observer who determines which questions hewill ask and thereby what kind of system prop-erty the interrogation will reveal and thus, ulti-mately, which kind of information he will acquire(although clearly he does not determine what thequestion outcome is). But relative to the observerthe system of qubits does not have propertiesother than those accessible to him. Pushing an informational interpretation of quan-tum theory to the extreme, one may speculatewhether the quantum state could also representa state of information in a gravitational or cosmo-logical context. For instance, is such an interpre-tation adequate for the ‘wave function of the uni-verse’ which is ubiquitous in standard approachesto quantum cosmology (e.g., see [95–97])? Suchan interpretation would require the existence ofan absolute and omniscient observer, an ideawhich we just abandoned.Alternatively, one could adopt one of the cen-tral ideas of relational quantum mechanics [28,29], according to which all physical systems canassume the role of an ‘observer’, recording infor-mation about other systems, thereby relieving theclear distinction used in this manuscript. Ex-tending this idea to a space-time context, onecould interpret the universe as an abstract net-work of subsystems/subregions, viewed as infor-mation registers, which can communicate and ex-change information through interaction (see fig-ure 7). In this background independent context,any information acquisition by any register is in-ternal , i.e. occurs within the network; a globalobserver outside the network becomes meaning-less. This may appear as a purely philosophicalobservation, but it implies concrete consequencesfor the description of the network: there shouldbe no global state (aka ‘wave function of the uni-verse’) for the entire network at once. Indeed,
PSfrag replacements absolute observer S S S S S S S communication Figure 7: The universe as an information exchange net-work of subsytems without absolute observer. admitting any register in the network to act as‘observer’, the self-reference problem [98, 99] im-pedes a given register to infer the global state ofthe entire network – including itself – from its in-teractions with the rest. Accordingly, relative toany subsystem, one could assign a state to therest of the network, but a global state and thusa global Hilbert space would not arise. The ab-sence of a global state in quantum gravity hasbeen proposed before [100–103] – albeit from adifferent, less informational perspective.This offers an operational alternative to theproblematic concept of the ‘wave function of theuniverse’ which is ubiquitous in quantum cosmol-ogy. However, while a ‘wave function of the uni-verse’ might not make sense as a fundamentalconcept, it could still be given meaning as an ef-fective notion, namely as the state of the largescale structure of our universe on whose descrip-tion ‘late time’ observers better agree.Clearly, if this was to offer a coherent pictureof physics, there would need to exist non-trivialconsistency relations among the different regis-ters’ descriptions such as, e.g., the requirementthat ‘late time observers’ agree on the state ofthe large scale structure. This is not a practicallyunrealistic expectation as a concrete playgroundfor this idea has recently been constructed froma different motivation [104]: a scalar field on thebackground geometry of elliptic de Sitter spacecan only be quantized in an observer dependentmanner. A global Hilbert space for the quantumfield does not exist, but consistency conditionsbetween different observers’ descriptions can bederived. (Observer consistency has also been dis-
Accepted in
Quantum2017-11-27, click title to verify
Quantum2017-11-27, click title to verify ussed in general terms within Relational Quan-tum Mechanics [28, 29] and QBism [64, 65].) Al-though not being a quantum gravity model, itmay serve as a platform for concretizing this pro-posal further. Acknowledgements
This work has benefited from interactions withmany colleagues. First and foremost, it is apleasure to thank Markus Müller for uncount-ably many insightful discussions, patient expla-nations of his own work and suggestions. I amgrateful to Chris Wever for collaboration on [1, 2]and important discussions, to Sylvain Carrozzafor drawing my attention to Relational Quan-tum Mechanics in the first place and many re-lated conversations, to Tobias Fritz and LucienHardy for asking numerous clarifying questions,and to Rob Spekkens for many careful com-ments and suggestions. I would also like tothank Caslav Brukner, Borivoje Dakic, MartinKliesch, Matt Leifer, Miguel Navascués, Mat-teo Smerlak and Rafael Sorkin for useful discus-sions. Research at Perimeter Institute is sup-ported by the Government of Canada through In-dustry Canada and by the Province of Ontariothrough the Ministry of Research and Innova-tion. The project leading to this publication hasalso received funding from the European Union’sHorizon 2020 research and innovation programmeunder the Marie Sklodowska-Curie grant agree-ment No 657661.
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