aa r X i v : . [ qu a n t - ph ] F e b If physics is an information science, what isan observer?
Chris Fields
21 Rue des Lavandi`eresCaunes Minervois, 11160 France fi[email protected] 23, 2018
Abstract
Interpretations of quantum theory have traditionally assumed a “Galilean” ob-server, a bare “point of view” implemented physically by a quantum system. Thispaper investigates the consequences of replacing such an informationally-impoverishedobserver with an observer that satisfies the requirements of classical automata the-ory, i.e. an observer that encodes sufficient prior information to identify the systembeing observed and recognize its acceptable states. It shows that with reasonableassumptions about the physical dynamics of information channels, the observationsrecorded by such an observer will display the typical characteristics predicted by quan-tum theory, without requiring any specific assumptions about the observer’s physicalimplementation.
Keywords : Measurement; System identification; Pragmatic information; Decoher-ence; Virtual machine; Quantum Darwinism; Quantum Bayesianism; Emergence
PACS : 03.65.Ca; 03.65.Ta; 03.65.Yz“
Information ? Whose information? Information about what ?”J. S. Bell ([1] p, 34; emphasis in original)
Despite over 80 years of predictive success (reviewed in [2]), the physical interpreta-tion of quantum states, and hence of quantum theory itself remains mysterious (forrecent reviews see [3, 4, 5]). Informally speaking, this mysteriousness results from the pparent dependence of the physical dynamics on the act of observation. ConsiderSchr¨odinger’s cat: the situation is paradoxical because the observer’s act of openingthe box and looking inside appears to cause the quantum state of the cat to “collapse”from the distinctly non-classical superposition | cat i = √ ( | alive i + | dead i ) to one ofthe two classical eigenstates | alive i or | dead i . The introduction of decoherence the-ory in the 1970s and 80s [6, 7, 8, 9, 10] transferred this mysterious apparently-causaleffect on quantum states from what the observer looks at - the system of interest -to what the observer ignores: the system’s environment (reviewed by [11, 12, 13]; seealso [3, 4, 5] for treatments of decoherence in a more general context and [14] for a lessformal, more philosophical perspective). Schr¨odinger’s poor cat, for example, inter-acts constantly with the environment within the box - stray photons, bits of dust, etc.- and via the walls of the box with the thermal environment outside. Components of | cat i thereby become entangled with components of the environmental state | env i , astate that spreads at the speed of light to encompass all the degrees of freedom of theentire universe (other than the cat’s) as the elapsed time t → ∞ . To an observer whodoes not look at the environment, this entanglement is invisible; the components ofthe environment can therefore be “traced out” of the joint quantum state | cat ⊗ env i to produce an ensemble of non-interfering, effectively classical states of just the cat,each with a well-defined probability. Such reasoning about what observers do notlook at is employed to derive effectively classical states of systems of interest through-out the applied quantum mechanics literature. For example, Martineau introducesdecoherence calculations intended to explain why the Cosmic Background Radiationdisplays only classical fluctuations with the remarks: “Decoherence is, after all, anobserver dependent effect - an observer who could monitor every degree of freedomin the universe wouldn’t expect to see any decoherence. However, our goal is to de-termine a lower bound on the amount of decoherence as measured by any observer... we trace out only those modes which we must ... and take our system to be com-posed of the rest” ([15] p. 5821). Noting that the setting for these calculations is theinflationary period immediately following the Big Bang, one might ask, “ Observer ? What observer? Looking at what ?”Ordinary observers in ordinary laboratories interact with ordinary, macroscopicapparatus in order to gain classical information in the form of macroscopically andstably recordable experimental outcomes. The reconceptualization of physics as aninformation science that developed in the last quarter of the 20 th century, motivatedby Feynman’s speculation that all of physics could be simulated with a quantumcomputer [16], Wheeler’s “it from bit” proposal that “all things physical ... mustin the end submit to an information-theoretic description” ([17] p. 349), Deutsch’sproof of the universality of the quantum Turing machine (QTM [18]) and Rovelli’sexplicitly information-theoretic derivation of relational quantum mechanics [19], re-formulated the problem of describing measurement as the problem of describing howobservers could obtain classical information in a world correctly described by thequantum mechanical formalism. Theoretical responses to this reconceptualizationcan be divided into two broad categories by whether they maintain the standardDirac - von Neumann Hilbert-space formalism as fundamental to quantum mechanicsand adopt information-theoretic language to its interpretation, or adopt information-theoretic postulates as fundamental and attempt to derive the Hilbert-space for- alism from them. Responses in the first category treat decoherence as a funda-mental physical process and derive an account of measurement from it; examplesinclude traditional relative-state (i.e. many-worlds or many-minds) interpretations[11, 20, 21, 22, 23, 24], the consistent histories formulation [25, 26, 27] and quantumDarwinism [12, 28, 29, 30, 31, 32]. Those in the second treat measurement as a fun-damental physical process; they are distinguished by whether they treat informationand hence probabilities as objective [33, 34, 35] or subjective [19, 36, 37, 38, 39, 40].While observers appear as nominal recipients of information in all interpretativeapproaches to quantum theory, the physical structure of an observer is rarely ad-dressed. Zurek [12], for example, remarks that observers differ from apparatus intheir ability to “readily consult the content of their memory” (p. 759), but nowherespecifies either what memory contents are consulted or what memory contents mightbe required, stating that “the observer’s mind (that verifies, finds out, etc.) consti-tutes a primitive notion which is prior to that of scientific reality” (p. 363-364). Hartle[26] characterizes observers as “information gathering and utilizing systems (IGUSes)”but places no formal constraints on the structure of an IGUS and emphasizes thatthe information gathered by IGUSes is “a feature of the universe independent of hu-man cognition or decision” (p. 983). Rovelli [19] insists that “The observer can beany physical system having a definite state of motion” (p. 1641). Schlosshauer [3]adopts the assumption that appears most commonly throughout the literature: “Wesimply treat the observer as a quantum system interacting with the observed system”(p. 361). Fuchs [37] treats observers as Bayesian agents, and not only rejects butlampoons the idea that the physical implementation of the observer could be theo-retically important: “would one ever imagine that the notion of an agent, the userof the theory, could be derived out of its conceptual apparatus?” (p. 8). Whilesuch neglect (or dismissal) of the structure of the observer is both traditional and prima facie consistent with the goal of building a fully-general, observer-independentphysics, it seems surprising in a theoretical context motivated by “it from bit” andthe conceptualization of physical dynamics as quantum computing.It is the contention of the present paper that the physical structure of the observeris important to quantum theory, and in particular that the information employed bythe observer to identify the system of interest as an information source must be takeninto account in the description of measurement. This contention is motivated by theintuition expressed by Rovelli, that “the unease (in the interpretation of quantum the-ory) may derive from the use of a concept which is inappropriate to describe the worldat the quantum level” ([19] p. 1638). On the basis of this intuition, Rovelli rejects theassumption of observer-independent quantum states, an assumption also rejected byquantum Bayesians [36, 37, 39, 40]. The present paper rejects an equally-deep assump-tion: the assumption of a “Galilean” observer, an observer that is simply “a quantumsystem interacting with the observed system” without further information-theoreticconstraints. As the analysis of Rovelli [19] demonstrates, measurement interactionsbetween a Galilean observer and a physical system can be described in terms of Shan-non information, but this can only be done from the perspective of a second observeror a theorist who stipulates what is to count as “observer” and “system.” The useof Galilean observers in an information-theoretic formulation of physical theory thusrequires that the identities of “systems” be given in advance. That this requirement s problematic has been noted by Zurek, who states that “a compelling explanation ofwhat the systems are - how to define them given, say, the overall Hamiltonian in somesuitably large Hilbert space - would undoubtedly be most useful” ([41] p. 1818), andrequires as “axiom(o)” of quantum mechanics that “(quantum) systems exist” ([12]p. 746; [31] p. 3; [42] p. 2) as objective entities. Zurek adopts Wheeler’s [43] viewthat the universe itself can be considered to be the “second observer,” and proposesfrom this “environment as witness” perspective that decoherence provides the phys-ical mechanism by which systems “emerge” into objectivity [12, 28, 29, 30, 31, 32].Decoherence is similarly proposed to be the mechanism by which quantum infor-mation becomes classical [44] and by which both Everett branches [22, 23] and theframeworks defining consistent histories [25, 26, 27] are distinguished. By rejectingthe assumption of Galilean observers, the present paper also rejects the idea thatthe objective existence of systems can be taken as given a priori , either by an axiomor by a physical process of emergence. Instead, it proposes that not just quantumstates but systems themselves are definable only relative to observers, and in partic-ular, that quantum systems are defined only relative to classical information encodedby observers. An alternative approach to understanding quantum theory in informa-tional terms is proposed, one that explicitly recognizes the requirement that observersencode sufficient information to enable the identification and hence the definition ofthe systems being observed.That ordinary observers in ordinary laboratories must be in possession of infor-mation sufficient to identify systems of interest as classical information sources, notjust instantaneously but over extended time, is uncontroversial in practice. It followsimmediately, moreover, from Moore’s 1956 proof that no finite sequence of obser-vations of the outputs generated by a finite automaton in response to given inputscould identify the automaton being observed ([45] Theorem 2; cf. [46] Ch. 6). Henceordinary observers are not Galilean. The information employed by an ordinary, non-Galilean observer to identify a system being observed is “pragmatic” information inthe sense defined by Roederer [47, 48], although as will be seen below, without Roed-erer’s restriction of such information to living (i.e. evolved self-reproducing) systems.That observers must encode such pragmatic information in their physical structuresfollows from the physicalist assumption - the complement of “it from bit” - that allinformation is physically encoded [49]. The notion of an “observer” as a physicaldevice encoding input-string parsers or more general input-pattern recognizers thatfully specify its observational capabilities underlies not only the design and imple-mentation of programming languages and other formal-language manipulation tools(e.g. [50, 51, 52]), but also computational linguistics and the cognitive neuroscienceof perception (e.g. [53, 54, 55, 56, 57, 58]).It is shown in what follows that when the pragmatic information encoded by or-dinary observers is explicitly taken into account, distinctive features of the quantumworld including the contexuality of observations, the violation of Bell’s inequalityand the requirement for complex amplitudes to describe quantum states follow nat-urally from simple physical assumptions. The next section, “Interaction and SystemIdentification” contrasts the description of measurement as physical interaction withits description as a process of information transfer, and shows how the problem ofsystem identification arises in the latter context. The third section, “Informational equirements for System Identification” formalizes the minimal information that anobserver must encode in order to identify a macroscopic system - a canonical measure-ment apparatus - that reports the pointer values of two non-commuting observables.It then defines a minimal observer in information-theoretic terms as a virtual ma-chine encoding this minimal required information within a control structure capableof making observations and recording their results. The following section, “PhysicalInterpretation of Non-commutative POVMs” considers the physical implementationof a minimal observer in interaction with a physical channel. It shows that if thephysical dynamics of the information channel are time-symmetric, deterministic, andsatisfy assumptions of decompositional equivalence and counterfactual definiteness,any minimal observer encoding POVMs that jointly measure physical action will ob-serve operator non-commutativity independently of any further assumptions aboutthe observed system. The fifth section, “Physical Interpretation of Bell’s Theorem,the Born Rule and Decoherence” shows that the familiar phenomenology of quantummeasurement follows from the assumptions of minimal observers and channel dynam-ics that are time-symmetric, deterministic, and satisfy decompositional equivalenceand counterfactual definiteness. It shows, in particular, that decoherence can be un-derstood as a consequence of hysteresis in quantum information channels, and thatthe use of complex Hilbert spaces to represent observable states of quantum systemsis required by this hysteresis. The sixth section, “Adding Minimal Observers to theInterpretation of Quantum Theory” reviews the ontology that naturally follows fromthe assumption of minimal observers, an ontology that is realist about the physicalworld but virtualist about “systems” smaller than the universe as a whole. It showsthat any interpretative framework that treats “systems” as objective implicitly as-sumes that information is free, i.e. implicitly assumes that the world is classical.The paper concludes by suggesting that the interpretative problem of interest is thatof understanding the conditions under which a given physical dynamics implementsa given virtual machine, i.e. the problem of understanding the “emergence” not of“classicality” but of observers. The extraordinary empirical success of quantum theory suggests strongly that quan-tum theory is the correct description of the physical world, and that classical physicsis an approximation that, at best, describes the appearance of the physical worldunder certain circumstances. Landsman [4] calls the straightforward acceptance ofthis suggestion “stance 1” and contrasts it with the competing view (“stance 2”) thatquantum theory is itself an approximation of some deeper theory in which the worldremains classical after all. This paper assumes the correctness of quantum theory;Landsman’s “stance 1” is thus adopted. In particular, it assumes minimal quantumtheory, in which the universe as a whole undergoes deterministic, unitary time evo-lution described by a Schr¨odinger equation. The question that is addressed is howthe formal structure of minimal quantum theory can be understood physically, as adescription of the conditions under which observers can obtain classical informationabout the evolving states of quantum systems. s emphasized by Rovelli [19], minimal quantum theory treats all systems, in-cluding observers, in a single uniform way. The interaction between an observer anda system being observed can, therefore, be represented as in Fig. 1a: both observerand observed system are collections of physical degrees of freedom that are embeddedin and interact with the much larger collection of physical degrees of freedom - the“environment” - that composes the rest of the universe. The present paper adoptsa realist stance about these physical degrees of freedom; they can be considered tobe the quantum degrees of freedom of the most elementary objects with which thetheory is concerned. The observer - system interaction is described by a Hamiltonian H O − S ; this Hamiltonian is well-defined to the extent that the boundaries separatingthe observer and the system from the rest of the universe are well-defined. In practice,however, neither the system - environment nor the observer - environment boundariesare determined experimentally. The degrees of freedom composing the system S aretypically specified by specifying a set {| s i i} of orthonormal basis vectors, e.g. by say-ing “let | S i = P i λ i | s i i .” The set {| s i i} is a subset of a set of basis vectors spanningthe Hilbert space H U of the universe as a whole; it defines a subspace of H U withfinite dimension d that represents S . The state of O , on the other hand, is typi-cally left unspecified, and the O − S interaction is represented not as a Hamiltonianbut as a measurement that yields classical information. Traditionally, measurementsare represented as orthonormal projections along allowed basis vectors of the system(e.g. [60]); distinct real “pointer values” representing distinct observable outcomesare associated with each of these projections. In current practice, the requirementof orthogonality is generally dropped and measurements are represented as positiveoperator-valued measures (POVMs), sets of positive semi-definite operators {E j } thatsum to the identity operator on the Hilbert space of S (e.g. [59] Ch. 2). As shown byFuchs [36], a “maximally informative” POVM can be constructed from a set of d pro-jections { Π j } on the Hilbert space spanned by {| s i i} . The first d components of sucha POVM are the orthogonal projections | s i ih s i | ; “pointer values” can be associatedwith these d orthogonal components in the usual way. a) “Observer” “Environment” H O − S “System”(b) Observer Information channelIntervention ✲✛ Outcome System
Fig. 1: (a) A physical interaction H O − S between physical degrees of freedom regarded ascomposing an “observer” O and other, distinct physical degrees of freedom regarded ascomposing a “system” S , all of which are embedded in and interact with physical degrees offreedom regarded as composing the “environment” E . Boundaries are drawn with broken linesto indicate that they may not be fully characterized by experiments. (b) A two-way informationtransfer between an observer O and a system S via a channel C . Replacing “physical interaction” with “informative measurement” and hence H O − S with {E j } effectively replaces Fig. 1a with Fig. 1b, in which a well-defined observerobtains information from a well-defined system. The surrounding physical environ-ment of Fig. 1a is abstracted into the information channel of Fig. 1b. This idea thatinformation is transferred from system to observer via the environment is made ex-plicit in quantum Darwinism [30, 31, 32]. However, it is implicit in the assumption ofstandard decoherence theory that the observer “ignores” the surrounding environmentand obtains information only from the system; an observer will receive informationfrom the system alone only if the observer - environment interaction transfers noinformation, i.e. only if the information content of the environment is viewed astransferred entirely through the system - observer channel.In the case of human observers of macroscopic systems, the information channelis in many cases physically implemented by the ambient photon field. If the systemof interest is stipulated to be microscopic - the electrons traversing a double-slitapparatus, for example, or a pair of photons in an anti-symmetric Bell state - theinformation channel is often taken to be the macroscopic measurement apparatusthat is employed to conduct the observations. For the present purposes, the systemwill be assumed to be macroscopic, and to comprise both the apparatus employedand any additional microscopic degrees of freedom that may be under investigation.As Fuchs has emphasized [36, 37], some intervention in the time-evolution of the ystem is always required to extract information; hence the channel is two-way asdepicted in Fig. 1b. The fact that the channel delivers classical information - realvalues of pointer variables computed by the component operators of POVMs - imposeson the observer an implicit requirement of classical states into which these classicalvalues may be recorded. Viewing observation as POVM-mediated information transferthus requires observers also to be effectively macroscopic. Consistent with the abovecharacterization of both system and observer as embedded in a “much larger” physicalenvironment, the number of states available to either system or observer will beassumed to be much smaller than the number of states within H U .Considering the channel through which information flows to be a physical andhence quantum system forcefully raises the question of how the observer identifiesas “ S ” the source of the signals that are received. This is the question that wasaddressed by Moore [45] in the general case of interacting automata. Moore’s answer,that no finite sequence of observations is sufficient to uniquely identify even a classicalfinite-state machine, calls into question the standard assumption that the observedsystem can be identified, either by the observer or by a third party, as a collectionof physical degrees of freedom represented by a specified set {| s i i} of basis vectors. Stipulating that the system can be so represented does not resolve the issue; it merelyreformulates the question from one of identifying the system being observed to oneof identifying and employing a POVM that acts on the stipulated system and not onsomething else. This latter question is eminently practical: it must be addressed inthe design of every apparatus and every experimental arrangement.By allowing both the degrees of freedom composing the system of interest and theoperators composing the POVM employed to perform observations to be arbitrar-ily stipulated, the standard quantum-mechanical formalism systematically obscuresthe question of system identification by observers. While it facilitates computations,placing the “Heisenberg cut” delimiting the domain that is to be treated by quantum-mechanical methods around a microscopic collection degrees of freedom further ob-scures the issue, as it introduces an intermediary - the apparatus - that must also beidentified. It has been shown, moreover, that decoherence considerations alone cannotresolve the question of system identification, as decoherence calculations require theassumption of a boundary that must itself be identified: a boundary in Hilbert spacethat specifies a collection of degrees of freedom, or a boundary in the space of allpossible frameworks or Everett branches that distinguishes the framework or branchunder consideration from all others [61, 62]. Absent a metaphysical assumption notjust of Zurek’s axiom(o), but of the specific a priori existence of all and only thesystems that observers actually observe, the only available sources of such boundaryspecifications are observers themselves. The next section examines the question ofwhat such specifications look like in practice.
A primary distinction between quantum mechanics and classical mechanics is thefailure, in the former but not the latter, of commutativity between physical observ- bles. Implicit in this statement is the phrase, “for any given system.” For example,[ˆ x, ˆ p ] = (ˆ x ˆ p − ˆ p ˆ x ) = 0 says that the position and momentum observables ˆ x and ˆ p do not commute for states of any particular, identified system S . An observationthat ˆ x and ˆ p do not commute for states of two spatially separated and apparentlydistinct systems S and S is prima facie evidence that S and S are not distinctsystems after all. If S and S are truly distinct, commutativity is not a problem:[ˆ x , ˆ p ] = [ˆ x , ˆ p ] = 0 for all states | S i and | S i operationally defines separability of S from S , and warrants the formal representation | S ⊗ S i = | S i ⊗ | S i of thestate of the combined system as separable. Hence quantum mechanics can only bedistinguished from classical mechanics by observers that know when they are observ-ing the same system S twice, as opposed to observing distinct systems S and S ,when they test operators for commutativity.The assumption that a single system S is being observed is indicated in the stan-dard quantum-mechanical formalism by simply writing down “ S ” and saying: “Let S be a physical system ...” In foundational discussions, however, such a facile andimplicit indication of sameness can introduce deep circularity. Ollivier, Poulin andZurek, for example, define “objectivity” as follows:“A property of a physical system is objective when it is:1. simultaneously accessible to many observers,2. who are able to find out what it is without prior knowledge aboutthe system of interest, and3. who can arrive at a consensus about it without prior agreement.”(p. 1 of [28]; p. 3 of [29])On the very reasonable assumption that knowing how to identify the system of inter-est counts as having knowledge about it - exactly what kind of knowledge is discussedin detail below - this definition is clearly circular: each observer must have “priorknowledge” to even begin her observations, and the observers must have a “prioragreement” that they are observing the same thing to arrive at a consensus aboutits properties [61, 62]. Hence while the assumption that observers can know thatthey are observing one single system over time is natural and even essential to ex-perimentation and practical calculations, both its role as a foundational assumptionand its relationship to other assumptions that are explicitly written down as axiomsof quantum theory bear examination.Let us fully specify, therefore, the information that an observer O must havein order to confirm that [ A , A ] = 0 for two observables A and A and somephysical system S . The situation can be represented as in Fig. 2: O is faced witha macroscopic system S , and at any given time t can measure a value for either A or A but not both. For example, S could be a Stern-Gerlach apparatus, includingion source, vacuum pump, magnet and power supply, and particle detectors. In thiscase, A and A are the spin directions ˆ s x and ˆ s z , the meters are event counters, andthe selector switch sets the position of a mask at either of two fixed angles. Let usexplicitly assume that O is herself a finite physical system, that O can make any finitenumber of measurements in any order, and that O has been tasked with recordingthe values for A or A along with the time t k of each observation. Let us, moreover, xplicitly assume that information is physical: that obtaining it requires finite timeand recording it requires finite physical memory. For simplicity, assume also thatthe information channel C from S to O has sufficient capacity to be regarded aseffectively infinite; as this channel is implemented by the environment surroundingthe experimental set-up, this assumption is realistic. b ✻ A A A b (cid:0)(cid:0)✒ A Fig. 2: A macroscopic system S with the observable A selected for measurement. Common sense as well as Moore’s theorem entail that in order to carry out obser-vations of S , O must encode information sufficient to (1) distinguish signals from S from other signals that may flow from the channel; (2) distinguish signals from S thatencode information about the positions of the A − A selector switch and the point-ers P and P from signals from S that do not encode this kind of information; and(3) distinguish between signals that encode different positions of the selector switchand different pointer values for P and P . For example, if S is a Stern-Gerlachapparatus, O must encode information sufficient to distinguish S from other sys-tems of similar size, shape and composition, such as leak detectors or general-purposemass spectrometers. Once O has identified S , she must be capable of identifying themask selector and the event counters, and determining both the position of the maskand the numbers displayed on the counters. As O is finite, all of the informationthat O can obtain about S , the selector switch, the pointers, and the values thatthe pointers indicate can be considered, without loss of generality, to be encoded byfinite-precision representations of real numbers. Assuming that one can talk about awell-defined physical state | C i of the channel C , the information that O must encodein order to identify and characterize S and its components can, therefore, be takento be encoded by four operators that assign (indicated by “ ”) fine-precision realnumbers to states | C i of C : S O ( | C i ) (cid:26) ( s , ..., s k ) if | C i encodes | S i NULL otherwisewhere the s , ..., s k are finite real values of a set of control variables of S ; P O ( | C i ) (cid:26) ( p , p ) if | C i encodes | S i NULL otherwisewhere ( p , p ) = (1 ,
0) if the selector switch points to “ A ” and ( p , p ) = (0 ,
1) if the elector switch points to “ A ”; A O ( | C i ) ( a ...a n ) if | C i encodes | P i AND p = 1NULL otherwisewhere a ...a n are finite real values, and; A O ( | C i ) ( a ...a m ) if | C i encodes | P i AND p = 1NULL otherwisewhere a ...a m are finite real values. In these expressions, “NULL” indicates that therelevant operator returns no value under the indicated conditions. The allowed valuesof a k and a k are the O -distinguishable “pointer values” for A and A respectively;they are guaranteed to be both individually finite and finite in number, irrespective ofthe size of the physical state space of S , by the requirement that a finite observer O records them with finite precision in a finite memory. Figure 3 illustrates the actionof these operators on | C i , assuming that S is in the state shown in Fig. 2.(a) b A A A b A (b) (c) ✻ (d) (cid:0)(cid:0)✒ Fig. 3: State information assigned by the operators (a) S O , (b) P O , (c) A O , and (d) A O on | C i .The operator S O assigns state information about all components of S other than the selectorswitch and pointers. The operator P O assigns state information about the selector switch only.The operators A O and A O , respectively, assign state information about the positions of the left-and right-hand pointers only. As illustrated in Fig. 3, the values of the control variables s , ..., s k are whatindicate to O that she is in fact observing S and not something else. In the case of theStern-Gerlach apparatus, these may include details of its size, shape and components,as well as conventional symbols such as brand names or read-out labels. In orderfor O to recognize these values, they clearly must be real and finite. The controlvariables must, moreover, take on “acceptable” values at t indicating to O that S isin a state suitable for making observations. A Stern-Gerlach apparatus, for example,must have an acceptable value for the chamber vacuum and the magnets and particledetectors must be turned on. The entire apparatus must not be disassembled, underrepair, or on fire. The existence, recognition by the observer, and acceptable values of uch control variables are being assumed whenever “ S ” is written down as the nameof a quantum system that is being observed. It is commonplace in the literature(e.g. [63] where this is explicit) to treat quantum systems as represented during themeasurement process by their pointer states alone, but as Figs. 3c and 3d illustrate,such a “bare pointer” provides no information by which the system for which itindicates a pointer value can be identified, much less be determined to be in anacceptable state for making observations.The operators S O , P O , A O and A O defined above assign finite values, i.e. donot assign “NULL,” only for subsets of the complete set of states of C . As discussedabove, the information channel C is physically implemented by the environment inwhich S and O are embedded. Let H C be the Hilbert space of this environment.As the environment of any experiment is contiguous with the universe as a whole,with increasing elapsed time the dimension dim ( H C ) ∼ dim ( H U ); H C can thereforebe considered to be much larger than the state spaces of either S or O , and in par-ticular much larger than the memory available to O . Let S O NULL , P O NULL , A O NULL and A O NULL , respectively, be operators defined on H C that assign a value of zeroto all states within H C that do not encode information about the states of S , theselector switch of S , P and P respectively, and “NULL” for states within H C thatdo encode such information. A POVM {S O k } acting on H C can then be defined asfollows: let S O = S O NULL , and for k = 0 let S O k be the component of S O that assignsthe value s k , normalized so that S O + P k = S O k = Id where Id is the identity op-erator for H C . The component S O of {S O k } is by definition orthogonal to the S O k with k = 0; however, these latter components are not, in general, required to beorthogonal to each other. The component of {S O k } that assigns the value “ready”to S , for example, will not in general be orthogonal to components that establishthe identity of S ; many parts of S must be examined to determine that it is readyfor use. Practical experimental apparatus are, nonetheless, generally designed toassure that many non-NULL components of {S O k } are orthogonal and hence distin-guishable and informationally independent. The vacuum gauge on a Stern-Gerlachapparatus, for example, is designed to be distinguishable from and independent ofthe ammeter on the magnet power supply or the readout on the event counter. Ingeneral, the distinguishability and informational independence of components is anoperational definition of their separability and hence of the appearance of classical-ity. The practical requirement that observer-identifiable systems have distinguishableand informationally-independent control and pointer variables is analogous to Bohr’srequirement [64] that measurement apparatus be regarded as classical.Additional POVMs {P O k } , {A O k } and {A O k } can be defined by including P O NULL , A O NULL and A O NULL as 0 th components. As in the case of {S O k } , these 0 th componentsare by definition orthogonal to the others. If S is assumed to be designed so as toallow only a single kind of measurement to be performed at any given time, and ifall observations are assumed to be carried out at maximum resolution, then the non-NULL components of {P O k } , {A O k } and {A O k } can also be taken to be orthogonal.For simplicity, orthogonality of these components will be assumed in what follows;the general case can be accomodated by assuming that the components of {P O k } thatindicate incompatible measurements are orthogonal, that components of {A O k } and {A O k } that assign values at maximum resolution are orthogonal, and by considering nly these orthogonal components when defining inverse images as described below.Regarding S O , P O , A O and A O respectively as POVMs {S O k } , {P O k } , {A O k } and {A O k } acting on H C is useful because it removes any dependence on an explicit spec-ification of the boundaries between C and S or between the selector switch, P , P and the non-switch and non-pointer components of S . These boundaries are replaced,from O ’s perspective, by the boundaries of the O -detectable encodings of S and itscomponents in H C . Let ǫ be O ’s detection threshold for encodings in C ; O is able torecord a value s k , for example, only if h C |S O k | C i ≥ ǫ . Because O is a finite observer, ǫ >
0; arbitrarily weak encodings are not detectable. Given this threshold, the en-coding of S can be defined, from O ’s perspective, as ∪ k ( Im − ( s k )), where Im − ( s k )is the inverse image in H C of the detectable value s k . Because S O is orthogonal toall of the S O k with k = 0, the intersection Im − ( S O ) ∩ ( ∪ k ( Im − ( s k ))) = ∅ ; indeedthese inverse images are separated by states for which 0 ≤ h C |S O k | C i ≤ ǫ for all S O k with k = 0. Let “ Im − {S O k } ” denote ∪ k ( Im − ( s k )); Im − {S O k } is then the propersubspace of H C containing vectors to which the POVM {S O k } assigns finite real valueswith probabilities greater than ǫ . The proper subspaces Im − {P O k } , Im − {A O k } and Im − {A O k } can be defined in an analogous fashion. As any state | C i that encodes anacceptable value of either the pointer position or the pointer value for either A O or A O also encodes acceptable values of the s k , it is clear that Im − {P O k } , Im − {A O k } and Im − {A O k } are properly contained within Im − {S O k } .Specifying {S O k } , {P O k } , {A O k } and {A O k } in terms of the values that they assignfor each state | C i of C completely specifies O ’s observational capabilities regarding S ;no further specification of S or its states is necessary. The notion that “systems exist”can, therefore, be dropped; all that is necessary for the description of measurement,other than observers equipped with POVMs, is that channels exist. By regardingall POVMs that identify systems or their components as observer-specific (hencedropping the superscript “ O ”), the minimal capabilities required by any observer canbe defined in purely information-theoretic terms. Given an information channel C , a minimal observer on C is a finite system O that encodes collections of POVMs {S ik } , {P ik } and {A ijk } within a control structure such that, for each i :1. The inverse images Im − {S ik } , Im − {P ik } and Im − {A ijk } for all j are non-empty proper subspaces of H C such that Im − {S ik } properly contains Im − {P ik } and the Im − {A ijk } for all j .2. The s i , ..., s i n i are accepted by the control structure of O as triggering the actionof the POVM {A ijk } for which p i j = 1.3. The control structure of O is such that the action on | C i with A ijk is followedby recording of the single non-zero value a ijk to memory.The control structure required by this definition consists of one “if - then - else” blockfor each POVM component, organized as shown in Fig. 4 for a minimal observerwith N POVMs {S ik } . Together with the specified POVMs and a memory alloca-tion process, this control structure specifies a classical virtual machine (e.g. [51]),i.e. a consistent semantic interpretation of some subset of the possible behaviors of acomputing device. Such a virtual machine may be implemented as software on anyTuring-equivalent functional architecture, and hence may be physically implemented y any quantum system that provides a Turing-equivalent functional architecture,such as a QTM [18] or any of the alternative quantum computing architectures prov-ably equivalent to a QTM [59, 65, 66, 67]. Constructing such an implementation usinga programming language provided by a quantum computing architecture is equivalentto constructing a semantic interpretation of the behavior of the quantum computingarchitecture that defines the virtual machine using the pre-defined semantics of theprogramming language. As in the classical case, programming languages for quan-tum computing architectures provide the required semantic mappings from formalcomputational constructs (e.g. logical operations or arithmetic) to the operationsof the underlying architecture (e.g. unitary dynamics for a QTM or a Hamiltonianoracle) [68, 69]; for any universal programming language, however, higher-level inter-pretations that define specific programs are independent of these lower-level semanticmappings. Hence from an ontological perspective, a minimal observer is a classicalvirtual machine that is physically implemented by a quantum system O that, if notuniversal, nonetheless provides a sufficient quantum computing architecture to realizeall the functions of the minimal observer. A physically-implemented minimal observerinteracts with and obtains physically-encoded information from a physically imple-mented information channel C . Laboratory data acquisition systems that incorporatesignal-source identification criteria and stably record measurement results are mini-mal observers under this definition. As is the case for all physical implementationsof classical virtual machines, and for all operations involving classically-characterizedinputs to or outputs from quantum computers, the semantic interpretation of a phys-ical (i.e. quantum) system as an implementation of a minimal observer requires,at least implicitly via the semantics of the relevant programming language, an in-terpretative approach to the quantum measurement problem. The consequences ofreplacing Galilean observers with minimal observers as defined here for interpretativeapproaches to the measurement problem are discussed in Sect. 6 below. Accept s , ..., s n ? b b b No ❄ p = 1? No bbb ❄ p m = 1? ✲ No b ❄ Accept s N , ..., s Nn N ? No ✲ b ❄ p N = 1? No bbb ❄ p Nm N = 1? ✲ No b ❄ Record a k = 0 ❄ ❄ Record a m k = 0 ❄ ❄ Record a N k = 0 ❄ ❄ Record a Nm N k = 0 ❄ Allocate new memory block
Fig. 4: Organization of “if - then - else” blocks in the control structure of a minimal observer.
A minimal observer as defined above, and as illustrated in Fig. 4, is clearly notGalilean; it is rather a richly-structured information-encoding entity. The informa-tion encoded by a minimal observer is relative to a specified control structure, and istherefore pragmatic , i.e. used for doing something [47, 48]. Hence a minimal observeris not just a “physical system having a definite state of motion” or “a quantum systeminteracting with the observed system.” Indeed, if considered apart from its physicalimplementation, a minimal observer as defined above is not a quantum system at all;it is a classical virtual machine, an entity defined purely informationally. One cannot,therefore, talk about the “quantum state” of a minimal observer. The traditionalvon Neumann chain representation ([60], reviewed e.g. by [3]), in which the observerbecomes entangled with the system of interest, after which the observer’s quantumstate must “collapse” to a definite outcome, cannot be defined for a minimal observer, nd the information encoded by a minimal observer cannot be characterized by a vonNeumann entropy. The physical implementation of a minimal observer can be char-acterized by a quantum state, and hence does have a von Neumann entropy; however, any physical implementation that provides a Turing-equivalent architecture and suf-ficient coding capacity will do. The history of compilers, interpreters, programminglanguages, and distributed architecures demonstrates that the emulation mappingfrom a virtual machine to its physical implementation can be arbitrarily complex,indirect, and de-localized in space and time; any straightforward interpretation ofvon Neumann’s principle of “psychophysical parallelism” as a constraint on the im-plementation of minimal observers is, therefore, undone by the architecture that vonNeumann himself helped devise two decades after the publication of MathematischeGrundlagen der Quantenmechanische .In consequence of their finite supplies of executable POVMs and finite memories,minimal observers display objective ignorance of two distinct kinds. First, a minimalobserver cannot, by any finite sequence of observations, fully specify the set of statesof C that encode states of any system S , regardless of the size of the state space of S . This form of objective ignorance follows solely from the large size of H C com-pared to memory available to O . A minimal observer cannot, therefore, determinewith certainty that any specification of the states of S derived from observations iscomplete. If the observational data characterizing S obtained by O are viewed asoutputs from an oracle, this failure of completeness can be viewed as an instance ofthe Halting Problem [51, 52]: O cannot, in principle, determine whether any oraclethat produces a specification of the states of S will halt in finite time. This firstform of objective ignorance blocks for minimal observers the standard assumption ofparticle physics that the states of elementary particles are specified completely bytheir observable quantum numbers, downgrading this to a “for all practical purposes”specification; it then extends this restriction to all systems, elementary or not. Thesecond form of objective ignorance is that required by Moore’s theorem: any system S ′ that interacts with C in a way that is indistinguishable using {S O k } , {P O k } , {A O k } and {A O k } from S will be identified by O as S . The information provided to O by {S O k } , {P O k } , {A O k } and {A O k } is, therefore, objectively ambiguous concerning thephysical degrees of freedom that generate the encodings in C on which these operatorsact. This second form of objective ignorance extends to all systems the indistinguisha-bility within types familiar from particle physics. Neither of these forms of objectiveignorance can be remedied by further data acquisition by O ; they thus differ funda-mentally from subjective or classical ignorance. As will be shown in the two sectionsthat follow, these two forms of objective ignorance together assure that the observa-tional results recorded by a minimal observer will display the typical characteristicspredicted by quantum theory, independently of any specific assumptions about theobserver’s physical implementation. Physical Interpretation of Non-commutativePOVMs
The definition of a minimal observer given above relies only on the classical concept ofinformation and the system-identification requirements placed on observers by clas-sical automata theory, the assumption that the channel C is physically implementedby the environment, the idea that information is physical, and the formal notion of aPOVM. It provides, however, a robust formal framework with which arbitrary mea-surement interactions can be characterized. This formal framework makes no mentionof “systems” other than O and C , requires no strict specification of the boundary be-tween the physical degrees of freedom that implement O and those that implement C ,and makes no assumption that O and C are separable. The physical interpretation ofinformation transfer by POVMs within this framework thus provides a “systems-free”interpretation of quantum mechanics with no a priori assumptions about the natureof quantum states. This interpretation does not violate the axiomatic assumptions ofminimal quantum mechanics in any way; hence it requires no changes in the standardquantum-mechanical formalism or its application in practice to specific cases.Let us drop temporarily the assumption of minimal quantum mechanics adoptedin Sect. 2, and assume only that the physical degrees of freedom composing thecoupled system O ⊗ C , where “ O ” here refers to the physical implementation of aminimal observer, evolve under some dynamics H that is time-symmetric and fullydeterministic. A natural, classical “arrow of time” is imposed on this dynamics,from the perspective of O , by the sequence of memory allocations executed by O ’sfunctional architecture. From a perspective exterior to O (e.g. the perspective of C ),the minimal observer O is only one of an arbitrarily large number of virtual machinesthat could describe the physical dynamics of its hardware implementation; hencethis O -specific arrow of time is unavailable from such an exterior perspective. Anyalternative minimal observer O ′ will, however, have its own arrow of time determinedby its own memory-allocation process.The large size of H C renders the physical degrees of freedom implementing C fine-grained compared to both the detection resolution ǫ and the memory capacity of anyminimal observer O ; in particular, these degrees of freedom are fine-grained comparedto the inverse images of the POVMs {S ik } , {P ik } and {A ijk } with which O obtainsinformation about an external system S . As illustrated in Fig. 1a, O is implementedby the same kinds of physical degrees of freedom that implement C ; the degrees offreedom implementing O are, therefore, also fine-grained compared to O ’s memory.Let us assume a weak version of counterfactual definiteness: that the fine-graineddegrees of freedom within the inverse images of the {S ik } , {P ik } and {A ijk } implementedby any O are well-defined at all times; this assumption is a natural correlate, if not aconsequence, of the realist stance toward physical degrees of freedom adopted in Sect.2. Note that this assumption of counterfactual definiteness does not apply to the statesof any “system” other than C , and that it applies to states of C without assumingthat C is separable from O . This assumption renders any physical interpretationbased on it a “hidden variables” theory. However, it does not violate the Kochen-Specker contextuality theorem [70]; indeed it provides a mechanism for satisfying t. The “hidden” fine-grained state variables of C are inaccessible in principle to O ,although they fully determine the course-grained measurement results that O obtains.As discussed above, no two instances of the execution of a {S ik } , {P ik } , {A ijk } tripleat times t and t ′ can be assumed by O to act on the same fine-grained state | C i ,nor is any measure of similarity or dissimilarity of channel states | C i and | C i ′ otherthan a {S ik } , {P ik } , {A ijk } triple available to O . All executions by O of a singlemeasurement {A ijk } are thus contextualized by prior executions of {S ik } and {P ik } ;executions of pairs {A ijk } and {A ilm } , commutative or otherwise, are contextualizedby two executions of {S ik } and {P ik } .Finally, let us assume that the dynamic evolution of C does not depend in anyway on the POVMs or the control structure implemented by O . Given that O isby definition a virtual machine, this is an assumption that the physical dynamics H is independent of its semantic interpretation by any observer. This assumption of decompositional equivalence assures that the allocation by H of fine-grained degreesof freedom to the inverse images of the {S ik } , {P ik } and {A ijk } are independent ofthe information O encodes, and hence of O ’s “expectations” about C or H . Thisassumption renders the interpretative framework free of “subjective” dependence onthe observer. By ruling out any dependence of H on system - environment boundariesdrawn by observers, it also renders the interpretative framework consistent with thecommon scientific practice of stipulating systems of interest ad hoc either demon-stratively by pointing and saying “that” or formally by specifying lists of degrees offreedom to be included within the boundaries of the stipulated systems.With these assumptions, the interpretation of O ⊗ C is both realist and objec-tivist about the fine-grained degrees of freedom implementing O ⊗ C , and free, viadecompositional equivalence, of any dependence on what observables and hence whatdescriptions of C or H are available to O . The physical interpretation of [ A ijk , A ilm ] = 0for POVM components A ijk and A ilm must, therefore, also be realist, objectivist, andindependent of the descriptions available to O . Suppose that at t , C is in a fine-grained state | C i such the action A ijk | C i would cause O to record a value a ijk and theaction A ilm | C i would cause O to record a value a ilm ; | C i at t is thus in the intersection Im − ( a ijk ) ∩ Im − ( a ilm ) of the inverse images of a ijk and a ilm . In this case, the failureof commutativity can be expressed intuitively (e.g. [3] Ch. 2) in terms of the physicaldynamics H by a pair of counterfactual conditionals:If | C i ∈ Im − ( a ijk ) ∩ Im − ( a ilm ) at t and O does nothing at t , then ata subsequent t + ∆ t , H| C i ∈ Im − ( a ijk ) ∩ Im − ( a ilm ); however, if | C i ∈ Im − ( a ijk ) ∩ Im − ( a ilm ) at t and O measures either A ijk or A ilm at t , thenat a subsequent t + ∆ t , H| C i / ∈ Im − ( a ijk ) ∩ Im − ( a ilm ).Figure 5 illustrates this situation, the familiar “dependence of the physical dynamicson the act of observation” mentioned in the Introduction. a) (b) tIm − ( a ijk ) | C i Im − ( a ilm ) ✲ t + ∆ tIm − ( a ijk ) H| C i Im − ( a ilm ) tIm − ( a ijk ) | C i Im − ( a ilm ) ✁✁✁✁✁✁✕✟✟✟✟✯❍❍❍❍❥❆❆❆❆❆❆❯ t + ∆ tIm − ( a ijk ) H| C i ? H| C i ? H| C i ? H| C i ? Im − ( a ilm ) Fig. 5: Dynamic evolution of | C i without (a) and with (b) O ’s measurement of A ijk or A ilm at t .Part (b) shows the four possible post-measurement locations of H| C i if [ A ijk , A ilm ] = Implicit in this intuitive formulation of non-commutativity as a counterfactualconditional, and in Fig. 5a, is the idea that the observer could “do nothing” at t ,thus avoiding the “perturbation” of | C i with either A ijk or A ilm . The definition ofa minimal observer, however, permits O to “do nothing” only if the control values s i , ..., s i n i are not accepted, i.e. only if (to use the usual language of external systemsmomentarily) the “system” S i is not identified as “ready” by the POVM {S ik } . If S i is identified by the action of {S ik } as ready, O deterministically makes an observationand records a value. The dynamics depicted in Fig. 5a is thus inconsistent withthe condition that | C i ∈ Im − {S ik } at t . Consistency with | C i ∈ Im − {S ik } at t requires that if | C i ∈ Im − ( a ijk ) ∩ Im − ( a ilm ) at t , | C i / ∈ Im − ( a ijk ) ∩ Im − ( a ilm ) atan immediately-previous t − ∆ t . This consistent situation is illustrated in Fig. 6, inwhich the uncertainties about the state of C before and after t are symmetric. Im − ( a ijk ) H − | C i ? H − | C i ? H − | C i ? H − | C i ? Im − ( a ilm ) t − ∆ t ❆❆❆❆❆❆❑ ❍❍❍❍❨ ✟✟✟✟✙ ✁✁✁✁✁✁☛ tIm − ( a ijk ) | C i Im − ( a ilm ) ✁✁✁✁✁✁✕✟✟✟✟✯❍❍❍❍❥❆❆❆❆❆❆❯ t + ∆ tIm − ( a ijk ) H| C i ? H| C i ? H| C i ? H| C i ? Im − ( a ilm ) Fig. 6: Dynamic evolution of | C i that is consistent at all times with | C i ∈ Im − {S ik } at t . ealism and objectivism demand that the forward and reverse dynamics of H depicted in Fig. 6 receive the same physical interpretation. Viewing the dynamicssymmetrically and considering O ’s control structure as shown in Fig. 4 makes thecausal structure of the sequence from t − ∆ t to t clear: if the physical evolution of O ⊗ C under the action of H results in | C i ∈ Im − ( a ijk ) ∩ Im − ( a ilm ) at t , either A ijk or A ilm will be executed by O at t , with precedence determined by O ’s control struc-ture. The control structure of O , however, is a virtual machine implemented by thecollection of physical degrees of freedom O , the time evolution of which are driven by H . Every action of O , therefore, is fully determined by H via the emulation mappingthat defines O as a physically-implemented virtual machine. Far from “dependenceof the physical dynamics on the act of observation,” the transition from t − ∆ t to t illustrates the deterministic dependence of the act of observation on the physicaldynamics. If the dynamics determines the observation from t − ∆ t to t , however, itmust determine the observation from t to t + ∆ t as well. There is nothing particu-lar to quantum mechanics in this claim: once information is viewed as physical, theconclusion that an interaction that transfers information from C to O also transfersinformation from O back to C follows straightforwardly from Newton’s Third Law.Given this physical interpretation of non-commutativity as a consequence of thereaction of O on C that is required by a time-symmetric, deterministic H , O willobserve non-commutativity between any pair of POVMs {A ijk } and {A ilm } with j = l for which the action of H on Im − ( a ijk ) alters the subsequent distribution of degreesof freedom into Im − ( a ilm ) for some m or vice versa. Commutativity of {A ijk } and {A ilm } thus requires that Im − ( a ijk ) and Im − ( a ilm ) are separable under the dynamics H for all k and m . Operators that jointly measure the action of H , in particular, willnever satisfy this condition; hence such operators cannot commute. It is impossible,moreover, for any minimal observer to predict the effect of H on a given Im − ( a ijk )and alter the choice of subsequent measurement to avoid the appearance of non-commutativity, as doing so would require an ability to represent the state of O ⊗ C ,a state about which minimal observers are objectively ignorant.The present framework offers, therefore, a straightforward answer to van Fraassen’s[71] question “How could the world possibly be the way a quantum theory says it is?”The world is a physically-implemented information channel, it evolves through theaction of a time-symmetric, deterministic dynamics that satisfies decompositionalequivalence and counterfactual definiteness, and it contains minimal observers imple-menting pairs of POVMs with non-separable inverse images, in particular pairs ofPOVMs that jointly measure action. Within the present framework, the more inter-esting question is the reverse of van Fraassen’s: what would the world have to be likefor classical mechanics to be true, i.e. for dynamics to be time-symmetric, determin-istic, satisfy decompositional equivalence and counterfactual definiteness, and for allpossible physical observables to commute? There are two answers. First, the worldwould be classical if information transfer required zero time. If information couldbe transferred instantaneously, multiple POVMs could act on a single channel state | C i without intervening reactions of O on C . Second, the world would be classicalif observers had effectively infinite coding capacity. With infinite coding capacity,observers could in principle realize the Laplacian dream of completely modeling H , nd hence designing time-dependent POVMs with inverse images that accurately pre-dicted the trajectory from any | C i to the unique subsequent H| C i . These conditionscould both be true if information was not physical. Hence the operator commuta-tivity required by classical mechanics could be true if information were not physical,and can be derived given a fundamental assumption that information is not physical,that information processing in principle costs nothing, is free (c.f. [49] where freeinformation is identified with classicality). What the empirical success of quantummechanics tells us is that information is physical: that information processing is not free. The previous section showed that, given reasonable, traditional, and not explicitlyquantum-mechanical assumptions about the dynamics driving the evolution of a phys-ical information channel, any physically-implemented minimal observer equipped withsufficiently high-resolution POVMs will discover one of the primary features of thequantum world: pairs of POVMs with mutually non-separable inverse images, includ-ing pairs of POVMs that jointly measure action, will not commute. This section willshow that minimal observers equipped with sufficiently high-resolution POVMs willalso discover several other canonically “quantum” phenomena. Before proceeding,however, it is useful to summarize, in Table 1, the meanings given to the fundamentalterms of the standard quantum-mechanical formalism by the formal framework fordescribing the O − C interaction developed in the last two sections. Standard quantum formalism Current framework
Quantum system S , a collectionof degrees of freedom Im − ( {S ik } ), the (non-NULL)inverse image in C of a POVMQuantum state | S i at t Im − ( a jk ) in C at t for value a jk of a POVM component A ijk Observable A , defined overstates of any quantum system {A } ... {A Njk } , a set of POVMsdefined over states of C Table 1: Meanings assigned to terms in the standard quantum formalism by the current framework.
As shown in Table 1, the fundamental difference between the current frameworkand the standard quantum formalism is the meaning assigned to the notion of a quan-tum system. In the standard quantum formalism, a quantum system is a collectionof physical degrees of freedom, and any quantum system is observable in principle. n the current framework, an observable quantum system is the non-NULL inverseimage, in a physical channel C , of a physically-implemented POVM with a finite num-ber of finite, real output values. The current framework thus limits quantum theoryby placing an observer-relative, information-theoretic restriction on what “counts” asan observable quantum system: the POVM {S ik } must be physically implemented byan observer O in order for the “quantum system” it detects to exist for O . Thus inthe current framework, to paraphrase Fuchs’ [37] paraphrase of de Finetti, “quantumsystems do not exist” as objective, “given” entities. This does not, clearly, mean thatthe stuff composing quantum systems does not exist; both C and O are implementedby physical degrees of freedom. What it means is that their boundaries do not exist.Systems are defined only by observer-imposed decompositions, and physical dynamicsdo not respect decompositional boundaries.Quantum states are, in the current framework, equivalence classes under the com-ponents of a POVM {A ijk } of states of C that are indistinguishable, in principle, by anobserver implementing {A ijk } . As discussed in Sect. 3, other than whether | C i is iden-tified by an available POVM {S ik } and the values a jk assigned by the {P ik } -selected j th available observable {A ijk } that are obtained in the course of a finite sequenceof measurements, observers in the current framework are objectively ignorant aboutquantum states. No physical state | C i of the channel, and therefore no physical stateof any “system” S , can be either fully characterized or demonstrated to be replicatedby any minimal observer, regardless of the amount of data that observer collects. Aworld in which no observer is able, in principle, to identify any quantum state as areplicate of any other quantum state is, however, equivalent from the perspective ofsuch an observer to a world in which quantum states cannot be replicated. The ob-servational consequences of objective ignorance regarding the replication of quantumstates are, therefore, equivalent to the observational consequences of the no-cloningtheorem [72], which forbids the replication of unknown quantum states. These conse-quences are realized objectively in the current framework for all quantum states, sinceall are “unknown” to all observers. In the current framework, the effective inabilityto clone quantum states is a consequence of the physicality of information and theboundarylessness of quantum systems defined as inverse images of POVMs.In the current framework, no-cloning renders all observational results observer-specific. Any two observers O and O ′ are objectively ignorant about whether theinverse images of any two POVMs {A O ijk } and {A O ′ ilm } are the same subsets of C ,whether these POVMs commute or not. Whether two observers share observablescan, therefore, at best be established “for all practical purposes” by comparing theresults of multiple observations. Hence it cannot be assumed, without qualifications,that two distinct observers have both measured a single observable such as ˆ x for asingle system S . This reflects laboratory reality: whether an observation has beensuccessfully replicated in all details is always subject to question.With these understandings of the familiar terms, the physical meaning of Bell’stheorem [73] for a minimal observer becomes clear. Consider an observer who mea-sures the same observable on two different “systems” S and S employing triples( {S k } , {P k } , {A jk } ) and ( {S k } , {P k } , {A jk } ) of POVMs at times t and t + ∆ t respec-tively. Between t and t + ∆ t , the state of C evolves from | C i to H| C i . Clearly | C i ∈ Im − ( {S k } ) at t and H| C i ∈ Im − ( {S k } ) at t + ∆ t ; otherwise the measure- ents could not be performed. What is relevant to Bell’s theorem is whether theseinverse images overlap, and in particular, whether Im − ( {A jk } ) evaluated at t inter-sects Im − ( {A lm ) evaluated at t + ∆ t for any j and l . If this intersection is empty,the measured “states” | S i and | S i are separable. However, the intersection of theinverse image Im − ( {A jk } ) at t and the inverse image Im − ( {A jk } ) at t + ∆ t isonly guaranteed to be empty if H respects the S - S boundary, and assuming that H respects the S - S boundary violates decompositional equivalence. Therefore,the default assumption must be that Im − ( {A jk } ) at t may overlap Im − ( {A jk } ) at t + ∆ t , and hence that | S i and | S i cannot be regarded as separable. That sep-arability between apparently-distinct systems cannot be assumed by default is theoperational content of Bell’s theorem, accepting the horn of the dilemma on whichcounterfactual definiteness and hence the ability to talk about the inverse images ofPOVMs is assumed.The problem with the classical reasoning that produces Bell’s inequality, on thecurrent framework, is that it assumes that observers can have perfect informationabout distant systems. If O is making a local measurement of S at t , and S has aspacelike separation from S at t , then O cannot be making a local measurement of S at t . If at some later time t + ∆ t O writes down a joint probability distributionfor particular states | S i and | S i at t , O must be in possession at t + ∆ t of dataobtained about | S i at t , such as a report of the state of S at t from some otherobserver, e.g. Alice, that is was local to S at t . The delivery of this report from Aliceto O requires a physical channel, with which O must interact, using an appropriatePOVM, in order to extract the information contained in the report. Writing downthe joint probability distribution for | S i and | S i at t therefore requires that O maketwo local measurements, one of | S i at t , and one of the report from Alice at the latertime t + ∆ t . Only if the inverse images of the two POVMs required to make thesetwo measurements are separable is the classical assumption of perfect informationtransfer from Alice to O warranted. In standard quantum-mechanical practice, O ’sinteractions with a macroscopic Alice at t + ∆ t are assumed to separable from Alice’sinteractions with S at t due to decoherence; any entanglement between Alice and S is assumed to be lost to the environment in a way that renders it inaccessible to O .This assumption, however, rests on an implicit assumption that O can distinguishAlice from the background of the environment without making a measurement ofAlice’s state [62], e.g. before asking for her report. If Alice is microscopic - forexample, if Alice is a single photon - this latter assumption is unwarranted, as isthe assumption that Alice is no longer entangled with S at t + ∆ t . A minimalobserver, however, cannot identify any system other than by making a measurementof that system’s state. A minimal observer cannot, therefore, assume that decoherencehas dissipated any previous entanglement into the environment; as will be describedbelow, for a minimal observer decoherence is a property of information channels,not an observer-independent property of system-environment interactions. Hence asdiscussed above, a minimal observer cannot assume that the inverse images of any twoPOVMs are separable; for a minimal observer, the default assumption must be thatany two systems are entangled. A minimal observer cannot, therefore, assume perfectinformation transfer from a distant source of data, and hence cannot derive Bell’sinequality for spacelike separated systems using classical conditional probabilities that ssume perfect information transfer. For a minimal observer, therefore, the failureof Bell’s inequality is expected, and the prediction of its failure by minimal quantummechanics is positive evidence for the theory’s correctness.Viewing both quantum systems and quantum states as inverse images of POVMsalso enables a straightforward physical interpretation of the Born rule. Observers areobjectively ignorant, at all times, of both the state | C i of the information channeland the dynamics H driving its evolution. By assuming decompositional equivalence,however, an observer can be confident that the future evolution of | C i will not de-pend on the locations or boundaries within the state space of C of the inverse imagesof the POVMs {S ik } , {P ik } or {A ijk } . Such an observer can, therefore, be confidentthat the probability of obtaining an outcome a ijk following a successful application of {S ik } , {P ik } and {A ijk } to | C i at some future time t will depend only on the numberof physical states within Im − ( a ijk ) relative to the total number of states within of Im − ( {S ik } ) at t . The Born rule expresses this confidence that H respects decompo-sitional equivalence.Let P ( a ijk | ij, t ) be the probability that O records the value a ijk at some futuretime t given that O has, immediately prior to t , identified a “system” S i by suc-cessful application of {S ik } and selected an observable {A ijk } by successful applicationof {P ik } . Given these conditions, O deterministically records some value a ijk , so P k P ( a ijk | ij, t ) = 1. If the POVM {A ijk } is restricted to only the components with k = 0 and hence considered to act only on the subspace Im − {A ijk } of H C , it can berenormalized so that P k A ijk is the Identity on Im − {A ijk } . Following the notationused by Zurek in his proof of the Born rule from envariance [42], let m k be the num-ber of states in Im − ( a ijk ) and M = P k m k be the number of states in Im − {A ijk } ; Im − {A ijk } then corresponds to the “counter” ancilla C in Zurek’s proof, each ofthe k components of which contains m k fine-grained states. What Zurek shows isthat (in his notation [42] but suppressing phases) if a joint system-environment state | ψ SE i has a Schmidt decomposition P Nk =1 a k | s k i| e k i with a k ∝ √ m k , an ancilla C of M fine-grained states can be chosen with k mutually-orthogonal components C k such that C = ∪ k C k and each C k contains m k fine-grained states. Using the C k tocount the number of fine-grained states available for entanglement with any givenjoint state | s k i| e k i , Zurek then shows that the probability p k of observing | s k i| e k i is m k /M , which equals | a k | by the definition of C , giving the Born rule.In the present context, the formalism of Zurek’s proof provides a constructivedefinition of the unknown future quantum state on which a POVM {A ijk } can act toproduce a k jk as a recorded outcome. The inverse image Im − {A ijk } is the subset of C that “encodes” the ”quantum state” of the “system” S i picked out by the POVM {S ik } ;the rest of C (i.e. C \ Im − {S ik } ) is the “environment” of S i . Hence Zurek’s “ | ψ SE i ” isa coarse-grained representation of | C i , where the coarse-grained basis vectors “ | s k i ”and “ | e k i ” span the subpaces Im − {A ijk } and C \ Im − {S ik } respectively. GivenZurek’s assumption that all system states are measureable, the | s k i can be readilyidentified as the Im − ( a ijk ) for the POVM {A ijk } ; the | e k i are notional, as they arefor Zurek. Hence the physical content of the Born rule is that, given decompositionalequivalence, the inverse images Im − ( a ijk ) can be regarded as coarse-grained basis ectors for Im − {A ijk } that together provide a complete specification of the state of Im − {A ijk } as measurable by O . This is in fact the role of the Born rule in standardquantum theory: it assures that the probabilities P ( a ijk | ij, t ) are exhausted by theamplitudes (squared) of the measureable basis vectors | s k i of the identified system ofinterest.Interpreting the Born rule in this way provides, in turn, a natural physical in-terpretation of decoherence. Observers, as noted in Sect. 3, are virtual machinesimplemented by physical degrees of freedom. Any “system” identified by a POVM {S ik } implemented by an observer is, therefore, itself a virtual entity: “quantum sys-tems do not exist” as objective entities. Decoherence must, therefore, be a virtualprocess acting on the information available to an observer, not a physical processacting on the degrees of freedom that implement C . Representing decoherence in thisway requires re-interpretating it as an intrinsic property of a (quantum) informationchannel. Such a re-interpretation can be motivated by noting that the usual physi-cal interpretation of decoherence relies on the identification of quantum systems overtime and is therefore deeply circular [61, 62].In standard quantum theory, decoherence occurs when a quantum system S is sud-denly exposed to a surrounding environment E . The S − E interaction H S − E rapidlycouples degrees of freedom of S to degrees of freedom of E , creating an entangled jointstate in which degrees of freedom “of S ” can no longer be distinguished from degreesof freedom “of E .” The phase coherence of the previous pure state | S i is dispersedinto the entangled joint system S − E . Under ordinary circumstances decoherenceis very fast; Schlosshauer ([3] Ch. 3) estimates decoherence times for macroscopicobjects exposed to ambient photons and air pressure to be many orders of magnitudeless than the light-transit times for such objects (e.g. 10 − s to spatially decohere a10 − cm dust particle at normal air pressure versus a light-transit time of 10 − s). Itis, therefore, safe to regard all ordinary macroscopic objects exposed to the ordinarymacroscopic environment as fully decohered.It is worth asking, however, what is meant physically by the supposition that S is“suddenly exposed” to E . If S is “suddenly exposed” to E at some time t , it must havebeen isolated from E before t . Call “ F ” whatever imposes the force required to isolate S from E . On pain of infinite regress, F must be in contact with E , in which casedecoherence theory tells us that F and E are almost instantaneously entangled. Theinteraction of F with S that imposes the force that keeps S isolated will, however, alsoentangle S with F . Unless F can be partitioned into separable components F1 and F2 that separately interact with S and E respectively, however, neither | F ⊗ S i nor | F ⊗ E i can be considered to be pure states, and nothing prevents the spread of entanglementfrom S to E . Hence unless F can be partitioned into separable components, S hasnever been isolated, and can never be “suddenly exposed.” In practice, F is often apiece of laboratory apparatus such as an ion trap, that interacts with an “isolated”system on one surface and the environment on another. The assumption that F can be partitioned into separable systems is, effectively, the assumption of an internalboundary within F that is not crossed by any entangling interactions. Such an internalboundary would, however, “isolate” everything inside it, and hence require anotherinternal boundary to enforce this isolation. Such an infinite regress of boundaries isimpossible; hence no such boundary can exist. hat this reasoning applies across the dynamical domains defined by the relationbetween the self and interaction Hamiltonians of S (e.g. [3, 4]) can be seen by con-sidering a high-energy cosmic ray that collides with the Earth. During its transitof interplanetary space and the upper atmosphere, the interaction of the cosmic raywith its immediate environment is small; it can be considered “isolated” as long asno measurements of its state are made. Its sudden collision with dense matter (e.g. ascintillation counter) “exposes” it to the local environment defined by that matter, alocal environment that is contiguous with the larger environment of the universe as awhole. This “sudden exposure” is, however, an artifact of the limited view of the cos-mic ray’s history just described. The cosmic ray was produced by a nuclear reaction,e.g. in the Sun. Prior to that reaction, its future components were fully exposed tothe local environment of the Sun, a local environment that, like the dense matter onEarth, was contiguous with the larger environment of the universe as a whole. Thepre-reaction entanglement between components of the future cosmic ray and othercomponents of the Sun, and hence with other components of the universe as a whole,is not physically destroyed by the formation and flight of the cosmic ray; it is merelyinaccessible to observers on Earth, who are only able to experimentally take note ofthe later, local entanglement between the cosmic ray and the Earth-bound matterwith which it collides. It is widely acknowledged that the notion of an “isolated sys-tem” is a holdover from classical physics; Schlosshauer, for example, notes that “theidealized and ubiquitous notion of isolated systems remained a guiding principle ofphysics and was adopted in quantum mechanics without much further scrutiny” ([3]p. 1). Yet if quantum systems are never isolated, if all physical degrees of freedom areentangled at all times with all other physical degrees of freedom, what is the physicalmeaning of decoherence?Standard quantum theory resolves this paradox formally. The formalism distin-guishes S from E by giving them different names. The representation | S ⊗ E i = P ij λ ij | s i i| e j i of the entangled joint state preserves this distinction, as does the jointdensity ρ = P ij | s i ih s j || e i ih e j | and its partial trace over E , ρ S = N P Nij =1 | s i ih s j |h e i | e j i .These representations all assume, implicitly, that S can be identified against the back-ground of E ; the partial trace additionally assumes, usually explicitly, that O is em-ploying an observable A ⊗ I that measures states of S in some basis but acts as theidentity operator on states of E . It is this latter assumption that is expressed by thestandard proviso that O cannot or does not observe the states of E . Given theseassumptions, however, the claim that decoherence explains O ’s ability to distinguish S from E by providing a physical mechanism for the “emergence of classicality” isclearly circular: the “emergence” is built in from the beginning by assigning the dis-tinct names S and E and assuming that they refer to different things. Indeed, the roleof decoherence in standard quantum theory appears to be that of an axiom, some-what more subtle that von Neumann’s axiom of wave-function collapse, stating thatobservers can distinguish quantum systems from their environments even though thetwo are always and inevitably entangled. The statement “decoherence is a physicalprocess” thus appears entirely equivalent to Zurek’s “axiom(o).”To see how “axiom(o)” is employed in practice, consider the now-classic cavity-QED experiments of Brune et al. [74] (reviewed in [3] Ch. 6), in which decoherence ofa mesoscopic “Schr¨odinger cat” created by coupling a well-defined excited state of a ingle Rb atom to a weak photon field inside a superconducting cavity is monitored asa function of time and experimental conditions. In the standard language of quantumsystems and states, the system S in this case provides two observables, the state e (excited) or g (ground) of an Rb atom after it has traversed the cavity, and thecorrelation P ij (∆ t ) between the states of successive atoms i and j arriving at thedetector with a time difference of ∆ t . The experimental outcomes are: (1) varyingthe coupling between the atomic state and the photon field varies the amount ofinformation about the traversing atom’s state that was stored in the field ([74] Fig.3); and (2) varying the time interval ∆ t varies the amount of information aboutthe i th atom’s state that could be extracted from the j th atom’s state ([74] Fig.5). The first result demonstrates that increasing the local interaction between two identified degrees of freedom (by increasing the coupling) increases the entanglementbetween those degrees of freedom. The second result demonstrates that after the localinteraction between the two identified degrees of freedom (after the i th atom leavesthe cavity), the entanglement between those degrees of freedom dissipates; the fieldin the cavity is also entangled with the atoms in the walls of the cavity, and thislatter entanglement decoheres the “information” about the i th atom’s state that “theatom leaves in (the cavity) C ” ([74] p. 4889). Critical to this explanation is the tacitassumption that the states of the atoms in the walls of the cavity are not themselvesobserved, or equivalently, that the atoms in the walls of the cavity are themselvesentangled with the general environment in which the apparatus is embedded. But,this assumption comes with the implicit proviso that this prior system - environmententanglement does not prevent the identification of quantum states of the individualRb atoms traversing the cavity. This assumption that the individual Rb atoms canbe regarded “objectively” even in the presence of system - environment entanglementis an instance of “axiom(o).”The current framework alters this standard account of the physics by re-castingit in informational terms and rejecting the tacit assumption that the i th and j th Rbatoms are distinguishable quantum systems. The “system” S B in this framework(“ B ” for Brune et al. ) is the inverse image of a POVM {S Bk } with control vari-ables s B , ..., s Bn B . Distinct acceptable sets of values of these variables describe distinctpreparation conditions for the system. This system can be considered an informationchannel from Im − ( {A Bg , A Be } ) to O , where the components of A B report the out-comes g and e respectively. In this representation, long-lived entanglement betweenthe atom traversing the cavity and the photon field within it causes delocalizationin time of the outcome: the values of the control variables s B , ..., s Bn B - specifically,those indicating the mirror separation and hence tuning of the cavity - can be ad-justed in a way that smears an outcome g (for example) out over pairs of applicationsof A B . Figure 7 illustrates this smearing in time using a simple circuit model, inwhich the (approximately) fixed “resistance” R represents information loss from thechannel (e.g. the approximately fixed coupling of the photon field to the cavity) andthe variable “capacitance” C represents the intrinsic memory of the channel (e.g. themanipulable coupling of the atomic beam to the photon field). An instantaneousinput impulse δ ( t − t ) at t = t results in an output ∝ e − t/RC for t > t at O . Thetime constant RC is the decoherence time; it is a measure of the channel’s memoryof each outcome. B = Im − ( {S Bk } ) Im − ( {A Bg , A Be } ) O ✲✟✟❍❍❍❍✟✟❍❍✟✟ R C (cid:0)(cid:0) (cid:0)(cid:0)(cid:0)✒
Figure 7: Simple circuit model of decoherence in an information channel Im − ( {S Bk } ) . The “capacitance” C in Fig. 7 is clearly a measure of the “quantum-ness” of thechannel; as C →
0, the channel appears classical. The condition C = 0 corresponds toinfinite temporal resolution for measurement events; hence it corresponds to the “freeinformation” (i.e. ~ →
0) assumption of classical physics discussed at the end of Sect.4. If C = 0, the channel stores no information about previous outcomes, so all pairsof POVMs, including those that jointly measure action commute. The “resistance” R measures the leakiness of the channel in either direction; as R →
0, the channelapproaches infinite decoherence time, i.e. perfect isolation, in the quantum (
C > C = 0) case.Given the representation of an information channel as an RC circuit, consider arandom sequence of measurements with the POVM {A Bg , A Be } . These measurementscorrespond to a random sequence of “states” of Im − ( {A Bg , A Be } ). The no-cloningtheorem requires that these “states” be non-identical, and hence that the collectionsof fine grained states | C ( t ) i that physically implement them be non-identical. The in-dividual measurement outcomes cannot, therefore, be “remembered” at C as identical;the “memory traces” of distinct | C ( t ) i and | C ( t ) i stored at C must interfere. From O ’s perspective, this interference can be represented formally by adding a randomphase factor e − iφ to each transmission through the channel. Without such interfer-ence, the signal at O would increase monotonically with time if measurements weremade with a time separation less that RC , since C would never fully discharge. Sucharbitrarily temporally-delocalized outcomes are never observed in practical experi-ments. Adding the random phase term assures that, for t ≫ RC , interference betweenmeasurements drives the time-averaged signal at O toward zero. In this purely infor-mational RC -circuit model of decoherence, therefore, no-cloning is what requires theuse of a complex Hilbert space to represent “states” in the inverse image Im − ( {A ijk } )of any observable associated with an identified system. Treating the Im − ( a ijk ) asnames of coarse-grained basis vectors for the “system” Im − ( {S ik } ) as discussed above,an unknown quantum state of Im − ( {S ik } ) as measured at a future time t using the j th available POVM A ijk can be written | ψ i j ( t ) i = P k α k e − iφ k | Im − ( a ijk ) i with α k eal, exactly as expected within standard quantum theory.A “quantum channel” defined solely by non-commutativity between observablesjointly measuring action is, therefore, a quantum channel as defined by standardquantum theory, provided that information is physical and the observer is a minimalobserver as defined in Sect. 3. If determinism, time-symmetry, counterfactual def-initeness and decompositional equivalence are assumed, observations made throughsuch channels satisfy the Kochen-Specker, Bell, and no-cloning theorems. The Bornrule emerges as a consequence of decompositional equivalence. Complex phases arerequired by objective ignorance of the physical states implementing the channel, i.e.by no cloning. Decoherence is understandable not as a physical process acting onquantum states, but as an intrinsic hysteresis in quantum information channels. Mea-surement, in this framework, is unproblematic; if minimal observers exist, the deter-minate, “classical” nature of their observations follows straightforwardly from theirstructure as classical virtual machines and the physics of a quantum channel. The fun-damental interpretative assumptions that must be added to quantum theory appear,then, to be that information is physical and that minimal observers exist. If Galilean observers are replaced by minimal observers as defined in Sect. 3, theinterpretation of quantum theory is radically simplified. The traditional problemsof why some measurement bases, such as position, are “preferred” and how super-positions can “collapse” onto determinate eigenstates of those bases are immediatelyresolved: a minimal observer “prefers” the bases in which she encodes POVMs, andis only capable of recording eigenvalues in these bases. The problem of the “emer-gence” of the classical world also vanishes: the classical world is the world of recordedobservations made by minimal observers. Minimal observers are virtual machines im-plemented by physical degrees of freedom; hence the classical world is a virtual world.What the current framework adds to previous proposals along these lines (e.g. [75])is a precisely formulated model theory: the model theory expressed by the POVMsimplemented by the minimal observer.From an ontological perspective, the current framework can be viewed as an in-terpolation between two interpretative approaches generally regarded as diametricalopposites: a “pure” relative-state interpretation such as that of Tegmark [22] andthe quantum Bayesianism (“QBism”) of Fuchs [37]. Like QBism, the current frame-work views quantum states as observer-specific virtual entities. However, instead of“beliefs” as they are in QBism, these virtual entities are inverse images of observer-specific POVMs in the space of possible states of the real physical world. Like a purerelative-state interpretation, the current framework postulates a deterministic, time-symmetric Hamiltonian satisfying counterfactual definiteness and decompositionalequivalence. However, “branching” into arbitrarily many dynamically-decoupled si-multaneous actualities is replaced by the classical notion that a sufficiently complexphysical system can be interpreted as implementing arbitrarily many semantically-independent virtual machines. Like QBism, the current framework rejects the in- erpretation of decoherence as a physical mechanism that generates actuality; unlikeQBism, it views the “classical world” as entirely virtual and rejects the observer-independent “real existence” of bounded, separable macroscopic objects. Like a purerelative-state interpretation, the current framework embraces non-locality as an in-trinsic feature of the universe; unlike a pure relative-state interpretation, it views non-locality as a temporal relationship between instances of observation, not as a spatialrelationship between objects. The current framework is, therefore, ontologically veryspare. It postulates as “real” only the in-principle individually unobservable physicaldegrees of freedom that implement both channel and observer. The virtual machinesthat are postulated are not in any sense physical; unlike Everett branches [22], thereis no sense in which virtual machines constitute parallel physical actualities. Thisstrongly Kantian ontology is similar to that of the recent “possibilist” extension [76]of the transactional interpretation [77, 78], but without the notion that transactions“actualize” quantum phenomena in an observer-independent way.What the current interpretative framework emphatically rejects is the notion thatthe “environment” is a witness that monitors quantum states and defines systems for observers. The idea that the environment preferentially encodes certain “objective”quantum states and makes information about these states and not others available toobservers is the foundation of quantum Darwinism [12, 28, 29, 30, 31, 32]. It is im-plicit, however, in all interpretative approaches in which the classical world “emerges”from the dynamics in an observer-independent way. The bounded and separable “realexistences” postulated by QBism [37], for example, are effectively the observations ofthe “rest of the universe” viewed as an observing agent [79]. The “witness” assump-tion can be found in interpretative approaches as distant in terms of fundamentalassumptions from both QBism and quantum Darwinism as the possibilist transac-tional interpretation, where an “experimental apparatus seems persistent in virtueof the highly probable and frequent transactions comprising it” ([80] p. 8) not fromthe perspective of an observer, but from the perspective of an observer-free universe.It is this assumption of emergence via environmental witnessing that enables, ex-plicitly or otherwise, the traditional and ubiquitous assumption of information-freeGalilean observers, mere points of view or (as “preparers” of physical systems) pointsof manipulation of a pre-defined objective reality.As pointed out in the Introduction, the logical coherence of Galilean observersmust be rejected on the basis of classical automata theory alone [45, 46]. It is useful,however, to examine the Galilean observer from the perspective of the “environmentas witness.” Consider the classic Wigner’s friend scenario [81], but with an omniscient“friend” who monitors not just an atomic decay but the states of all possible “systems”in the universe. An observer can then obtain information about the state of any systemby asking his friend, i.e. by interacting with the local environment as envisaged byquantum Darwinism. A minimal observer asks his friend in language , by executinga POVM. The information that such an observer can obtain from the environment,whether viewed as a communication channel or as an omniscient oracle, is limited bythe observer’s repertoire of POVMs; a minimal observer can obtain no informationabout a system he cannot describe, and cannot “observe” that a system is in a statehe cannot represent and record. A Galilean observer, in contrast, stores no priorinformation and hence has no language. Having an omniscient friend does not help Galilean observer; they have no way to communicate. The assumption that aGalilean observer can nonetheless obtain any information encoded by the environmentis, effectively, the assumption that the observer has the same encoding capacity asthe environment: what is “given” to the omniscient environment is also “given” tothe Galilean observer. This assumption was encountered at the end of Sect. 3; it isthe familiar, classical assumption that information is free.Replacing Galilean observers with minimal observers replaces the intractable philo-sophical problem of why observers never observe superpositions - a pseudoproblemthat results from the informationally-impoverished and hence unconstrained natureof the Galilean observer - with two straightforwardly scientific problems. The firstis a problem in quantum computer science: what classical virtual machines can beimplemented by a given quantum computer, e.g. by a given Hamiltonian oracle [65]?One answer to this question is known: a quantum Turing machine [18] can implementany classical virtual machine. A second, more practical, answer is partially known:the quantum systems, whatever they are, that implement our everyday classical com-puters are Turing equivalent. What we do not know is how to describe these familiarsystems quantum mechanically, or how to approach the analysis of an arbitrary quan-tum system capable of implementing some limited set of classical virtual machines.The second problem straddles the border between machine intelligence and biopsy-chology. It is the question of what physically-realized virtual machines share POVMs,and of how these systems came to share them. If we are to understand how multipleobservers can reach an agreement that they are observing the same properties of thesame thing, it is this question that we must be able to answer.
This paper has investigated the consequences of replacing the Galilean observer tra-ditionally employed in interpretations of quantum theory with an observer that fullysatisfies the requirements of classical automata theory. It has shown that if boththe observer and the information channel with which it interacts are implemented byphysical degrees of freedom, the state space of which admits a linear measure enablingthe definition of POVMs, and if the temporal dynamics of these physical degrees offreedom are deterministic, time symmetric, and satisfy decompositional equivalenceand counterfactual definiteness, then the observations made by the observer are cor-rectly described by standard quantum theory. Quantum theory does not, therefore,require more than these assumptions. The unmotivated and ad hoc nature of theformal postulates that have been employed to axiomitize quantum theory, both tra-ditionally [60] and more recently (e.g. [34, 39]) can be seen as a side-effect of theassumption of Galilean observers and the compensatory, generally tacit assumptionof “axiom(o).”The introduction of information-rich minimal observers into quantum theory bringsto the fore the distinction between Shannon or von Neumann information definedsolely by the dynamics and pragmatic information defined relative to an emulationmapping that specifies a control structure and hence a virtual machine. A determin-istic, time-symmetric Hamiltonian conserves fine-grained dynamic information; the on Neumann entropy of the channel C is zero. Nonetheless, the pragmatic informa-tion - the list of observational outcomes - recorded by a minimal observer with anapproximately ideal memory increases monotonically with time. Pragmatic informa-tion appears, therefore, not to be conserved; “history” appears actual, objective andgiven. This apparent asymmetry is, however, illusory. Pragmatic information is onlydefinable relative to an emulation mapping, a semantic interpretation of C . Everyclassical bit encoded by a minimal observer must be computed when such an emu-lation mapping is specified. Hence pragmatic information is not free; it is balancedby the computational effort required to specify emulation mappings. This effort is“expended” by H as dynamic evolution unfolds; minimal observers and the outcomesthat they record are the result. “It from bit” is thus balanced by “bit from it.” Acknowledgement
Thanks to Eric Dietrich, Ruth Kastner and Juan Roederer for stimulating discussionsof some of the ideas presented here. Three anonymous referees provided helpfulcomments on the manuscript.
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