PPhase Space Logic
Niklas Johansson, Felix Huber, Jan- ˚Ake Larsson
12 February 2021
Abstract
We propose a phase space logic that can capturethe behavior of quantum and quantum-like systems. Theproposal is similar to the more generic concept of epistemiclogic: it encodes knowledge or perhaps more correctly, pre-dictions about outcomes of future observations on some sys-tems. For a quantum system, these predictions are state-ments about future outcomes of measurements performedon specific degrees of freedom of the system. The proposedlogic will include propositions and their relations includingconnectives, but importantly also transformations betweenpropositions on different degrees of freedom of the systems.A key point is the addition of a transformation that allows toconvert propositions about single systems into propositionsabout correlations between systems. We will see that sub-tle choices of the properties of the transformations lead todrastically different underlying mathematical models; onechoice gives stabilizer quantum mechanics, while anotherchoice gives Spekkens’ toy theory. This points to a crucialbasic property of quantum and quantum-like systems thatcan be handled within the present phase space logic by ad-justing the mentioned choice. It also enables a discussion onwhat behaviors are properly quantum or only quantum-like,relating to that choice and how it manifests in the systemunder scrutiny.
Niklas JohanssonDept of Electrical Engineering, Link¨oping University, SE-581 83Link¨oping, SWEDEN, E-mail: [email protected] HuberAtomic Optics Department, Jagiellonian University, PL-30-348Krak´ow, POLAND, E-mail: [email protected] ˚Ake LarssonDept of Electrical Engineering, Link¨oping University, SE-581 83Link¨oping, SWEDEN, E-mail: [email protected]
Due to the current fast development of technology towardsquantum information processing, there is great interest indifferent tools for understanding the behavior of quantumsystems. To model this processing, often a digital informa-tion representation is used. This calls for a suitable associ-ated logic. Of course, there are already well-known logics tochoose from.Quantum logic was proposed already by Birkhoff andvon Neumann (1936), by extending the language of stan-dard logic to encompass Hilbert space structure. In their pro-posal, they start from the Hilbert space description itself anddescribe a logic that, for example, replaces logical negationwith orthogonality in the Hilbert space. More recently, therehas been a drive to avoid using any Hilbert space structureas a postulate, and to look instead at other ways of build-ing up a mathematical structure that eventually arrives atquantum mechanical behavior. This has grown into an entirefield of scientific investigation (Hardy 2001; Pawłowski etal. 2009; Chiribella, D’Ariano, and Perinotti 2011; Masanesand M¨uller 2011, to give a few examples). Part of the discus-sion is the difficulty to avoid a direct postulate of an under-lying Hilbert space. Here we will make yet another attemptto build up a structure that avoids inserting Hilbert spacestructure by hand. We will, however, still retain propositionsthat relate to phase space as in Birkhoff and von Neumann(1936), but not the direct relation used there where proposi-tions are subsets of phase space.Instead, we will aim for a more epistemic approach,by encoding knowledge or predictions of future measure-ment outcomes, where the measurements are associatedwith the appropriate phase space. Naturally, epistemic logichas also already been studied, as a subfield of epistemol-ogy concerned with logical approaches to knowledge, be-lief and related notions. While any logic with an epistemic a r X i v : . [ qu a n t - ph ] F e b Niklas Johansson, Felix Huber, Jan- ˚Ake Larsson interpretation may be called an epistemic logic, the mostwidespread type of epistemic logic is that of modal log-ics (von Wright 1951). Epistemic modal logic is concernedwith agents, their knowledge, and beliefs. It formally en-codes for example, one agent’s belief about another agent’sknowledge. Unfortunately this is not well adapted to thequantum-mechanical situation, where the so-called quantumcontextuality (Kochen and Specker 1967) prohibits assign-ing a consistent set of values to propositions representing anagents knowledge.For this reason, the logic we aim for here is intendedto capture fundamental uncertainty: that statements not onlyhave an unknown truth value, but simply do not possess anyintrinsic truth value. In this sense it is no longer a questionof incomplete knowledge about an existing truth value, butthere are composite propositions that are fundamentally un-predictable. In contrast to epistemic modal logic which addsknowledge predicates on top of standard logic, we intend tocapture this within the logic itself. The phase space logic wepropose here will allow but does not force quantum behav-ior, and will capture the notion of incomplete knowledge inthe strongest possible sense: it will contain fundamentallyundecidable composite of propositions.The connection to the phase space is the following: inthis logic, propositions will concern predictions of quantum-like measurement outcomes on physical systems. Logicpropositions then discretize the measurement outcomes intotrue/false, in general being sentences on the form “the mea-surement outcome will lie between a and b ”. Here, we willrestrict ourselves to dichotomic measurement outcomes,without loss of generality. Such a dichotomic measurementis usually associated with a spin measurement along the Z axis in three-dimensional space, forming the first coordinatein a two-dimensional discrete phase space (Wootters 2003).The second coordinate is usually associated with a spin- X measurement, and diagonal lines in the discrete phase spaceare associated with spin- Y measurement.Our proposed phase space logic will contain these de-grees of freedom, and importantly also transformations be-tween them, not only on single systems but also transforma-tions involving correlations between systems. This makesour approach different from other three-valued logics thatattempt to capture aspects of quantum mechanics, but donot include phase space degrees of freedom as an essentialpart, nor transformations between them, see e.g., Reichen-bach (1944). In addition, the inclusion of these transfor-mations enables a logic-language basis for stabilizer quan-tum mechanics complete with Clifford-group transforma-tions (Calderbank et al. 1998; Gottesman 1998a; Gottesman1998b).Stabilizer quantum mechanics is an important tool inquantum information processing. Our construction will beable to reproduce the behavior of this subset of quan- tum mechanics, but will avoid postulating an underlyingHilbert space. It will also enable a logic-language basis forSpekkens’ toy theory and extensions of it (Spekkens 2007;Wallman and Bartlett 2012; Blasiak 2013; Johansson andLarsson 2019; Lillystone and Emerson 2019) that are usefulin foundational considerations on these issues. Let us nowturn to the explicit construction. We start with something that looks reasonably familiar; athree-valued logic where one truth value will denote an in-determinate outcome. Note that we immediately deviate byadding several degrees of freedom intending to capture theidea that not all measurements are compatible, meaning si-multaneously predictable. Standard propositional logic con-tains propositions, usually denoted p and q , that can havetruth values in { false,true } or { } . In the new logic anatomic proposition is a prediction for the outcome of a mea-surement, that is, a statement on the form “measurement ofthe Z degree of freedom would give the outcome 0”, denoted (cid:104) Z (cid:105) . We choose “outcome 0” because this gives a naturalcorrespondence to the stabilizer formalism where the nota-tion (cid:104)·(cid:105) is used for a stabilizer , a transformation that keepsa quantum state unchanged. There, (cid:104) Z (cid:105) is the notation forthe stabilizer of the quantum state | (cid:105) , which gives a directconnection to the proposition above.Similarly, we add propositions to two other degrees offreedom in phase space whose corresponding propositions (cid:104) X (cid:105) and (cid:104) Y (cid:105) are statements on predictions about these mea-surement outcomes. Propositions of this kind are atomicmuch like simple propositions of ordinary propositionallogic, and serve as building blocks for more complex ex-pressions below. Statements on measurement outcomes ofdifferent systems will be denoted by numerical indexes Z j or by position in a string of symbols.Propositions can be true or false just as in standardpropositional logic. They may also hold no truth valuemeaning that the measurement outcome is indeterminate,uncertain, unknown, or cannot be predicted, denoted “?”. Insome presentations of propositional logic one can see “T”for true (1) and “F” for false (0), so we could add “I” forindeterminate (Reichenbach 1944), in what follows we willuse “?” for readability. Importantly, the phase space propo-sitional logic we construct here will contain statements that cannot all be known simultaneously. We will also add trans-formations between propositions, corresponding to physicaltransformations of these systems.Just as in standard logic we can form connectives. Instandard logic, the unary connective NOT ( ¬ ) converts aproposition into its negation. It is natural to extend this no-tion such that the phase space logic NOT gives truth value hase Space Logic 3 ∧ ):it gives truth value 0 if one of the constituent propositionshas truth value 0, and 1 if both the constituent propositionshave truth value 1. It is natural to extend this notion so thatthe phase space logic conjunction has exactly this behaviorand that it gives indeterminate “?” otherwise, i.e., in the casewhen at least one constituent is indeterminate and the otheris either indeterminate or 1. For example, we denote “[mea-surement of the Z degree of freedom would give the out-come 0] AND [measurement of the X degree of freedomwould give the outcome 0]” by (cid:104) Z (cid:105) ∧ (cid:104) X (cid:105) ; or sometimes (cid:104) ZI (cid:105) ∧ (cid:104) IX (cid:105) or (cid:104) ZI , IX (cid:105) .Phase space logic disjunction (OR, ∨ ) can be definedsimilarly by extending the standard truth table to caseswith indeterminate constituent propositions. Thus, the phasespace logic disjunction gives truth value 1 if one of the con-stituent propositions has truth value 1, truth value 0 if boththe constituent propositions have truth value 0, and gives in-determinate “?” otherwise, i.e., if one of the constituents isindeterminate and the other is either indeterminate or 0. Wesee that the phase space logic conjunction and disjunctionmirror each other just as they do in standard logic, as alsocan be seen in Table 1. This is more commonly known as deMorgan’s laws, see below.Finally, phase space logic exclusive disjunction (XOR, (cid:89) ) is also an extension of the standard logic truth value 1 ifone of the constituent propositions has truth value 1 and theother 0, and truth value 0 if both the constituent propositionshave truth value 0 or 1. We simply let it give indeterminate“?” if either or both of the constituents are indeterminate. We have now come to the phase space logic material con-ditional and material biconditional ( → and ↔ ), where the Table 1
Phase space logic connectives p q ¬ p p ∧ q p ∨ q p (cid:89) q p → q p ↔ q qualifier “material” is used to distinguish the connective ( → )from the “formal” conditional ( ⇒ ) as Russell (1903) writes,or the “logical” conditional as it is now more commonlyknown. Perhaps jokingly, one could add that a better namewould be immaterial conditional instead of material condi-tional when describing quantum and quantum-like systems.We will not press the issue more here, after all, quantumsystems are the dreams that stuff is made of .To construct material conditional and biconditional thatincorporates an intrinsic uncertainty, we will take a closerlook at two alternatives. One is to construct condition-als such that they follow identities from ordinary proposi-tional logic. For example, then p ↔ q would be identicalto ( p ∧ q ) ∨ ( ¬ p ∧ ¬ q ) . However, this does not capture thedesired behavior well: the latter expression gives an inde-terminate value “?” if both p and q are indeterminate, whilethe material biconditional should compare if the truth valuesare equal. Thus, in our view, a better alternative is to have thematerial biconditional (EQUIVALENT TO, ↔ ) compare thetwo constituent propositions and have p ↔ q be 1 if p and q have the same truth value, and 0 otherwise. In particular, p ↔ q is 1 if both p and q are indeterminate. The logical bi-conditional between two composite propositions p ⇔ q cannow be defined as usual, namely as the situation in which p ↔ q is always true. This now coincides with the statementthat p and q have identical truth table entries.The material conditional (IMPLIES, → ) can similarly beextended to have p → q be 1 if p is 0, if q is 1, or (in addi-tion to the standard definition) if p and q have the same truthvalue, and 0 otherwise. The additional clause captures thenotion of implication in the case when both p and q are inde-terminate, encoding that when p is indeterminate we cannotdraw any conclusion about q so that an indeterminate truthvalue is acceptable by the connective. This completes the listof connectives in Table 1, and allows us to define the logicalconditional between two composite propositions p ⇒ q asthe situation in which p → q is always true.Using the notation (cid:104)¬ Z (cid:105) for the statement “measurementof the Z degree of freedom would give the outcome 1” wenote that ¬(cid:104) Z (cid:105) ⇔ (cid:104)¬ Z (cid:105) , (1)where the equivalence holds also for indeterminate truth val-ues. The propositions (cid:104) I (cid:105) and (cid:104)¬ I (cid:105) represent tautology andcontradiction, respectively, corresponding to trivial mea-surements that always have the outcome 0 or 1. Table 2
Truth table for the inverse law p ¬ p p ∨ ¬ p (cid:104) I (cid:105) p ∧ ¬ p (cid:104)¬ I (cid:105) In standard logic there are a number of logical equivalencerelations and implication laws. Some of these are importantproperties of logic expressions like reflexivity, symmetry,and transitivity, and these are retained by phase space logicas defined here. However, some equivalences that are com-mon tools in standard propositional logic no longer hold.The most prominent example is the “inverse law,” which instandard propositional logic tells you that ( p ∨ ¬ p ) is a tau-tology and that ( p ∧ ¬ p ) is a contradiction. These relationsfail because of the indeterminate values involved, see therelevant truth tables in Table 2. It follows that seeminglyinnocuous simplifications are no longer available to us, forexample, (cid:104) ZI , IX (cid:105)∨(cid:104)¬ ZI , IX (cid:105) ⇔ (cid:16) (cid:104) ZI (cid:105)∨(cid:104)¬ ZI (cid:105) (cid:17) ∧(cid:104) IX (cid:105) (cid:54)⇔ (cid:104) IX (cid:105) . (2)It is a simple exercise to verify that for generic propo-sitions p , q , and r , the following equivalences hold, and donot hold, details in Appendix A.(E1) Double negation ¬¬ p ⇔ p (E2) De Morgan’s laws ¬ ( p ∧ q ) ⇔ ¬ p ∨ ¬ q ¬ ( p ∨ q ) ⇔ ¬ p ∧ ¬ q (E3) Commutative laws p ∧ q ⇔ q ∧ pp ∨ q ⇔ q ∨ p (E4) Associative laws p ∧ ( q ∧ r ) ⇔ ( p ∧ q ) ∧ rp ∨ ( q ∨ r ) ⇔ ( p ∨ q ) ∨ r (E5) Distributive laws p ∧ ( q ∨ r ) ⇔ ( p ∧ q ) ∨ ( p ∧ r ) p ∨ ( q ∧ r ) ⇔ ( p ∨ q ) ∧ ( p ∨ r ) (E6) Idempotence p ∧ p ⇔ pp ∨ p ⇔ p (E7) Identity laws p ∧ (cid:104) I (cid:105) ⇔ pp ∨ (cid:104)¬ I (cid:105) ⇔ p (E8) Domination laws p ∧ (cid:104)¬ I (cid:105) ⇔ (cid:104)¬ I (cid:105) p ∨ (cid:104) I (cid:105) ⇔ (cid:104) I (cid:105) (E9) Inverse laws p ∧ ¬ p (cid:54)⇔ (cid:104)¬ I (cid:105) p ∨ ¬ p (cid:54)⇔ (cid:104) I (cid:105) (E10) Absorption laws p ∧ ( p ∨ q ) ⇔ pp ∨ ( p ∧ q ) ⇔ p (E11) Implication law p → q (cid:54)⇔ ¬ p ∨ q (E12) Contrapositive law p → q ⇔ ¬ p → ¬ q (E13) Equivalence law p → q ⇔ ( p → q ) ∧ ( q → p ) An example of how the inverse law breaks in quantummechanics is the double slit experiment, where the propo-sition p = “the particle passes through the right-hand slit”corresponds to a dichotomic measurement. If interference isdesired from the experiment, it must be arranged so that p is indeterminate “?”, in which case p ∧ ¬ p is indeterminate“?” so not a contradiction, and p ∨ ¬ p is indeterminate “?”so not a tautology. We find that this describes the situationbetter than the popular-science “the particle passes throughboth of the slits (both p and ¬ p are true so p ∧ ¬ p is true) and simultaneously none of the slits (both p and ¬ p are falseso p ∨ ¬ p is false),” which at best is just confusing.For the inverse laws only the trivial implications remain,while the implication law does not hold in either direction. Asimilar exercise for logical implications gives the followinglist, details in Appendix A.(I1) Modus ponens ( p → q ) ∧ p ⇒ q (I2) Law of syllogism ( p → q ) ∧ ( q → r ) ⇒ ( p → r ) (I3) Modus tollens ( p → q ) ∧ ¬ q ⇒ p (I4) Conjunctive simpl. p ∧ q ⇒ p (I5) Disjunctive ampl. p ⇒ p ∨ q (I6) Disjunctive syllogism ( p ∨ q ) ∧ ¬ q (cid:54)⇒ p (I7) Proof by contradiction ( ¬ p → (cid:104)¬ I (cid:105) ) ⇒ p (I8) Proof by cases ( p → r ) ∧ ( q → r ) ⇒ ( p ∨ q ) → r Most of the standard rules that we use when provingtheorems do still hold, the exception is disjunctive syllo-gism, that succumbs to the same problem as the implicationlaw: when q is indeterminate, we cannot draw a conclusionabout p . Note that modus tollens still holds, encompassing aslightly stronger requirement. We now arrive at a crucial point in the construction, thevery reason to include the indeterminate value for the propo-sitions used. So far, the indeterminate value could corre-spond to lack of knowledge about the “actual” ontic (exist-ing) value of the property being measured. But when makingstatements about quantum systems, one should take into ac-count that the standard mathematical description does notcontain such ontic values, but rather, only allows calcula-tion of probabilities after specifying which property is to bemeasured. The debate goes back to the founding fathers ofquantum theory (Einstein, Podolsky, and Rosen 1935; Bohr1935). We do not wish to take a stance on that particularissue here, merely describe a truly epistemic logic, that en-compasses the possibility that indeterminate values “?” notjust denote lack of knowledge but may be fundamentally un-certain .In quantum mechanics, not all measurement outcomescan simultaneously be predicted with certainty (probabil-ity 1 as EPR 1935 write). Also in quantum-like systems, likeSpekkens’ toy theory, not all measurement outcomes can besimultaneously predicted with certainty. It is then naturalto require that in the present phase space logic that not allpropositions can hold truth values simultaneously. In partic-ular, the uncertainty principle holds in the form of a boundon the descriptional power of this phase space propositionallogic concerning atomic propositions, i.e., propositions onsingle degrees of freedom of single systems. hase Space Logic 5
Postulate: Bound on descriptional power
No more than a single atomic proposition can betrue or false for any single system.We can immediately conclude that some compositepropositions on a single system cannot have a definite truthvalue, for example, (cid:104) Z (cid:105) (cid:89) (cid:104) X (cid:105) ⇔ ? . (3)This bound on descriptional power corresponds directly tothe uncertainty relation in quantum mechanics (Heisenberg1927) and to the knowledge balance principle of Spekkens’toy theory (Spekkens 2007). A set of propositions that can have truth values simultaneously we will call compatible .Since not all propositions can have simultaneous truth val-ues, not all measurements are compatible.The postulate has consequences on what we can sayabout systems in general, for example about correlations be-tween systems. However, to arrive at a precise statement wewill need transformations between propositions; these cor-respond to physical operations performed on the physicalsystems that we study. Our propositions concern predicted outcomes of measure-ments performed on physical systems. As such, one of oursystems can be subjected to a range of physical transforma-tions, the simplest case is 180° rotation of the system aroundone of the phase space coordinate axes. Such a rotation willconserve the proposition that concerns the axis, but performa negation of the other two. It is therefore natural to use theconserved axis as label, in complete parallell with the stabi-lizer formalism: ϕ X (cid:104) X (cid:105) ⇔ (cid:104) X (cid:105) , ϕ X (cid:104) Y (cid:105) ⇔ (cid:104)¬ Y (cid:105) , ϕ X (cid:104) Z (cid:105) ⇔ (cid:104)¬ Z (cid:105) , (4) ϕ Y (cid:104) X (cid:105) ⇔ (cid:104)¬ X (cid:105) , ϕ Y (cid:104) Y (cid:105) ⇔ (cid:104) Y (cid:105) , ϕ Y (cid:104) Z (cid:105) ⇔ (cid:104)¬ Z (cid:105) , (5) ϕ Z (cid:104) X (cid:105) ⇔ (cid:104)¬ X (cid:105) , ϕ Z (cid:104) Y (cid:105) ⇔ (cid:104)¬ Y (cid:105) , ϕ Z (cid:104) Z (cid:105) ⇔ (cid:104) Z (cid:105) . (6)Since these are equivalence relations, the transformationspreserve truth values and compatibility relations. The“Phase rotation” S comes in two variants. The quantumphase rotation corresponds to a 90° rotation around the Z axis, so that ϕ S (cid:104) X (cid:105) ⇔ (cid:104) Y (cid:105) , ϕ S (cid:104) Y (cid:105) ⇔ (cid:104)¬ X (cid:105) , ϕ S (cid:104) Z (cid:105) ⇔ (cid:104) Z (cid:105) . (7a)An alternative transformation is used in Spekkens’ toy the-ory (Pusey 2012) where the rotation is followed by an inver-sion along the Z axis, ϕ S (cid:104) X (cid:105) ⇔ (cid:104) Y (cid:105) , ϕ S (cid:104) Y (cid:105) ⇔ (cid:104)¬ X (cid:105) , ϕ S (cid:104) Z (cid:105) ⇔ (cid:104)¬ Z (cid:105) . (7b)The effect of this difference is small, note that in both cases ϕ S ϕ S = ϕ Z . We now arrive at a crucial step in the construction,the “Hadamard” transformation. This transformation alsocomes in two variants with a seemingly small difference inthe transformation itself, but this difference will instead havevery important consequences for the type of model that canbe used to describe the system, we will expand on this below.The quantum-mechanical Hadamard corresponds to a physi-cal rotation around an axis 45° between X and Z , giving herea transformation that interchanges X and Z and inverts Y , ϕ H (cid:104) X (cid:105) ⇔ (cid:104) Z (cid:105) , ϕ H (cid:104) Y (cid:105) ⇔ (cid:104)¬ Y (cid:105) , ϕ H (cid:104) Z (cid:105) ⇔ (cid:104) X (cid:105) . (8a)The alternative Hadamard transformation (Pusey 2012) usedin Spekkens’ toy theory corresponds to an mirror operationover the plane spanned by the mentioned axis 45° between X and Z and the Y axis, i.e., a transformation that interchanges X and Z and preserves Y , ϕ H (cid:104) X (cid:105) ⇔ (cid:104) Z (cid:105) , ϕ H (cid:104) Y (cid:105) ⇔ (cid:104) Y (cid:105) , ϕ H (cid:104) Z (cid:105) ⇔ (cid:104) X (cid:105) . (8b)Finally, the identity transformation ϕ I leaves all proposi-tions unchanged. For both versions of the Phase rotation andHadamard, we arrive at a non-Abelian group generated by ϕ S and ϕ H , since ϕ S ϕ S = ϕ Z , ϕ H ϕ Z ϕ H = ϕ X and ϕ S ϕ X ϕ − S = ϕ Y . (9)This also generates all single-system propositions from one,say (cid:104) Z (cid:105) . For composite systems, there remains to handle joint mea-surements, which enable statements about correlation be-tween measurement outcomes without making statementsabout the individual measurement outcomes. The statement“a joint measurement of the XOR between Z and Z wouldgive outcome 0” can be written (cid:104) Z (cid:89) Z (cid:105) , or to conform withstandard stabilizer notation (cid:104) ZZ (cid:105) , we will use the latter no-tation below.To avoid confusion please note that (cid:104) ZZ (cid:105) corresponds toa joint measurement of a single dichotomic value that givesthe XOR, it does not correspond to individual measure-ment of two dichotomic values followed by calculation ofthe XOR between them, this would be denoted (cid:104) ZI (cid:105) (cid:89) (cid:104) IZ (cid:105) .These are different procedures, and have different conse-quences, in particular (cid:104) ZI (cid:105) (cid:89) (cid:104) IZ (cid:105) ⇒ (cid:104) ZZ (cid:105) but (cid:104) ZI (cid:105) (cid:89) (cid:104) IZ (cid:105) (cid:54)⇐ (cid:104) ZZ (cid:105) . (10)There are some simple equivalences since the conjunction ofa statement about a single system and a statement about itscorrelation to another system, is equivalent to a conjunctionof individual statements about the two systems, for example, (cid:104) XI , XX (cid:105) ⇔ (cid:104) XI , IX (cid:105) , (cid:104)¬ XI , XX (cid:105) ⇔ (cid:104)¬ XI , ¬ IX (cid:105) . (11) Niklas Johansson, Felix Huber, Jan- ˚Ake Larsson
This simplification applies to conjunctions. We will nowadd another transformation that maps a single propositionon the outcome of a correlation measurement to a proposi-tion on some outcome of a single-system measurement. Onenatural choice is the CZ (“controlled- Z ”), because its actionis symmetric on both subsystems, and can be summarized as ϕ CZ (cid:104) IX (cid:105) ⇔ (cid:104) ZX (cid:105) , ϕ CZ (cid:104) IY (cid:105) ⇔ (cid:104) ZY (cid:105) , ϕ CZ (cid:104) IZ (cid:105) ⇔ (cid:104) IZ (cid:105) , ϕ CZ (cid:104) XI (cid:105) ⇔ (cid:104) XZ (cid:105) , ϕ CZ (cid:104) Y I (cid:105) ⇔ (cid:104)
Y Z (cid:105) , ϕ CZ (cid:104) ZI (cid:105) ⇔ (cid:104) ZI (cid:105) , ϕ CZ (cid:104) ZZ (cid:105) ⇔ (cid:104) ZZ (cid:105) , ϕ CZ (cid:104) XX (cid:105) ⇔ (cid:104) YY (cid:105) , ϕ CZ (cid:104) XY (cid:105) ⇔ (cid:104)¬ Y X (cid:105) . (12)The motive for this definition (which (12) should be viewedas) is the standard gate-based description of a CZ , that ap-plies a ϕ Z transformation on the second system if “measure-ment of the Z degree of freedom would give the outcome1”, and ϕ I if “measurement of the Z degree of freedomwould give the outcome 0”. It is immediate that (cid:104) II (cid:105) , (cid:104) IZ (cid:105) , (cid:104) ZI (cid:105) , and (cid:104) ZZ (cid:105) are unaffected by ϕ Z on either system. To de-rive the effect of this map on the proposition “measurementof X would give the outcome 0,” (cid:104) IX (cid:105) , we have (cid:40) ϕ CZ (cid:104) ZI , IX (cid:105) ⇔ ϕ II (cid:104) ZI , IX (cid:105) ⇔ (cid:104) ZI , IX (cid:105) , ϕ CZ (cid:104)¬ ZI , IX (cid:105) ⇔ ϕ IZ (cid:104)¬ ZI , IX (cid:105) ⇔ (cid:104)¬ ZI , ¬ IX (cid:105) . (13)Since we are looking for the unique statement that (cid:104) IX (cid:105) maps to, we use the equivalences in Eqn. (11) to obtain (cid:40) ϕ CZ (cid:104) ZI , IX (cid:105) ⇔ (cid:104) ZI , ZX (cid:105) , ϕ CZ (cid:104)¬ ZI , IX (cid:105) ⇔ (cid:104)¬ ZI , ZX (cid:105) , (14)so that, for each measurement outcome of Z the map trans-forms “measurement of X would give the outcome 0” into“measurement of the XOR between Z and X would givethe outcome 0”. Even though the “inverse law” is not avail-able, which would have directly given us ϕ CZ (cid:104) IX (cid:105) ⇔ (cid:104) ZX (cid:105) ,this is the only remaining possibility for a well-defined map ϕ CZ . Similar reasoning and the symmetry of CZ gives thetransformation output of the remaining single-system propo-sitions of (12).Some further elaboration is needed for the two finalentries in the table. These can be derived from the trans-formation property that compatible propositions are trans-formed into compatible propositions, using only three ofthe just established single-system transformation outputs.We know that (cid:104) IX (cid:105) and (cid:104) XI (cid:105) are compatible, and there-fore (cid:104) ZX (cid:105) ⇔ ϕ CZ (cid:104) IX (cid:105) and (cid:104) XZ (cid:105) ⇔ ϕ CZ (cid:104) XI (cid:105) are compati-ble. Conversely, (cid:104) IX (cid:105) and (cid:104) IY (cid:105) are incompatible, and there-fore (cid:104) ZX (cid:105) ⇔ ϕ CZ (cid:104) IX (cid:105) and (cid:104) ZY (cid:105) ⇔ ϕ CZ (cid:104) IY (cid:105) are incompat-ible. It is now possible to use single-system transforma-tions to deduce whether a given pair of two-system proposi-tions are compatible or not. For example, (cid:104) ZX (cid:105) is compati-ble with (cid:104) ZI (cid:105) , (cid:104) IX (cid:105) , (cid:104) XZ (cid:105) , (cid:104) YY (cid:105) , (cid:104) XY (cid:105) , (cid:104) Y Z (cid:105) , and no othertwo-system propositions. It is of particular interest that the pair {(cid:104) ZX (cid:105) , (cid:104) XZ (cid:105)} is only compatible with (cid:104) YY (cid:105) (and (cid:104) II (cid:105) ),which implies that the set {(cid:104) IX (cid:105) , (cid:104) XI (cid:105) , (cid:104) XX (cid:105)} of three pair-wise compatible propositions must be transformed into theset {(cid:104) ZX (cid:105) , (cid:104) XZ (cid:105) , (cid:104) YY (cid:105)} or possibly {(cid:104) ZX (cid:105) , (cid:104) XZ (cid:105) , (cid:104)¬ YY (cid:105)} .The latter choice will give inconsistencies, for details seeAppendix B.Both quantum mechanics and Spekkens’ toy theory usethe choice ϕ CZ (cid:104) XX (cid:105) ⇔ (cid:104) YY (cid:105) . (15)Then, the identity ϕ S ϕ Z = ϕ S ϕ S ϕ S = ϕ Z ϕ S fixes ϕ CZ (cid:104) XY (cid:105) ⇔ ϕ CZ ϕ IS (cid:104) XX (cid:105) ⇔ ϕ IS (cid:104) YY (cid:105) ⇔ (cid:104)¬ Y X (cid:105) (16)This finishes the construction of ϕ CZ , and enables generat-ing the whole Clifford group, e.g., ϕ CNOT = ϕ IH ϕ CZ ϕ IH .Furthermore, we can now reproduce the behavior of stabi-lizer quantum mechanics if the choices of phase rotation andHadamard are made as in Eqn. (7a) and (8a); we reproducethe behavior of Spekkens’ toy theory if the choices are as inEqn. (7b) and (8b). To transform any non-“identity” proposition on a many-system correlation into a single-system proposition, it isenough to follow these steps.(i) WLOG there is a nontrivial letter at the first position.Apply ϕ S (transforms Y to ¬ X ) and ϕ H (transforms X to Z ) to transform the first position to X and the followingpositions to identity I or Z .(ii) Then use ϕ CZ repeatedly to reduce the last n − Z to I , to create a string with a single X at the first position.We will call this a Clifford reduction of a joint measurement.As an example, ϕ ISIIIS (cid:104)
XY ZIZY (cid:105) ⇔ (cid:104)
XXZIZX (cid:105) ϕ IHIIIH (cid:104)
XXZIZX (cid:105) ⇔ (cid:104)
XZZIZZ (cid:105) ϕ CZ ϕ CZ ϕ CZ ϕ CZ (cid:104) XZZIZZ (cid:105) ⇔ (cid:104)
XIIIII (cid:105) . (17)A simultaneous Clifford reduction of two propositionsinto single-system propositions can also be performed.(i) Reduce the first proposition and perform the same trans-formations on the second. The first proposition nowreads (cid:104) XI . . . I (cid:105) . Either the second proposition has alsobeen reduced to the form (cid:104)· I . . . I (cid:105) in which case we aredone, or WLOG there is a nontrivial letter at the secondposition.(ii) Reduce the second proposition excluding the very firstindex, to put it on the form (cid:104)· XI . . . I (cid:105) . hase Space Logic 7 (iii) If the first index reads I , we have two compatible single-system expressions (cid:104) XII . . . I (cid:105) and (cid:104) IXI . . . I (cid:105) . If the firstindex reads X , the two propositions can be reduced tocompatible single-system propositions using Eqn. (11).If the first index reads Y or Z , the two propositions can bereduced to two incompatible single-system propositions,using Hadamards on both systems followed by CZ .Then we are done and the two propositions have beenreduced to either compatible or incompatible single-system propositions.We have now shown that two propositions are always si-multaneously reducible to one-system propositions. The re-duction is to two compatible propositions (for different sys-tems) if and only if the two original propositions are compat-ible (but not equivalent), and otherwise the reduction givestwo incompatible propositions (for the same system).The previous discussion shows that if two propositionsare compatible, then they can be simultaneously reducedto the form (cid:104) XII . . . I (cid:105) and (cid:104) IXI . . . I (cid:105) . The same is true for n compatible propositions: they can be simultaneously re-duced to contain one X and identities I on the remainingpositions. This is proven by induction as follows. Supposethat m compatible propositions have been reduced to one-index X ’s on the first m systems using a simultaneous Clif-ford reduction. If m < n , consider an additional proposition P m + compatible with the previously reduced propositions,to which the the same Clifford operations have been appliedas used to reduce the m first propositions. Since P m + is com-patible with the previously reduced propositions, all indices i ≤ m only contain I or X , because otherwise P m + wouldbe incompatible with some P i , i ≤ m . The indices i ≤ m thatcontain X can now be converted into I using Eqn. (11). Nowreduce P m + excluding the m first indices, putting it on theform (cid:104) I . . . IXI . . . I (cid:105) . This gives us a set of m + m + = n we are done,otherwise repeat the process.If the string length equals n the last step is trivial, reduc-ing P n excluding the first n − We have already seen that some conjunctions are equivalentto other conjunctions. This also enables us to make predic-tions. The equivalence in Eqn. (11) is the basic tool, for ex-ample, given the proposition (cid:104) XI , IX (cid:105) , the measurement of (cid:104) XX (cid:105) will lead to no new knowledge; it can be obtained fromthe conjunction (cid:104) XI , IX (cid:105) already. It is also true that (cid:104) XZ , ZX (cid:105) ⇔ ϕ CZ (cid:104) XI , IX (cid:105) ⇒ ϕ CZ (cid:104) XX (cid:105) ⇔ (cid:104) YY (cid:105) (18)which is perhaps less straightforward to see by inspection.There is a direct link to the stabilizer formalism but we willnot comment more on that link here.We will call a logical consequence of a conjunction ofpropositions a prediction . In general, to obtain all predic-tions from a collection of propositions, one would need toperform a joint Clifford reduction, generate the set of pre-dictions by repeated use of Eqn. (11), and then invert theClifford reduction to perform a Clifford expansion . This willrestore the initial collection and create the full set of predic-tions. As a consequence, a collection of n systems allowsfor at most 2 n propositions on single systems or correlationsto be simultaneously true. To see why, recall that at most n independent single-system propositions can be simultane-ously true. Every pair of single-system or correlation propo-sitions yields a new single-system or correlation prediction;thus one arrives at in total 2 n propositions.
10 Measurements affect propositions
For a single system, no more than a single atomic propo-sition can be true or false at the same time. For multiplesystems this generalizes to the statement that only mutu-ally compatible propositions can simultaneously be known:at most 2 n general propositions of which there are n inde-pendent ones.We thus need to specify what happens when some de-gree of freedom is measured that belongs to an incompatibleproposition. Here we take the approach that any measure-ment must yield a definite outcome, and that measurementenables prediction of subsequent measurement outcomes.Given that there is a descriptional power bound, the ques-tion is how to incorporate this new prediction. It is reason-able that any previous proposition that is compatible with thenew one remains, whereas incompatible propositions cannotcoexist with the new proposition and must therefore becomeindeterminate.More formally, suppose that the conjunction P = p ∧· · · ∧ p m holds. Now measure some degree of freedom ofthe system. Since by our assumption the outcome enablesprediction of a future measurement outcome, we capture thatin a proposition q . Denote by P q = (cid:94) p i compatible with q p i . (19)After measurement, we have the conjunction P q ∧ q , (20) Niklas Johansson, Felix Huber, Jan- ˚Ake Larsson and all predictions possible from it. Note that if the measure-ment gives a proposition q that is a prediction of P , nothingchanges. This captures the behavior of both quantum me-chanics and Spekkens’ toy theory. We arrive at the follow-ing consequence of the postulate of bounded descriptionalpower: Consequence: Action of Measurements
A measurement determines the truth value of aproposition of the system and renders all incom-patible propositions indeterminate.
11 Contextuality, and noncontextuality
The phase space logic constructed here allows treatment ofquantum and quantum-like systems within the same for-malism, capturing not only fundamental uncertainty, butalso transformations between different degrees of freedom.These transformations are needed for the emergence ofproperly quantum behavior, in particular transformationsfrom knowledge on one system into knowledge about cor-relation between systems. It is the addition of the lattertype of transformation that makes stabilizer quantum me-chanics accessible, but it is in fact one of the single-systemtransformations that differentiate between properly quan-tum, and merely quantum-like, behavior. It also makes ourthree-valued logic more restrictive. To see this we need toconsider a properly quantum phenomenon, here we will usethat of quantum contextuality of the Peres-Mermin square(PM, Peres 1993; Mermin 1993): (cid:104) ZI (cid:105) (cid:104) IZ (cid:105) (cid:104) ZZ (cid:105)(cid:104) IX (cid:105) (cid:104) XI (cid:105) (cid:104) XX (cid:105)(cid:104) ZX (cid:105) (cid:104) XZ (cid:105) (cid:104) YY (cid:105) . (21)The rows and columns in this square consist of compatiblepropositions. In fact, we can make predictions for the lastitem in rows and columns from the preceding items, usingEqns. (11) and (18), (cid:104) ZI (cid:105) ∧ (cid:104) IZ (cid:105) ⇒ (cid:104) ZZ (cid:105) ; (cid:104) IX (cid:105) ∧ (cid:104) XI (cid:105) ⇒ (cid:104) XX (cid:105) ; (cid:104) ZI (cid:105) ∧ (cid:104) IX (cid:105) ⇒ (cid:104) ZX (cid:105) ; (cid:104) IZ (cid:105) ∧ (cid:104) XI (cid:105) ⇒ (cid:104) XZ (cid:105) ; (cid:104) ZX (cid:105) ∧ (cid:104) XZ (cid:105) ⇒ (cid:104) YY (cid:105) . (22)The exception is the final column where the derivation of theprediction (see Sec. 9) involves a Hadamard. The two dif-ferent Hadamards give different predictions. The quantum-mechanical Hadamard gives (cid:104) XX , ZZ (cid:105) ⇔ ϕ HI (cid:104) ZX , XZ (cid:105) ⇒ ϕ HI (cid:104) YY (cid:105) ⇔ (cid:104)¬ YY (cid:105) , (23a)while Spekkens’ toy theory Hadamard instead gives (cid:104) XX , ZZ (cid:105) ⇔ ϕ HI (cid:104) ZX , XZ (cid:105) ⇒ ϕ HI (cid:104) YY (cid:105) ⇔ (cid:104) YY (cid:105) . (23b)The latter prediction (23b) allows simultaneous assignmentof truth values (other than indeterminate “?”) to all the propositions in the PM square (21). The former prediction(23a) does not.This is known as quantum contextuality (Kochen andSpecker 1967), where the word “context” here refers towhich conjunction a proposition is contained in, e.g., row orcolumn in the PM square. For a system with the quantum-mechanical Hadamard transformation, one needs to acceptone of the following alternatives(i) some proposition in the PM square (21) must be indeter-minate “?”,(ii) some proposition in the PM square (21) must possessdifferent values in different contexts.The former alternative captures fundamental uncertainty,while the latter alternative would give a contextual onticmodel. Spekkens’ toy theory, on the other hand, is a non-contextual ontic model where propositions do not possessdifferent values in different contexts.
12 Conclusions
We have here constructed phase space logic , that includesnot only an extension to standard logic introducing an in-determinate “?” truth value, but also several phase spacedegrees of freedom for each individual system. Crucially,it also includes transformations between propositions thatconcern different degrees of freedom, and transformationsbetween single-system propositions and correlation proposi-tions. The construction is intended to capture not only quan-tum uncertainty but more properly quantum properties of thedescribed systems, for which the transformations are crucial.The introduction of a limitation in predictive power, andthe implied action of measurements, does make the logiccontain quantum-like elements. But it is really the choice ofHadamard transformation, in particular its transformation ofthe proposition (cid:104) Y (cid:105) , that decides if the behavior is quantumor merely quantum-like. From the point of view of phasespace logic, it is the Hadamard transformation which is re-sponsible for the emergence of quantum contextuality.In contrast to quantum logic, the approach presentedhere does not force contextuality into the logic through pos-tulating Hilbert space structure. It is instead a generic frame-work that allows contextuality to occur, or not occur, as de-sired. We aim to use this simple language to describe otherquantum and quantum-like phenomena, and hope that theframework will be useful to others in the same line of work. References
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Some equivalence laws are very simple to check, such as (E1) and(E3)–(E8), and the inverse law (E9) is contained in the main text. Whatremains are the following.
Table 3 (E2) de Morgan’s laws: ¬ ( p ∧ q ) ⇔ ¬ p ∨ ¬ q ; ¬ ( p ∨ q ) ⇔¬ p ∧ ¬ q p q p ∧ q ¬ p ∨ ¬ q p ∨ q ¬ p ∧ ¬ q Table 4 (E10) absorption laws: p ∨ ( p ∧ q ) ⇔ p ; p ∧ ( p ∨ q ) ⇔ pp q p ∧ q p ∨ ( p ∧ q ) p ∨ q p ∧ ( p ∨ q ) Table 5 (E11) the implication law: ( p → q ) (cid:54)⇔ ( ¬ p ∨ q ) p q p → q ¬ p ∨ q ( p → q ) → ( ¬ p ∨ q ) ( p → q ) ← ( ¬ p ∨ q ) Table 6 (E12) the contrapositive law: p → q ⇔ ¬ q → ¬ pp q p → q ¬ p ¬ q ¬ q → ¬ p Table 7 (E13) the equivalence law: p ↔ q ⇔ ( p → q ) ∧ ( p ← q ) p q p ↔ q p → q p ← q Table 8 (I1) Modus ponens: ( p → q ) ∧ p ⇒ qp q p → q ( p → q ) ∧ p [( p → q ) ∧ p ] → q Table 9 (I2) law of syllogism: ( p → q ) ∧ ( q → r ) ⇒ p → rp q r p → q q → r ( p → q ) ∧ ( q → r ) p → r Table 10 (I3) Modus tollens: ( p → q ) ∧ ¬ q ⇒ ¬ pp q p → q ( p → q ) ∧ ¬ q [( p → q ) ∧ ¬ q ] → ¬ p Table 11 (I4) Conjunctive simplification: p ∧ q ⇒ p ; and (I5) Disjunc-tive strengthening: p ⇒ p ∨ qp q p ∧ q ( p ∧ q ) → p p ∨ q p → ( p ∨ q ) Table 12 (I6) Disjunctive syllogism: ( p ∨ q ) ∧ ¬ q (cid:54)⇒ pp q p ∨ q ( p ∨ q ) ∧ ¬ q [( p ∨ q ) ∧ ¬ q ] → p Table 13 (I7) proof by contradiction: ( ¬ p → (cid:104)¬ I (cid:105) ) ⇒ pp ¬ p → (cid:104)¬ I (cid:105) ( ¬ p → (cid:104)¬ I (cid:105) ) → p Table 14 (I8) proof by cases: ( p → r ) ∧ ( q → r ) ⇒ ( p ∨ q ) → rp q r p → r q → r ( p → r ) ∧ ( q → r ) ( p ∨ q ) → r B Choice of CZ transformation This appendix contains a derivation of an inconsistency that wouldarise if the CZ transformation is chosen so that (cid:101) ϕ CZ (cid:104) XX (cid:105) ⇔ (cid:104)¬ YY (cid:105) , (24)where the tilde is used to distinguish the transformation used in themain text from the one considered here. The inconsistency arises whenmaking predictions from the conjunction (cid:104)¬ YY I , ¬ IYY (cid:105) . To generatepredictions we first use the procedure of Section 9. This starts with thesimultaneous Clifford reduction (cid:104)¬
YY I , ¬ IYY (cid:105) ⇔ ϕ SSS (cid:104)¬
XXI , ¬ IXX (cid:105)⇔ ϕ SSS ϕ IHH (cid:104)¬
XZI , ¬ IZZ (cid:105)⇔ ϕ SSS ϕ IHH (cid:101) ϕ CZ (cid:104)¬ XII , ¬ IZZ (cid:105)⇔ ϕ SSS ϕ IHH (cid:101) ϕ CZ ϕ IHI (cid:104)¬
XII , ¬ IXZ (cid:105)⇔ ϕ SSS ϕ IHH (cid:101) ϕ CZ ϕ IHI (cid:101) ϕ CZ (cid:104)¬ XII , ¬ IXI (cid:105) . (25)We have the simple prediction (cid:104)¬ XII , ¬ IXI (cid:105) ⇒ (cid:104)
XXI (cid:105) . (26)Clifford expansion now gives ϕ − SSS ϕ IHH (cid:101) ϕ CZ ϕ IHI (cid:101) ϕ CZ (cid:104) XXI (cid:105)⇔ ϕ − SSS ϕ IHH (cid:101) ϕ CZ ϕ IHI (cid:104)
XXZ (cid:105)⇔ ϕ − SSS ϕ IHH (cid:101) ϕ CZ (cid:104) XZZ (cid:105)⇔ ϕ − SSS ϕ IHH (cid:104)
XIZ (cid:105)⇔ ϕ − SSS (cid:104)
XIX (cid:105)⇔ (cid:104)
Y IY (cid:105) , (27)which gives the prediction (cid:104)¬ YY I , ¬ IYY (cid:105) ⇒ (cid:104)
Y IY (cid:105) . (28)The above prediction does not depend on the choice between ϕ CZ and (cid:101) ϕ CZ , although predictions in general might depend on the choice, as weshall see below. One could note that the choice does not affect Cliffordreduction, since the CZ is only applied to propositions that contain oneor more Z entries and single X entries.A second prediction from (cid:104)¬ YY I , ¬ IYY (cid:105) starts from (cid:104)
XXI , IXX (cid:105) ⇒ (cid:104)
XIX (cid:105) , (29)that can also be obtained as above. We now apply three (cid:101) ϕ CZ transfor-mations to reach (cid:104)¬ YY I , ¬ IYY (cid:105) ⇔ (cid:101) ϕ CZ (cid:101) ϕ CZ (cid:101) ϕ CZ (cid:104) XXI , IXX (cid:105)⇒ (cid:101) ϕ CZ (cid:101) ϕ CZ (cid:101) ϕ CZ (cid:104) XIX (cid:105) ⇔ (cid:104)¬
Y IY (cid:105) . (30)The two predictions (28) and (30) can hold jointly only if (cid:104) Y IY (cid:105) is al-ways indeterminate “?”, which in turn forces the left-hand side propo-sitions (cid:104)¬
YY I (cid:105) and (cid:104)¬
IYY (cid:105) to be incompatible. But there are situa-tions where we can predict (cid:104)