aa r X i v : . [ qu a n t - ph ] F e b Nonlocality with less Complementarity
Tobias Fritz ∗ ICFO – Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain (Dated: November 26, 2018)In quantum mechanics, nonlocality (a violation of a Bell inequality) is intimately linked to com-plementarity, by which we mean that consistently assigning values to different observables at thesame time is not possible. Nonlocality can only occur when some of the relevant observables donot commute, and this noncommutativity makes the observables complementary. Beyond quantummechanics, the concept of complementarity can be formalized in several distinct ways. Here wedescribe some of these possible formalizations and ask how they relate to nonlocality. We par-tially answer this question by describing two toy theories which display nonlocality and obey theno-signaling principle, although each of them does not display a certain kind of complementarity.The first toy theory has the property that it maximally violates the CHSH inequality, althoughthe corresponding local observables are pairwise jointly measurable. The second toy theory alsomaximally violates the CHSH inequality, although its state space is classical and all measurementsare mutually nondisturbing: if a measurement sequence contains some measurement twice with anynumber of other measurements in between, then these two measurements give the same outcomewith certainty.
I. INTRODUCTION
Since the groundbreaking work of Bell [3], it has beenrecognized that quantum theory cannot be completed toa theory with local hidden variables; this result is knownas
Bell’s theorem , or almost synonymously as quantumnonlocality . Bell’s theorem is usually proven by derivinginequalities for the correlations between observables lo-cated at spatially separated sites, which are satisfied forany theory having local hidden variables, but violatedby many quantum-mechanical models. See [12] or [5] forbackground on Bell’s theorem, including a clear overviewof properties of hidden variable theories and the preciseassumptions needed for the proof of Bell’s theorem.The fact that Bell inequalities can be violated in manyquantum-mechanical models stems from the fact that,due to non-commutativity, a simultaneous joint measure-ment of these observables is not possible; quantum ob-servables display complementarity . But what does thismean, exactly? And what happens beyond quantum me-chanics? From a theory-independent perspective, how dononlocality and complementarity relate? This is the kindof question we are concerned with in this paper.Recent work by Oppenheim and Wehner [9] hasbrought to light a rather general theory-independentquantitative relationship between uncertainty relationsand nonlocality. The absence of an uncertainty relationrules out nonlocality. They have also considered a certaindifferent notion of complementarity (our property (c))and noted that the absence of such complementarity alsorules out nonlocality.In this paper, we would like to define two other con- ∗ [email protected] In Section D of [9]. Note that Oppenheim and Wehner do notregard the existence of an uncertainty relation as complementar-ity. cepts of complementarity. In the particular case of quan-tum theory, both are equivalent to non-commutativityof observables. For each of these two notions of com-plementarity, we then ask whether there are no-signalingtheories which display nonlocality, although they do notdisplay this kind of complementarity. We answer thisin the positive by finding unphysical toy theories whichhave all the required properties.
II. WHAT IS COMPLEMENTARITY?
So what actually is complementarity? In particular,once we leave the framework of quantum mechanics, whatdoes it mean to say that observables A and B are com-plementary? Bohr [4] has coined this term and used it torefer to the practical impossibility of a joint measurementof A and B :“[. . . ] it is only the mutual exclusion ofany two experimental procedures, permit-ting the unambiguous definition of comple-mentary physical quantities, which providesroom for new physical laws, the coexistenceof which might at first sight appear irrecon-cilable with the basic principles of science. Itis just this entirely new situation as regardsthe description of physical phenomena, thatthe notion of complementarity aims at char-acterizing.”This reduces the problem of defining complementarityto the problem of defining joint measurability.Clearly, the concept of joint measurability should betaken to mean joint measurability in theory rather thanjoint measurability in practice ; for if it were to mean thelatter, than this would let the complementarity of A and B depend on the current state of the art in experimentand on the skill of the experimenter.One can attribute different operational meanings tothe concept of joint measurability of A and B . We willconsider the following four:(a) Joint distribution: there is an observable C fromthe measurement of which one can deduce both thevalue of A and the value of B (meaning that theprobability distribution over outcomes of C con-tains those of A and B as marginals).(b) Symmetric nondisturbance of measurements: inthe measurement sequence ABA , the two measure-ments of A always give the same result. Similarlyfor the sequence BAB .(c) Asymmetric nondisturbance of measurements [10,Sec. 7-4. C ]: on any initial state, the measurementsequence AB gives the same probability distribu-tion over outcomes of B as a direct measurementof B .(d) No uncertainty relation: there is no non-trivial un-certainty relation between the outcome distributionof A and that of B . This can be formalized in sev-eral inequivalent ways, e.g. in terms of entropic un-certainty relations [8] or in terms of the fine-graineduncertainty relations of [9]. Lemma 1.
In the case of quantum theory with projectivemeasurements, properties (a), (b) and (c) are equivalentto commutativity of A and B . This is straightforward to prove, but we neverthelessinclude a proof sketch for the sake of completeness.
Proof.
We work in terms of the spectral decompositions B = P λ ∈ sp( A ) λP λ and B = P λ ∈ sp( B ) λQ λ . Each P λ can be written as a polynomial in A , and each Q λ as apolynomial in B . So if A and B commute, then P λ Q µ = Q µ P λ for all λ, µ ; the converse is also clear.To see that property (a) follows from commutativity,choose an injection j : sp( A ) × sp( B ) ֒ → R and define C = P λ ∈ sp( A ) , µ ∈ sp( B ) j ( λ, µ ) P λ Q µ . By injectivity of j ,a measurement of C automatically also measures both A and B . Conversely, if C corresponds to a joint mea-surement of A and B as in property (a), then there arepolynomials f and g such that A = f ( C ) and B = g ( C ),and therefore A commutes with B .Property (b) and property (c) are each equivalent to P λ P λ Q µ P λ = Q µ and P µ Q µ P λ Q µ = P λ . The firstequation implies that P λ Q µ P ′ λ = 0 for λ = λ ′ . Therefore, P λ Q µ = X λ ′ P λ Q µ P λ ′ = X λ P λ Q µ P λ = X λ ′ P λ ′ Q µ P λ = Q µ P λ which implies commutativity of A and B . The same cal-culation also shows the converse implication. Despite this equivalence in the quantum case, beyondquantum theory these four notions of complementarityare conceptually very different. For example, proper-ties (a) and (d) do not care about any post-measurementstates under measuring either A or B , but only about thedistribution of outcome probabilities; for property (c),this only applies to measurement A ; while for prop-erty (b), the post-measurement states of both A and B are relevant. Further discussion on the interrelations be-tween these properties is beyond the scope of this work. III. NONLOCALITY REQUIRESCOMPLEMENTARITY?
It seems to be intuitive that complementarity is a nec-essary requirement for displaying nonlocality. One maybelieve that any no-signaling theory which does not dis-play complementarity admits a local realistic description.Indeed, this can be made precise for property (c) as fol-lows [9, D.1]: if property (c) holds for all pairs A , B ,then it follows that the outcome distribution of any ob-servable A does not depend on whether other A is mea-sured directly, or other observables are measured before A . This implies that one can assign a joint probabilitydistribution to all the observables at once, and hence theexistence of a local realistic model of the resulting cor-relations. For bounds on certain kinds of nonlocality interms of uncertainty relations as in property (d), we alsorefer to [9].One might now conjecture that also the properties (a)and (b), together with the no-signaling principle, are like-wise sufficient in order to exlucde nonlocal behavior.In the following, we will show that this is not nec-essarily so. In Section IV (resp. V), we will describe ano-signaling toy theory which has property (a) (resp. (b))for every pair of measurements A, B , but maximally vi-olates the Bell inequality of Clauser, Horn, Shimony andHolt [6] (CHSH inequality). These two toy theories aremore classical than quantum mechanics in the sense ofhaving less complementarity, but nevertheless display ahigher degree of nonlocality.There are several things that should be kept in mindwhile reading this paper. First and most importantly,our constructions are toy theories , which means that wedo not ascribe any physical significance to them. In fact,they can easily be seen to be unphysical. Any toy theorydisplaying nonlocality with “less” complementarity thanquantum mechanics is necessarily unphysical—precisely because of the very feature of displaying less complemen-tarity than quantum mechanics, which is our current the-oretical framework for (microscopic) physics. Therefore,the present investigations should not be regarded as ac-tual physics in the sense of describing reality, but ratheras theoretical investigations around the foundations ofBell’s theorem.Second, the systems which make up our toy theoriesare nonlocal systems, in the sense that their states andobservables are only defined globally in the whole “toyuniverse”, and not locally. This is just like in quantummechanics, where wave functions are likewise nonlocal en-tities. For simplicity, our small toy universe only consistsof two independent worldline segments called “Alice” and“Bob”, which we take to be spacelike separated.Third, what we regard as a “theory” comprises the def-inition of both states and measurements. While in someframeworks, the set of measurements may be determinedby the set of states (and/or vice versa), we do not assumethis to be the case. Our toy theories can be formalizedin the language of generalized probabilistic theories andthen satisfy the Assumptions 1, 2, 3 and 7 of [2], butnot its Assumptions 4, 5 and 6. See also footnote 4 andSection VIII.A in [2].Fourth, while our toy theories are obviously tailoredto achieve maximal nonlocality in the CHSH scenario, itmakes perfect sense to consider them independently ofany considerations involving Bell inequalities or nonlo-cality, and in particular independently of any particularBell scenario.Sections IV and V can be read independently.
IV. NONLOCALITY WITH JOINTLYMEASURABLE OBSERVABLES
We now describe the first toy theory which maximallyviolates the CHSH inequality by reproducing the corre-lations of the Popescu-Rohrlich box (PR-box [11], Ta-ble I), but nevertheless satisfies property (a) for all rel-evant pairs of observables. The basic idea is that thePR-box correlations between Alice’s observables A , A and Bob’s observables B , B do not prevent A and A from having a joint distribution; and likewise for B and B . (What it does prevent is the existence of a jointdistribution for all four observables together [7].) More-over, for each Alice and Bob one can define an observablewhich probes this joint distribution.We will define the state space of the toy theory in termsof four “basic observables” A , A , B , B . We assumethese four observables to be ± P A ,A ( a , a ) , P A ,B ( a , b ) , P A ,B ( a , b ) ,P A ,B ( a , b ) , P A ,B ( a , b ) , P B ,B ( b , b ) , (1)satisfying the marginal conditions X y P X,Y ( x, y ) = X y P X,Y ′ ( x, y ) A A +1 − − B +1 − B +1 −
12 12 TABLE I. Table of joint outcome probabilities for thePopescu-Rorhlich box. for all observables
X, Y, Y ′ ∈ { A , A , B , B } and anyoutcome x . When X is an observable of Alice and Y, Y ′ are Bob’s, or vice versa, these are precisely theno-signaling equations.Since the set of all these states is convex, it is automati-cally closed under probabilistic mixtures. In the abstractframework of [1], this state space is the set of empiri-cal models over four measurements A , A , B , B withmeasurement contexts precisely all the pairs of measure-ments.This definition should make it clear that we wouldlike to regard the four basic observables A , A , B , B as measurements which can all be performed pairwisejointly, but not triplewise jointly. Following this idea, theset of observables of the toy theory is defined to consistof the basic observables together with the four-outcomeobservables representing pairwise joint measurements.The physical picture behind joint measurability of allobservable pairs is as follows: there is a source of entan-gled particle pairs which distributes them to Alice’s andBob’s labs, respectively. In Alice’s lab, she can choose tomeasure either A or A ; in Bob’s lab, he can choosebetween B and B . Moreover, as an additional op-tion, there is a third experimenter, who has access to thesource of entangled pairs; let us call him Charlie. Char-lie can decide to let an entangled pair pass by and reachthe labs of Alice and Bob; or he can decide to applya measurement of one of two four-outcome observables C A and C B . In this latter case, the entangled pair getsdestroyed due to the measurement process. The threeexperimenters find out that the observable C A behaveslike a joint distribution of A and A , while C B behaveslike a joint distribution of B and B , in the sense ofproperty (a).This toy theory has the following features: • Among the basic observables A , A , B , B , allpairs are jointly measurable in the sense of prop-erty (a); • The theory contains states which display PR-boxcorrelations: defining a state through the joint dis-tributions of Table I together with any joint distri-bution of A and A with uniform marginals, andlikewise between B and B , gives the desired re-sult.We conclude that joint measurability of local observ-ables together with the no-signaling principle is not suffi-cient to guarantee the existence of a local hidden variabledescription. V. NONLOCALITY WITH MUTUALLYNONDISTURBING MEASUREMENTS
We will now define our second toy theory in terms of afinite number of states and observables. Similar to before,the observables are all ± A , A , B , B ; we will think of A and A asobservables located on the subsystem “Alice”, while B and B are observables on the subsystem “Bob”. Themodel will have the following properties:(a) All measurements are perfectly mutually nondis-turbing: sequential measurements of the same ob-servable always give the same value, no matterwhich other measurements were conducted in themeantime.(On the other hand, for each measurement, thereare some states on which the outcome is random.Post-measurement states are in general differentfrom pre-measurement states.)(b) Signaling is impossible: the outcome probabilitydistribution of any measurement sequence of Bobdoes not depend on the actions of Alice, and viceversa.(c) The correlations between A i and B j constitute aPopescu-Rohrlich box [11], thereby maximally vio-lating the CHSH inequality [6].Before diving into the details, we describe a differentmodel which is operationally almost equivalent to ourtoy theory. Suppose that Alice and Bob share a PR-box, and in addition each party has a memory of twobits available. The purpose of Alice’s two bits is to storeany previously measured values of A and A , and like-wise for Bob’s two bits. Now upon the first measurementcarried out on each subsystem, the PR-box is used, andthe measurement outcome is stored in the correspond-ing bit, while the other local bit is initialized randomlyand independently. All subsequent measurements onlyreproduce the values of the corresponding local bit. Thissimple protocol already is a model having property (b)while displaying PR-box correlations.The following toy theory is essentially an elaborationon this idea. It has the additional feature of having aclassical state space. Also, it does not artificially distin-guish the first measurement from subsequent ones. due to Matthew Pusey. outcome − A a ⊖ a ∅ b − b − a ⊕ a ∅ b +1 b +2 A a ∅ a ⊖ b − b +2 a ∅ a ⊕ b +1 b − B a − a − b ⊖ b ∅ a +1 a +2 b ⊕ b ∅ B a − a +2 b ∅ b ⊖ a +1 a − b ∅ b ⊕ TABLE II. Table of post-measurement states for the modeldescribed in Section V with pre-measurement state a ∅ a ∅ b ∅ b ∅ .For each observable, each outcome has probability 1 /
2, andthe post-measurement state is given by the table entry corre-sponding to that outcome.
A pure state in the toy theory is defined to be a formalexpression of the form a j a k b l b m , where each of the upperindices is an element of the set of hidden values V = {∅ , − , + , ⊖ , ⊕} . An example of a pure state is a +1 a ∅ b ⊖ b − . The elements of V correspond to properties of the associated observableas follows. When the state is of the form a ∅ . . . , then wesay that the hidden value of the observable A is unde-termined ; on states of the form a − . . . , we say that A has a potential value of −
1, while on those which looklike a +1 . . . , the observable A has a potential value of+1. On states like a ⊖ . . . , we associate to A an actualvalue of −
1, while on a ⊕ . . . , the observable A has anactual value of +1. The same applies to the observables A , B and B . We will soon explain the significance ofundetermined values, potential values and actual values.A mixed state is defined to be a probabilistic ensembleof pure states. This means that the theory has a classi-cal state space in the sense that every state is a unique probabilistic mixture of pure states. We will mostly workon the level of pure states rather than ensembles; whennot explicitly mentioned otherwise, “state” means “purestate”.We now need to define the measurements. Sincenow we are interested in measurement sequences, wealso need to consider post-measurement states. There-fore, a measurement definition consists of assignments ofprobabilities for each outcome on each pre-measurementstate, as well as a (probabilistic) assignment of post-measurement state for each combination of outcome andpre-measurement state which occurs with non-zero prob-ability. We take these data to be given as follows:(a) The outcome probabilities for any observable aredetermined by the hidden value of that observable.If this value is undetermined, both outcomes occurequally likely with probability 1 /
2; otherwise, thepotential value or actual value is reproduced withcertainty.(b) The post-measurement states are defined by spec-ifying how the hidden values of the observableschange. For all pre-measurement states except a ∅ a ∅ b ∅ b ∅ , we define this to work as follows. Thehidden value of the observable always turns intothat actual value which corresponds to the mea-surement outcome. All other hidden values stayuntouched, except when the other observable of thesame party has a potential value; in this case, thispotential value flips its sign with a probability of1 / a ∅ a ∅ b ∅ b ∅ are as inTable II.For example, on the pre-measurement state a +1 a ∅ b ⊖ b − ,a measurement of A gives a definite +1 outcome andyields the post-measurement state a ⊕ a ∅ b ⊖ b − ; a measure-ment of A gives a random outcome; if this outcome is+1, then the post-measurement state is randomly cho-sen between a +1 a ⊖ b ⊖ b − and a − a ⊖ b ⊖ b − ; if it is −
1, thenbetween a +1 a ⊕ b ⊖ b − and a − a ⊕ b ⊖ b − . Measuring B givesa definite − a +1 a ∅ b ⊖ b − and a +1 a ∅ b ⊖ b +2 are equally likely. Fi-nally, a measurement of B has a definite − a +1 a ∅ b ⊖ b ⊖ .This ends the definition of the model. We can nowturn to the study of its properties.One characteristic property of classical systems is thatmeasurements do not change the state of the system.This is not always the case in our model; for example,when the pre-measurement state is x = a ∅ a ∅ b ∅ b ∅ , thenthe post-measurement is always different from x , no mat-ter what the measurement and its outcome are. Never-theless, the model has several nice properties which makeit look classical in some respects. One of these is the fol-lowing: Lemma 2.
For any pre-measurement state, all measure-ments in this model are mutually nondisturbing: if . . . , A i , . . . , A i , . . . is any sequence of measurements containing A i at leasttwice, then these two measurements of A i have the sameoutcome with probability . Likewise for B j .Proof. By definition of post-measurement states, afterthe first measurement of A i the state has an actual valueon a i , and this value never changes in any subsequentmeasurement. Hence all further measurements of A i willreproduce this value with certainty. Proposition 3.
The model does not allow signaling inthe following sense: for any initial state, the outcomedistribution of any measurement sequence of Bob doesnot depend on how many and which measurements Aliceconducts in the meantime, and conversely.Proof.
On any initial state which is not a ∅ a ∅ b ∅ b ∅ , this isclear since the measurement rules are “local” in the sensethat measuring A or A can only change the hiddenvalues of a and a , and likewise for B or B .On the initial state a ∅ a ∅ b ∅ b ∅ , one can reason as fol-lows. We may assume without loss of generality that Bob’s measurement sequence contains both B and B at least once. Then due to lemma 2, it is actually enoughto assume that he measures both of them exactly once.Then we claim that all four outcome sequences ( ± , ± / A , and perfectly anticorrelatedif she started with A . In either case, this correlationgets erased due to Bob’s first measurement, which mayflip one of the two potential values, so that all four pos-sible sign combinations are equally likely, independentlyof what Alice did.For this last part of the proof, the distinction betweenpotential values and actual values is crucial: if we wouldregard all potential values as actual values which neverchange in any measurement, then signaling would be pos-sible since Bob could measure whether B and B are per-fectly correlated or perfectly anticorrelated, and therebyhe would know whether Alice measured A or A . Proposition 4.
On the initial state a ∅ a ∅ b ∅ b ∅ , the cor-relations between A i and B j coincide with those of thePR-box (Table I).Proof. This can be directly checked from the rules spec-ified above together with Table II, which has been con-structed precisely in such a way that the correlationsof the Popescu-Rohrlich box (Table I) can be repro-duced.
Reply to potential criticism.
There are several defi-ciences which one may deem this toy theory to have andwhich have been pointed out to us in discussion. Wewould like to address some of these now.First, one may remark that the theory is unphysical:it has almost trivial dynamics and contradicts quantummechanics. This is certainly correct, and it ought to bekept in mind that this is the whole point of this investiga-tion: how classical can a theory be while still displayingnonlocality?Second, one may object that the theory is very un-natural: the state a ∅ a ∅ b ∅ b ∅ is a state that can neverbe prepared within the theory starting from any otherstate. While this is correct, one can easily amend the the-ory by extra bells and whistles which remedy this prob-lem and similar ones. For example, one can add a newmany-outcome observable X which randomly preparesany state and whose outcome is the numerical represen-tation of this prepared state.Third, one may wonder what would happen when Al-ice and Bob measure simultaneously? It seems that jointmeasurements have not been defined in the theory. Butthanks to Proposition 3, this is irrelevant. If Alice mea-sures A i and Bob simultaneously measures B j , one candescribes this in the toy theory as either the measure-ment sequence A i B j or as the measurement sequence B j A i . While these two time orderings are inequivalentas transformations on states, Proposition 3 implies thatthe observational predictions coincide. VI. DISCUSSION
The notion of complementary in fundamental physicsrefers to the phenomenon that to some pairs of observ-ables, it is impossible to consistently assign values to bothof them at the same time. It is generally believed that ano-signaling theory without complementary observablescannot display nonlocality. Along these lines, a quan-titative relationship between uncertainty relations andcertain kinds of nonlocality has been worked out in [9].In the work, we have pointed out that nonlocality doesnot necesssarily require complementarity.In Section II we have discussed several ways to for-malize the notion of joint measurability as properties (a)to (d), and noted that a no-signaling theory with prop-erty (c) or (d) for all pairs of observables always allowslocal hidden variables and therefore cannot display non-locality. In Section IV, we have discussed a simple toyexample of a generalized probabilistic theory which maxi- mally violates the CHSH inequality by displaying PR-boxcorrelations, although the corresponding local observ-ables are jointly measurable in the sense of property (a).In Section V, we have described a no-signaling toy theorywhich also displays PR-box behavior, although its statespace is classical and it satisfies property (b) for all pairsof observables.While we expect our results to be valid not only formaximal violations of the CHSH inequality, but moregenerally for any no-signaling violation of any Bell in-equality, we have not considered these more general casesso far. For the sake of maximal concreteness, we havepreferred to consider a specific example only.
ACKNOWLEDGEMENTS
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