Consequences and applications of the completeness of Hardy's nonlocality
aa r X i v : . [ qu a n t - ph ] F e b Consequences and applications of the completeness of Hardy’s nonlocality
Shane Mansfield ∗ School of Informatics, University of Edinburgh, Informatics Forum,10 Crichton Street, Edinburgh EH8 9AB, United Kingdom (Dated: November 5, 2018)Logical nonlocality is completely characterised by Hardy’s “paradox” in ( , , l ) and ( , k , ) scenarios. Weconsider a variety of consequences and applications of this fact. (i) Polynomial algorithms may be given fordeciding logical nonlocality in these scenarios. (ii) Bell states are the only entangled two-qubit states whichare not logically nonlocal under projective measurements. (iii) It is possible to witness Hardy nonlocalitywith certainty in a simple tripartite quantum system. (iv) Non-commutativity of observables is necessary andsufficient for enabling logical nonlocality. I. INTRODUCTION
Since the fundamental insight of Bell [1, 2], it is known thatquantum mechanics gives rise to stronger-than-classical, non-local correlations. Under seemingly natural assumptions oflocality and realism, it can be shown that any empirical cor-relations should satisfy certain Bell inequalities, which can beviolated quantum-mechanically, from which Bell’s conclusionfollows.A more intuitive, logical approach to nonlocality proofswas pioneered by Heywood and Redhead [3], Greenberger,Horne, Shimony and Zeilinger [4, 5] and Hardy [6, 7] [8].This kind of nonlocality proof disregards the precise valuesof the probabilities for the various outcome events and onlyrefers to events as being possible (with probability greaterthan zero) or impossible (having probability zero). This turnsout to be sufficient for demonstrating nonlocality in quantummechanics. We refer to these as logical nonlocality proofs.Probabilistic nonlocality, as witnessed by violations of Bellinequalities and logical nonlocality are the first two levels ofa qualitative hierarchy of nonlocality introduced in [9] [10],the highest level of which is strong nonlocality, which ariseswhen even at the level of possibilities the model cannot befactored into a local and a nonlocal part.We work within a general framework, introduced in [11],for logical nonlocality proofs in ( n , k , l ) scenarios —i.e., Bell-type scenarios in which n is the number parties or sites, k is themaximum number of measurement settings available at eachsite, and l is the maximum number of potential outcomes forthese measurements. Our framework bears some similarity tothe relational hidden variable framework of Abramsky [12], aswell as a combinatorial framework due to Degorre and Mhalla[13], and while not as general could be considered a precur-sor to the unified sheaf-theoretic [9] and combinatorial [14]frameworks for nonlocality and contextuality [15]. The ad-vantage of the present framework is that it comes with a par-ticular representation for n = n = empirical models ; i.e., probabilityor possibility tables for the various joint outcomes in a givenscenario. ∗ smansfi[email protected] Hardy’s logical nonlocality proof or “paradox” [6, 7] is of-ten considered to be the simplest of all quantum mechanicalnonlocality proofs. In [11], the author and Fritz proved com-pleteness results which establish that Hardy’s paradox is anecessary and sufficient condition for logical nonlocality inall ( , k , ) and ( , , l ) scenarios (thereby subsuming all otherlogical nonlocality proofs or “paradoxes” in these scenarios).For the ( , , ) [11] and ( , , ) [16] scenarios, it is knownthat this no longer holds.In this article, we explore a variety of consequences andapplications of the completeness of Hardy nonlocality. To be-gin with, we will see that in the relevant scenarios they leadto explicit algorithms for deciding logical nonlocality whichare polynomial in l and k . They also lead to a constructiveproof that the Popescu-Rohrlich box [17] is the only stronglynonlocal ( , , ) empirical model.Next, we obtain a proof that Bell states are not logicallynonlocal under projective measurements. Surprisingly, theseare the only entangled qubit states with this property: allother entangled two-qubit states have been shown to admit aHardy paradox [7], and all entangled n -qubit states have alsobeen shown to be logically nonlocal [18], both via appropriatechoices of local projective measurements. In this sense, theBell states are anomalous in the landscape of entangled states,in spite of the fact that they are among the most studied andutilised of these.Much of the literature on Hardy’s paradox is concernedwith the paradoxical probability ; i.e., the probability of wit-nessing the particular outcome event from which the logicalargument follows. This is often considered to be an indica-tor of the quality of Hardy nonlocality. For Hardy’s fam-ily of quantum mechanical, nonlocal empirical models, themaximum paradoxical probability that can be achieved is ( √ − ) / ≈ .
09. It has been shown, however, that itis possible to achieve a paradoxical probability of 0 .
125 for ageneralised version of Hardy’s paradox in a tripartite quantumsystem [19], and it has also been shown that a “ladder” versionof Hardy’s paradox, which allows k measurement settings toeach party, can give rise to a higher paradoxical probabilitywhich approaches 0 . k → ∞ .More recently, Chen et al. found that another generalisa-tion of Hardy’s paradox can be witnessed with probability ≈ . et al. paradox contains within it many different Hardy para-doxes. Moreover, we will see that their “paradoxical prob-ability” might more accurately be described as the sum ofthe paradoxical probabilities for these Hardy paradoxes, allof which occur within the one model.Using the completeness of Hardy nonlocality we willachieve a rather comprehensive improvement on these results,demonstrating by a much simpler argument that if such a sum-ming of paradoxical probabilities is considered, it is possi-ble to witness Hardy nonlocality with certainty for a tripar-tite quantum system. Interestingly, the argument relies onthe same state and measurements as the Greenberger-Horne-Zeilinger (GHZ) experiment [5]. We also show that Hardynonlocality can be achieved with certainty for a particularnon-quantum, no-signalling ( , , ) model, which turns outto be the Popescu-Rohrlich no-signalling box [17].Moreover, the notion of witnessing logical nonlocality withcertainty corresponds precisely to the notion of strong nonlo-cality, the highest level in the qualitative hierarchy of nonlo-cality (the hierarchy also applies more generally to contextu-ality) introduced in [9].Finally, we employ the completeness results in order toprove that incompatibility of observables is necessary and suf-ficient for logical nonlocality, thus extending to the logicalsetting a result due to Wolf, Peres-Garcia and Fernandez [21]which establishes that incompatibility is necessary and suffi-cient for (probabilistic) nonlocality. II. LOGICAL NONLOCALITY AND HARDY PARADOXES
The possibility table (or possibilistic empirical model ) usedfor the original Hardy nonlocality proof, Table I (a), concernsthe ( , , ) scenario. Each of the two parties can make oneof two measurements on their subsystem, giving rise to out-comes which we label here {↑ , ↓} for the first measurementand { R , G } for the second. A 1 in the table signifies that itis possible (with probability greater than zero) to obtain thecorresponding joint outcome, and a 0 signifies that it is notpossible. The precise probabilities of obtaining the variousjoint outcomes are not required to prove the nonlocality of themodel. Any probabilistic empirical model can be transformedinto a possibilistic empirical model of this kind in a canonicalway via possibilistic collapse [9, 11]: the process by which allnon-zero probabilities are conflated to 1, with zero probabili-ties mapping to 0. Definition II.1.
Any empirical model which is nonlocal at thelevel of its possibilistic table is said to be logically nonlocal . Proposition II.2 ([11]) . A possibilistic empirical model is(logically) nonlocal if and only if it cannot be realised as aunion of local deterministic models; or, equivalently, if thereexists a in its possibility table which cannot be completed toa deterministic grid. Local deterministic models are empirical models for whichthe outcome at each site is determined uniquely by the mea-surement at that site, and in the tabular representation take the TABLE I: (a) A possibilistic empirical model containing aHardy paradox. This is a possibility table in which 1 denotes“possible” and 0 denotes “impossible”. The blank entries arenot relevant and may each take either of the values. (b) A“deterministic grid” or local deterministic model.BobAlice ↑ ↓
R G ↑ ↓ R G ↑ ↓ R G ↑ ↓ R G deterministic grids ; e.g. Table I (b). Deterministicgrids correspond to global sections of the event sheaf in thesheaf-theoretic approach [16], and indeed logical nonlocalityis a special case of the general notion of contextuality as con-sidered in [9], which is also proved there to be equivalent tothe failure of a model to be realisable by a factorisable hiddenvariable model.In the case of the Hardy paradox, it is clear that the 1 in Ta-ble I cannot be completed to a deterministic grid, regardless ofthe unspecified entries. However, depending on the scenario,this is just one way in which a model might exhibit nonlocalityat the possibilistic level [11, 16]. Definition II.3.
Up to re-labelling of measurements and out-comes, any possibilistic ( , , ) empirical model containingthe arrangement of 1’s and 0’s shown in Table I (a) is said to contain a Hardy paradox (i.e., it admits Hardy’s logical nonlo-cality proof) and we say that the joint outcome ( ↑ , ↑ ) witnessesHardy nonlocality .Definition II.3 defines Hardy nonlocality for ( , , ) sce-narios. It is also possible to extend the definition to ( , , l ) scenarios simply by course-graining outcomes; see Table II.Furthermore, one may define Hardy nonlocality in ( , k , l ) models as arising whenever some 2 × k measurements at each site)contains a Hardy paradox; see Table III. Definition II.4.
Any possibilistic ( , k , l ) empirical modelcontaining a 2 × (coarse-grained)Hardy paradox .Wang and Markham have described a generalisation ofHardy’s logical nonlocality proof to ( n , , ) scenarios, whichthey have used to demonstrate that all symmetric n -partitequbit states for n > ( , , l ) scenario with a coarse-grained Hardyparadox. o ′ · · · o ′ l o · · · o m o m + · · · o l o ′ · · · o ′ l o ... o m · · · · · · o m + ... o l ( , k , ) scenario containing a Hardy paradox.1 · · · · · · p ( o | m ) = o when joint mea-surement m is made, and p ( o | m ) = ( n , , ) scenarioswe also let measurements and outcomes both be labelled by { , } at each site, though note that these 0’s and 1’s simplyplay the role of labels. Definition II.5.
For any ( n , , ) scenario, an n-partite Hardyparadox occurs if (up to re-labelling of measurements and out-comes) the following possibilistic conditions are satisfied.• p ( , . . . , | , . . . , ) = p ( π ( , , . . . , ) | π ( , , . . . , ) ) = π ∈ S n • p ( , . . . , | , . . . , ) = n = n = n = ( , k , l ) scenariosa possibilistic empirical model is local if and only if every1 in its table can be completed to a deterministic grid. Thischaracterisation generalises in the obvious way to the three-dimensional representation for n = ( , , ) model containing this arrangementof 1’s and 0’s, or red and blue boxes, is logically nonlocal.It is known that Hardy nonlocality completely characteriseslogical nonlocality in a variety of scenarios. The followingtheorem combines the completeness results of [11]. Theorem II.6 (Mansfield and Fritz [11]) . For any ( , k , ) or ( , , l ) scenario, an empirical model is logically nonlocal ifand only if it contains a (coarse-grained) Hardy paradox. We also rephrase the definition of strong nonlocality as in-troduced in [9] within the present framework.
Definition II.7.
An empirical model is strongly nonlocal ifand only if no < Hardy < logical < strong , (1)where membership of any of these classes implies member-ship of all lower classes. At the lowest level, a model is prob-abilistically nonlocal if and only if it violates some Bell in-equality. The hierarchy is in general strict: for each class, em-pirical models can be found which do not belong to any higherclass. For measurement scenarios in which Theorem II.6 ap-plies, however, the Hardy and logical classes coincide. III. COMPLEXITY OF LOGICAL NONLOCALITY
Theorem II.6 is relevant to the computational complexity ofdeciding logical nonlocality in ( , , l ) and ( , k , ) scenarios,where it is equivalent to deciding whether a Hardy paradoxoccurs. The fact was mentioned in [11]; here we find explicitpolynomial algorithms. Proposition III.1.
Polynomial algorithms can be given fordeciding nonlocality in ( , , l ) and ( , k , ) models.Proof. For ( , k , ) scenarios, deciding whether a model in thetabular form is local or nonlocal simply amounts to checkingall 2 × (cid:0) k (cid:1) sub-tables, which is O ( k ) . For ( , , l ) scenarios, one has tocheck each 1 in the table to see whether it can be completedto a deterministic grid. There are 4 l entries in the table, andeach check is O ( l ) , so again we have an algorithm that ispolynomial in the size of the input.It was conjectured in [11] that decidability of logical non-locality with k as the free input is NP-hard when n > , l ≥ n ≥ , l >
2, as is known to be the case for probabilisticmodels [25]. The problem was shown to be NP by Abramskyin [12], and the it has since been proved to be NP-completeby Abramsky, Gottlob and Kolaitis [26]. This gives strongreason to suspect that it is not possible to obtain a classifica-tion of conditions that are necessary and sufficient for logicalnonlocality in full generality.
IV. STRONG NONLOCALITY AND THE PR BOX
Recall from (1) that strong nonlocality is strictly strongerthan logical nonlocality. Theorem II.6 can be used to givea constructive proof of a result originally proved by case-analysis by Lal [9, 27] that the only strongly nonlocal ( , , ) models are the Popescu-Rohrlich no-signalling boxes [17],whose probability table up to re-labelling is given in Ta-ble VII. Proposition IV.1.
The only strongly nonlocal no-signalling ( , , ) models are the PR boxes. Before we prove this proposition, recall that by TheoremII.6 strong nonlocality is equivalent to the property that every1 in its possibility table witnesses a Hardy paradox. We willsimply use this property together with the requirement thatthe model satisfies no-signalling to derive our result. An il-lustration of no-signalling in the possibilistic sense is the fol-lowing. We see from Table I (a) that if Alice and Bob eachmake their {↑ , ↓} measurement then it is possible for Alice toobtain the outcome ↑ . Now in order to make sure that Bobcannot instantaneously signal to Alice who is assumed to bespacelike separated from him it must be the case that it wouldalso be possible for Alice to obtain the outcome ↑ had Bobmade his { R , G } measurement. We can therefore deduce that TABLE IV: Stages in the proof of proposition IV.1.1 1 011 0 10 1 1 1 0 1 00 0 11 0 0 10 1 1 1 0 1 00 1 0 11 0 0 10 1 1 0(a) (b) (c)since the event ( ↑ A , G B ) is not possible the event ( ↑ A , R B ) mustbe possible. More generally in the tabular representation, no-signalling translates to the condition that whenever a 1 occursin a table, the outcome row and column the event belongs tomust each contain at least one 1 per measurement setting, forotherwise the possibility of witnessing a particular outcomefor one party could depend on the measurement choice of theother (see [11] for a more detailed discussion). Proof.
For any choice of measurements there must be some possible outcome. This possible assignment is represented bya 1 in the table, and it must witness a Hardy paradox. After re-labelling as necessary, we can represent the model as in TableI (a). For this to be a no-signalling model, it is necessary tofill in 1’s as in Table IV (a). Using the fact that the 1’s in thelower-right box must also witness Hardy paradoxes, we mustfill in 0’s as in Table IV (b). By no-signalling, the remainingunspecified entry in the upper-left box must be a 1, and bythe fact that it must witness a Hardy paradox, the remainingentry in the lower-right box must be a 0. We thus arrive atTable IV (c), and the unique no-signalling probabilistic em-pirical model whose possibility table has this form is the PRbox.
V. BELL STATE ANOMALY
It is known how to prescribe projective measurements foralmost all entangled two-qubit states such that the resultingempirical model will contain a Hardy paradox [7], the excep-tion being the maximally entangled states; i.e., the familiarBell states. This naturally raises the question of whether thereexist any projective measurements that can be chosen for themaximally entangled states such that the resulting empiricalmodel contains a Hardy paradox. Indeed, in light of Theo-rem II.6 we know that this is equivalent to asking whether themaximally entangled states are logically nonlocal under pro-jective measurements. Some previous failed attempts at find-ing a logical nonlocality proof for the Bell states are describedin [28].We answer this question in the negative, and show that noprojective measurements can be chosen that lead to a Hardyparadox (and thus logical nonlocality). A result showing thatif the same pair of local measurements are available at eachqubit then it is impossible to realise a Hardy paradox wasproved independently by Abramsky and Constantin [29], butthe theorem we are about to present holds for any number ofmeasurements per qubit, and without the restriction that thesame set of local measurements be available at each qubit.In fact, Bell states are the only entangled n -qubit states, forany n , which are not logically nonlocal under projective mea-surements, since for n > n -qubit entangled states whichgive rise to logical nonlocality [18]. In this sense, despite be-ing among the most studied and utilised states in the fieldsof quantum information and computation, the Bell states areactually anomalous in the landscape of entangled states. Theorem V.1.
Bell states are not logically nonlocal underprojective measurements.Proof.
We prove the statement for the Bell state (cid:12)(cid:12) φ + (cid:11) = √ ( | i + | i ) . Since all other maximally entangled states are equivalent tothis one up to local unitaries, which can easily be incorpo-rated into the local measurements, the proof will extend to allmaximally entangled states.Any quantum mechanical empirical model obtained bymaking local projective measurements on | φ + i will necessar-ily give rise to a ( , k , ) model. By Theorem II.6 we knowthat Hardy’s paradox completely characterises logical nonlo-cality for such scenarios, and that logical nonlocality wouldtherefore imply the occurrence of a Hardy paradox in some ( , , ) sub-model. It therefore suffices to show that for anyobservables A and A for the first qubit and B and B forthe second qubit the resulting model does not contain a Hardyparadox.The + − | i i = cos θ i | i + e i φ i sin θ i | i| i i = sin θ i | i + e − i φ i cos θ i | i where { ( θ i , φ i ) } i ∈{ , , , } label the coordinates of the + (cid:10) j k | φ + (cid:11) = √ (cid:18) cos θ j θ k + e − i ( φ j + φ k ) sin θ j θ k (cid:19)(cid:10) j k | φ + (cid:11) = √ (cid:18) cos θ j θ k + e − i ( φ j − φ k ) sin θ j θ k (cid:19)(cid:10) j k | φ + (cid:11) = √ (cid:18) sin θ j θ k + e i ( φ j − φ k ) sin θ j θ k (cid:19)(cid:10) j k | φ + (cid:11) = √ (cid:18) sin θ j θ k + e i ( φ j + φ k ) cos θ j θ k (cid:19) where j ∈ { , } and k ∈ { , } . We see that (cid:10) j k | φ + (cid:11) = e − i ( φ j + φ k ) (cid:10) j k | φ + (cid:11) and (cid:10) j k | φ + (cid:11) = (cid:10) j k | φ + (cid:11) for each TABLE V: Stages in the proof of Theorem V.1.1 11 1 1 11 11 11 1 1 1 00 1 11 11 1 0 1 B B A A p ( | AB ) = p ( | AB ) , (2) p ( | AB ) = p ( | AB ) . (3)Note that the PR box (Table VII), which we know fromProposition IV.1 to be the only strongly nonlocal ( , , ) model (up to re-labellings), satisfies these symmetries. How-ever, it is also known that the PR box is not quantum-realisable[17, 30], so while it satisfies the symmetries it neverthelesscannot be realised by measurements on | φ + i .Next, we show that there is a unique possibilistic ( , , ) model (up to re-labelling) which satisfies the symmetries (2)and (3) and is logically but not strongly nonlocal. If a modelis not strongly nonlocal then there exists at least one globalassignment compatible with the model, or in tabular form atleast one deterministic grid. Up to re-labelling this is repre-sented in Table V (a). By the symmetry (3) there must exist asecond global assignment, as in Table V (b). It is clear fromthe configuration of the table that none of the entries that havealready been specified can witness a Hardy paradox. If themodel is logically nonlocal, therefore, at least one of the un-specified entries in Table V (b) must witness a Hardy paradox.Up to re-labelling, this can be represented as in Table V (c).By the symmetry (2) the table must be completed to Table V(d). This (up to re-labelling) is the only possibilistic empiricalmodel that respects the symmetries and is logically nonlocalwithout being strongly nonlocal. The question now is whetherit can be realised by measurements on | φ + i .Consider the measurement statistics for the joint measure-ment A B required by Table V (d). If these are to arisefrom quantum observables A and B , then h φ + | i = h φ + | i = √ and h φ + | i = h φ + | i =
0. So, ei-ther | i = | i = | i and | i = | i = | i up to an overallsign or vice versa. The eigenvectors of both observables are {| i , | i} , so they must simply be Pauli X operators (up toa common sign, which would allow for re-labelling the out-comes): A = B = ± X . (4)A similar argument applies for the joint measurements A B and A B , showing that A = B = ± X , (5) A = B = ± X . (6)Eqs. (4)–(6) imply that A = A = B = B = ± X ;but therefore the measurement statistics for A B must be thesame as for each of the other joint measurements, and Ta-ble V (d) is not realised. This completes the proof that noquantum mechanical logically nonlocal empirical model canbe obtained by considering (any number of) local projectivemeasurements on the Bell state.Symmetry is important here: the symmetry of the underly-ing state manifests itself as a symmetry of the probabilities ofoutcomes for each joint measurement, (2) and (3). By The-orem II.6, logical nonlocality also requires a particular rela-tionship between certain probabilities in each of these dis-tributions (a Hardy paradox). However, quantum mechani-cally, there cannot exist local projective measurements thatrealise these correlations and respect the symmetries at thesame time. On the other hand, there exists a whole familyof no-signalling empirical models which are logically nonlo-cal and respect the symmetries. These are the no-signallingmodels with support as in Table V (d), along with the PR box.These models have some interesting properties in their ownright [16]: despite not being realisable quantum mechanically,they may lie within the Tsirelson bound, coming arbitrarilyclose to the local polytope. They can be seen, however, toviolate information causality, which has been proposed as aphysical principle that might characterise quantum correla-tions [31] or “almost quantum” correlations [32], by meansof the same protocol described in [31]. In fact, similar fami-lies of models to this one have already been considered in thiscontext in [33].We also note that Fritz [34] has considered quantum ana-logues of Hardy’s paradox. These are not realisable quantummechanically, but can arise in more general no-signalling em-pirical models. An interesting point is that Table V (d) con-tains two such paradoxes, and so the fact that any model withthis support is not quantum-realisable also follows more di-rectly from this observation. VI. HARDY SUBSUMES OTHER PARADOXES
An immediate consequence of Theorem II.6 is that in therelevant scenarios Hardy’s paradox subsumes all other para-doxes, in the sense that any model which can be demonstratedto be logically nonlocal necessarily contains a Hardy paradox.For instance, the ladder paradox [35] has been proposed as ageneralisation of the original Hardy paradox and was used forexperimental tests of quantum nonlocality [36]. Up to symme-tries, there is one ladder paradox for any number of settings k ; i.e., for each ( , k , l ) scenario. It was observed in [11] that,by Theorem II.6, any ladder paradox necessarily contains a TABLE VI: The Chen et al. paradox occurs when at leastone of the starred entries is non-zero. The relevant outcomesfor each joint measurement are either those above or thosebelow the diagonal.* · · · * 0 · · ·
0. . . ... . . . ...* 00 · · ·
0. . . ... 00 ... . . .0 · · · et al. [20]for an alternative generalisation of Hardy’s paradox for high-dimensional (qudit) systems (see Table VI); this will also berelevant to the discussion in Sec. VII. In the present terminol-ogy, the argument applies to ( , , l ) Bell scenarios.
Proposition VI.1.
The occurrence of a Chen et al. paradox(Table VI) implies the occurrence of a Hardy paradox.Proof.
This follows directly from Theorem II.6, but one canalso prove the proposition more directly. Suppose one of thestarred entries corresponding to outcomes ( o ′ i , o j ) of Table VIis non-zero. We write p ( i , j ) > p ( r , j ) = r > ( l − j ) . Similarly, for the measurement represented by the lower-left box, p ( i , s ) = s > ( l − i ) . In the lower-right box,we have p ( r , s ) = r ≤ ( l − j ) and s ≤ ( l − i ) . Thisdescribes a ( , , l ) Hardy paradox.The proof shows that every non-zero starred entry in Ta-ble VI witnesses a (coarse-grained) Hardy paradox.
VII. HARDY NONLOCALITY WITH CERTAINTY
While Hardy’s paradox is considered to be an “almostprobability free” nonlocality proof, much of the literature onHardy’s paradox has been concerned with the value of the paradoxical probability (e.g. [19, 20, 24, 35]); i.e., the prob-ability of obtaining the particular outcome that witnesses aHardy paradox (Definition II.3). This is motivated as beingespecially relevant for experimental tests. In this section, wewill show how Hardy nonlocality can be demonstrated in sucha way that even this probability becomes irrelevant.We note that similar argument was put forward by Cabello[37], but stress that the results contained in this section has theTABLE VII: The PR box. ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ p max = √ − ≈ . . (7)A model has also been found for which the tripartite Hardyparadox can be witnessed with probability 0 .
125 [19], andin [24, 38, 39] it is demonstrated that for a generalised no-signalling theory it is possible to witness a ( , , ) Hardyparadox with probability 0 .
5. It was shown that the laddergeneralisation of Hardy’s paradox could achieve a paradoxicalprobability approaching 0 . ( , k , ) scenarios, as k → ∞ .For the ( , , l ) scenario, Chen et al. [20] (cf. Sec. VI) haveclaimed that it is possible to achieve a paradoxical probabilityof ≈ . d limit for two qu d it systems with theparadox presented in Table VI. From our Proposition VI.1, itis clear that strictly speaking this comes about by summingthe probabilities of witnessing a number of different (coarse-grained) Hardy paradoxes; ( l − ) / ( , , ) empirical model, whichturns out to be the PR box. Proposition VII.1.
The PR box witnesses Hardy nonlocalitywith certainty.Proof.
The probabilistic version of the PR box is given in Ta-ble VII. We have already observed in the proof of PropositionIV.1 that every joint outcome that has non-zero probabilitywitnesses a Hardy paradox. Therefore, each non-zero entryin the table represents a joint outcome that witnesses Hardynonlocality with paradoxical probability 0 .
5, and so it is clearthat for each joint measurement the probability of obtainingan outcome that witnesses a Hardy paradox is 1. TABLE VIII: The relevant portion of the GHZ-Merminpossibilistic empirical model. The suppressed rows of thetable { XXY , XY X , Y XX , YYY } have full support. SeeFig. 2 (a) for the three-dimensional representation of themodel.000 001 010 011 100 101 110 111 XXX
XY Y
Y XX
Y Y X . every joint outcome witnesses a Hardy paradox in the presentexample, the arguably more relevant parameter, the proba-bility of witnessing Hardy nonlocality, is actually 1 for anychoice of measurements.Nevertheless, it is not possible to use this method of sum-ming paradoxical probabilities to witness Hardy nonlocalitywith higher probability than (7) for any ( , k , ) empiricalmodel which can be realised by projective measurements on aBell state. Proposition VII.2.
For any ( , k , ) empirical model arisingfrom projective measurements on a Bell state, the probabilityof witnessing Hardy nonlocality cannot be improved by sum-ming the paradoxical probabilities for different paradoxes oc-curring within the same model.Proof. First, we note that it suffices to prove the propositionfor ( , , ) models, since a ( , k , ) model contains a Hardyparadox if and only if some ( , , ) sub-model contains aHardy paradox. In order to obtain an improvement in theprobability of witnessing Hardy nonlocality it would have tobe the case that, for some joint measurement, more than oneHardy paradox could be witnessed. Working in the presentframework, it is clear that any such empirical model is eitherthe PR box or belongs to the family of models with supportgiven by Table V (d), up to re-labelling of measurements andoutcomes, as discussed in the proof of Theorem V.1. Indeed,in this family, for the joint measurement A B , the probabil-ity of witnessing Hardy nonlocality is 1. However, it was alsoshown in the proof of Theorem V.1 that no model in the familyis quantum-realisable.We now consider the ( , , ) empirical model used in theGHZ-Mermin logical nonlocality proof [5, 40]. It should benoted that that the original nonlocality argument based on thisempirical model was not of the tripartite Hardy form men-tioned in Sec. II. Here, we need only consider a subset of themeasurement contexts, shown in Table VIII in more orthodoxnotation, and three-dimensional representation in Fig. 2 (a).FIG. 2: (a) The GHZ model. We represent only the red,impossible outcomes; all other entries are possible. (b)Hardy’s paradox within the GHZ model; the blue outcome ispossible.(a) (b) Proposition VII.3.
The GHZ model witnesses Hardy nonlo-cality with certainty.Proof.
The three-dimensional representation makes it easyto identify a tripartite Hardy paradox, which is shownin Fig. 2 (b). It can also be expressed as follows.• p ( , , | Y , Y , Y ) > p ( , , | Y , Y , X ) = p ( , , | Y , X , Y ) = p ( , , | X , Y , Y ) = p ( , , | X , X , X ) = n -partite Hardy para-dox we met in Sec. II. Moreover, it can similarly be demon-strated that any joint outcome for the measurement context YYY witnesses a Hardy paradox (this may be seen by inspec-tion, but a detailed and more general treatment can also befound in the proof of Proposition IX.1 in the appendix to thisarticle). The paradoxical probability is p paradox = p ( , , | Y , Y , Y ) = . . However, since every outcome to the measurement
YYY wit-nesses some Hardy paradox, then it is again the case thatthe combined probability of witnessing Hardy nonlocality is1. This provides a much simpler tripartite Hardy argumentthan that of Ghosh, Kar and Sarkar [19], using a simpler em-pirical model (theirs also used the GHZ state, but with alter-native measurements on this state), while still obtaining thesame value of 0 .
125 for the individual paradoxical probabil-ities. Again, perhaps more importantly, in our model everypossible outcome event for the joint measurement
YYY wit-nesses some Hardy paradox, and therefore Hardy nonlocal-ity is witnessed with certainty. The model considered here isexactly the GHZ-Mermin model, given that the observables available at each subsystem are simply the X and Y operators.As a result, it can be said that the GHZ experiment [5] wit-nesses Hardy nonlocality with certainty. Corollary VII.4.
The GHZ experiment [5] witnesses Hardynonlocality with certainty.
Mermin gave logical nonlocality proofs for n -partite gen-eralisations of the GHZ state [42] for all n >
2. Again, hisarguments were not of the Hardy form, but we can generaliseProposition VII.3 to some of the GHZ( n ) models (see the ap-pendix). VIII. MEASUREMENT INCOMPATIBILITY ISSUFFICIENT FOR LOGICAL NONLOCALITY
In [21] it was shown that a pair of observables are incom-patible in the sense of not being jointly observable if and onlyif they enable a Bell inequality violation. Subsequent workshave also considered how the degree of incompatibility relatesto the degree of nonlocality [43, 44]. Here, we show that, inthe basic case of projective or sharp measurements, incompat-ibility is necessary and sufficient for logical nonlocality [45].
Proposition VIII.1.
A pair of projective measurements en-ables a logical nonlocality argument if and only if it is incom-patible.Proof.
In [7], it was shown that any non-maximally entangledtwo-qubit pure state can be written in the form | Ψ i = N (cid:0) − α ∗ β ∗ | uu ⊥ i − α ∗ β ∗ | u ⊥ u i + α | u ⊥ u ⊥ i (cid:1) (8)for some orthonormal basis {| u i , | u ⊥ i} and complex α , β such that α + β = α >
0, where N is simply a nor-malisation factor. Logical nonlocality in the form of the Hardyparadox is realised by local projective measurements on eachqubit in the directions | u i and | d i : = α | u i + β | u ⊥ i .A pair of non-commuting Hermitian operators has at leastone pair of non-commuting spectral projections, say P = | u i h u | and Q = | d i h d | for some | u i and | d i . For the momentlet us not assume any relation to the vectors considered in theprevious paragraph. The projections are used to build a pairof non-commuting two-outcome observables ˜ P : = P − and˜ Q : = Q − . Essentially, these correspond to course-grainingthe probabilities of all outcomes not corresponding to | u i or | d i , respectively. We may assume that | d i = α | u i + β | u ⊥ i for some | u ⊥ i orthogonal to | u i and complex α , β such that α + β = α >
0, for otherwise the projections P and Q would commute. Now suppose we have a bipartite systemin which each party may choose to measure P or Q . Hav-ing defined | u i and | u ⊥ i in this way, logical nonlocality in theform of a coarse-grained Hardy paradox is realised on the en-tangled state specified by Eq. (8). IX. CONCLUSION
Theorem II.6, which combines the completeness resultsproved by the author and Fritz in [11], has been seen in this ar-ticle to lead to an abundance of consequences and applicationswhich we now briefly recap.The polynomial algorithms for deciding logical nonlocalityin ( , , l ) and ( , k , ) scenarios given in Sec. III are of par-ticular relevance since the problem is known in general to beNP-complete [26]. Further scenarios have been shown to betractable elsewhere [46].It was already known that PR boxes are the only stronglynonlocal ( , , ) models [9, 27], but the proof obtained inSec. IV provides more insight than the previously existingcomputational proof: in particular it is seen that the result isa straighforward consequence of the completeness of Hardynonlocality and the property of no-signalling.Given that all n -partite entangled states admit logical non-locality proofs via projective measurements for n > with certainty is understood to be equivalent to theproperty of strong nonlocality.While previous works have considered how measurementincompatibility relates to nonlocality in terms of Bell inequal-ity violations, Proposition VIII.1 provides initial progress onthe question of how incompatibility relates to other classes ofnonlocality in the qualitative hierarchy, which will be a topicfor future research.As a final open question, we note that a correspondencehas been established between possibilistic empirical modelsand relational database theory [48]. It remains to be exploredwhether Theorem II.6 might find applications in database the-ory, or indeed whether similar results already exist in that fieldthat might lead to further insights in the study of nonlocality. ACKNOWLEDGEMENTS
The author thanks Samson Abramsky, Rui Soares Barbosa,Tobias Fritz, Lucien Hardy, Ray Lal, Leon Loveridge, An-drew Simmons and Jamie Vicary for comments and discus-sions, Johan Paulsson for invaluable help with figures, andgratefully acknowledges financial support from the Fonda-tion Sciences Math´ematiques de Paris, postdoctoral research grant eotpFIELD15RPOMT-FSMP1, Contextual Semanticsfor Quantum Theory. This work was partially carried out atl’Institut de Recherche en Informatique Fondamentale, Uni-versit´e Paris Diderot - Paris 7, the Simons Institute, Universityof California, Berkeley as the Logical Structures in Computa-tion programme, and the Department of Computer Science,University of Oxford.
APPENDIX: GHZ( n ) Mermin gave logical nonlocality proofs for n -partite gener-alisations of the GHZ state [42] for all n >
2. These argumentsare not of the Hardy form, but we will now show how to gen-eralise Proposition VII.3 to some of the GHZ( n ) models.The GHZ( n ) states are: | GHZ ( n ) i : = √ ( | · · · i + | · · · i ) , (9)where n is the number of qubits. Note that for n = | φ + i Bell state. For n >
2, Merminconsidered models in which each each party can make Pauli X or Y measurements. With a little calculation, it is possible toconcisely describe the resulting empirical models in a logicalform [49].The eigenvectors of the X operator are | x i = √ (cid:0) | i + e i | i (cid:1) , | x i = √ (cid:0) | i + e i π | i (cid:1) . (10)The vector | x i has eigenvalue + | x i haseigenvalue −
1. These are more usually denoted | + i and |−i ,respectively, but we use an alternative notation to agree withthe { , } labelling of outcomes used in this article. Thephases have been made explicit since they will play the crucialrole in the following calculations. Similarly, the + − Y operator are (cid:12)(cid:12) y (cid:11) = √ (cid:16) | i + e i π / | i (cid:17) , (cid:12)(cid:12) y (cid:11) = √ (cid:16) | i + e − i π / | i (cid:17) . (11)The various probabilities for these quantum-mechanicalempirical models can be calculated as |h GHZ ( n ) | v . . . v n i| , where the v i are the appropriate eigenvectors. This evaluatesto (cid:12)(cid:12)(cid:12)(cid:12) + e i φ √ n + (cid:12)(cid:12)(cid:12)(cid:12) = n ( + cos φ ) , (12)where φ is the sum of the phases of the v i . From the phasesof the possible eigenvectors, (10) and (11), it is clear that wemust have φ = k π / k ∈ Z , the four element cyclicgroup. For k = √ n − ; for k = √ n ; and for k = k x is the num-ber of | x i eigenvectors, k x is the number of | x i eigenvectors,and so on, then k = k y + · k x + · k y ( mod 4 )= (cid:0) k y + k y (cid:1) + · (cid:0) k x + k y (cid:1) ( mod 4 ) . • For contexts containing an odd number of Y ’s, everyoutcome is possible with equal probability √ n , since k = Y ’s, outcomes are pos-sible if and only if they contain an even number of 1’s.For these outcomes, k = √ n − . If there were an odd number of 0’s in theoutcome then k = Y ’s, out-comes are possible if and only if they contain an oddnumber of 1’s. Again, the non-zero probabilities are √ n − .Though the probabilities are seen to be easily be calculatedin this way, we need only concern ourselves with the possi-bilistic information in what follows. Proposition IX.1.
All GHZ(n) models for n = wit-ness an n-partite Hardy paradox with certainty. Proof. Proposition VII.3 showed that this holds for n =
3. Let o = ( o , . . . , o n ) be any binary string of length n , let γ i be thefunction that changes the i th entry of a binary string, and let o − denote the binary string of length n which differs in everyentry from o . We show that every outcome o to the measure-ments ( Y , . . . , Y ) witnesses a Hardy paradox. We deal with thecases that o has an even or an odd number of 1’s separately.Suppose o has an even number of 1’s: p ( o | Y , . . . , Y ) >
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