Non-local games and optimal steering at the boundary of the quantum set
Yi-Zheng Zhen, Koon Tong Goh, Yu-Lin Zheng, Wen-Fei Cao, Xingyao Wu, Kai Chen, Valerio Scarani
aa r X i v : . [ qu a n t - ph ] J un Non-local games and optimal steering at the boundary of the quantum set
Yi-Zheng Zhen,
1, 2
Koon Tong Goh, Yu-Lin Zheng,
1, 2
Wen-FeiCao,
1, 2
Xingyao Wu, Kai Chen,
1, 2 and Valerio Scarani
3, 4 Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
The boundary between classical and quantum correlations is well characterised by linear con-straints called Bell inequalities. It is much harder to characterise the boundary of the quantum setitself in the space of no-signaling correlations. For the points on the quantum boundary that violatemaximally some Bell inequalities, Oppenheim and Wehner [Science , 1072 (2010)] pointed outa complex property: the optimal measurements of Alice steer Bob’s local state to the eigenstate ofan effective operator corresponding to its maximal eigenvalue. This effective operator is the linearcombination of Bob’s local operators induced by the coefficients of the Bell inequality, and it canbe interpreted as defining a fine-grained uncertainty relation. It is natural to ask whether the sameproperty holds for other points on the quantum boundary, using the Bell expression that defines thetangent hyperplane at each point. We prove that this is indeed the case for a large set of points, in-cluding some that were believed to provide counterexamples. The price to pay is to acknowledge thatthe Oppenheim-Wehner criterion does not respect equivalence under the no-signaling constraint: foreach point, one has to look for specific forms of writing the Bell expressions.
I. INTRODUCTION
The violation of Bell inequalities [3, 4] is one of theclearest examples in which quantum resources outper-form classical ones: the outcomes of local measurementson a shared entangled quantum state cannot in generalbe reproduced by reading shared classical information.In other words, the quantum set of probability distribu-tions is strictly larger than the (classical) local set . It isalso known that shared entanglement cannot be used ata later time to exchange a message, but quantum theorydoes not achieve all the possible no-signaling probabilitydistribution [24].The resources “shared quantum states” thus hangsomehow between two easily formulated types of re-sources. This observation has triggered the effort to finda physical or information-theoretical principle that wouldidentify the quantum set. Some of these tentative prin-ciples are device-independent: they can be formulated atthe level of probability distributions [17, 22]; they failedto reach the quantum set [15]. Others rely on the quan-tum formalism and extrapolate from it. The observationmade by Oppenheim and Wehner (OW) [20], which isthe focus of this paper, belongs to the latter category.In a nutshell, OW recast Bell inequalities in terms ofsteering from Alice to Bob, in a way that highlights thatviolation can come from two contributions: the capac-ity of Alice to steer Bob’s state, and the “certainty” ofthe statistics of Bob’s different measurements. Then,for several examples including the iconic Clauser-Horne-Shimony-Holt (CHSH) inequality [8], they proved thatthe point of maximal quantum violation is characterizedby both full steerability and an amount of certainty ashigh as allowed by some uncertainty relations. A link between bipartite nonlocality and local uncer-tainty relations is definitely an observation worth closerscrutiny. In particular, it is natural to ask whether theproperty holds beyond the examples studied in Ref. [20],and ultimately perhaps for the whole of the quantumboundary. A few months ago, Ramanathan and cowork-ers answered negatively to the latter conjecture, by pro-ducing Bell expressions for which the OW property doesnot hold [25]. Though correct, their conclusion overlooksa subtle feature of the OW property that was known bysome but had never been highlighted in a publication.We need to explain this feature to motivate our contri-bution (more details will be given in II B below).If I ≤ I L is a Bell inequality, for every constant k it is clear that I + k ≤ I L + k is an equivalent way ofwriting the same inequality (in particular, the point onthe quantum boundary that reaches the quantum max-imum I Q is going to be the same). Now, the constant k can be written in various ways exploiting the normal-isation of probabilities and the no-signaling constraints.For instance, it is known that the so-called CH inequal-ity [7] is equivalent to CHSH for no-signaling correlations;and everyone who has worked with the Collins-Linden-Gisin-Massar-Popescu (CGLMP) inequalities knows thatvarious authors adopt different ways of expressing them[9, 27, 30]. The algorithmic classification of Bell inequali-ties by Rosset and coworkers [26] makes a systematic useof this equivalence. Now, it turns out that the OW way ofdealing with Bell inequalities does not respect this equiv-alence : if the OW property holds for a Bell expression I , it may well not hold for an equivalent Bell expression I + k if the constant k is written in a way that exploitsthe no-signaling constraint.On the one hand, this features makes the OW approachsignificantly weaker than one might have hoped. On theother hand though, to prove that the OW property doesnot hold for some points on the quantum boundary, it isnot sufficient to work with specific Bell expressions: oneshould prove that, given a point on the quantum bound-ary, OW holds for no Bell expression that is maximisedby that point. In short, since the OW property is at besta “there exist” statement, its negation must take the formof a “for all” statement.In this paper, we are going to show that the OW linkholds for several known examples of points on the quan-tum boundary, including those of the alleged counterex-amples just mentioned. No counterexample having beenfound, it remains an open conjecture whether the OWproperty holds on the whole of the quantum boundary.
II. METHODOLOGYA. Definitions and notations
Consider a bipartite ( m A , m B , n A , n B ) -scenario: Al-ice’s measurements are indexed by x ∈ { , ..., m A } andtheir outcomes are labeled a ∈ { , ..., n A } ; Bob’s mea-surements are indexed by y ∈ { , ..., m B } and their out-comes are labeled b ∈ { , ..., n B } . A probability point P will be described by all the P ( ab | xy ) . A Bell inequal-ity, denoted by I ( P ) , can be written as a linear sum ofconditional probabilities upper bounded by its maximumvalue achievable by local realistic resources, I L . I ( P ) := X abxy V abxy P ( a, b | x, y ) ≤ I L (1)For a given Bell inequality, the values of V abxy are notuniquely defined. Hence, different V abxy , or Bell expres-sions, are equivalent Bell inequalities which are maxi-mally violated by the same point on the quantum bound-ary. By rewriting the left-hand side of equation (1): X abxy V abxy P ( a, b | x, y ) = X ax P ( a | x ) X by V abxy P ( b | y, a, x ) ≡ X ax P ( a | x ) (cid:10) ˆ B ( x, a ) (cid:11) ρ B | x,a (2)where ˆ B ( x, a ) = P by V abxy Π yb . Denoting λ ( x,a ) the largest eigenvalue of ˆ B ( x, a ) , it obviously holds (cid:10) ˆ B ( x, a ) (cid:11) ρ B | x,a ≤ λ ( x,a ) . Inspired by the approach ofOppenheim and Wehner [20], we call OW-game a Bellexpression for which these inequalities are saturated atthe boundary of the quantum set: I ( P ) = I Q = ⇒ (cid:10) ˆ B ( x, a ) (cid:11) ρ B | x,a = λ x,a ∀ x, a (3)where I Q is the maximum value of I ( P ) achievable byquantum resources. In words: in an OW-game, with heroptimal measurements Alice steers Bob’s state precisely to the eigenvector of the effective operator B ( x, a ) , for allinputs x and outputs a . Oppenheim and Wehner provedthat several XOR games, including the one based on theCHSH inequality, are OW-games.Apart from the contents of subsections III C and III D,the results of this paper are obtained in the (2 , , , Bell scenario, so we introduce a convenient notation. La-belling x, y ∈ { , } and a, b ∈ { , } , we represent Bellexpressions as tables I = V V V V V V V V V V V V V V V V . (4) B. The OW property does not respect equivalenceunder no-signaling
In the (2 , , , scenario, the only tight Bell inequality(facet of the local polytope) is the famous CHSH inequal-ity [8]. Its maximal violation defines a single point on thequantum boundary. The basic results of the Oppenheim-Wehner paper is that (3) holds for that point. What isnot explicit from the paper is that the conclusion dependson the Bell expression that is used: the XOR CHSH gameis an OW-game, but other Bell expressions that define thesame inequality (and are in particular maximised by thesame point) may not be OW-games. This observation isthe basis of all this work, so let us provide an explicitexample.We start with the CHSH XOR game I CHSH = P x,y =0 P ( a x ⊕ b y = xy ) , where P ( a x ⊕ b y = xy ) = P a P ( a, b = xy ⊕ a | xy ) . It has I LCHSH = 2 and I QCHSH = 2 + √ . The state and measurements thatachieve I Q can be uniquely written, up to local isome-tries, as | Φ + i = ( | i + | i ) / √ , and A = σ z , A = σ x , B = ( σ z + σ x ) / √ , B = ( σ z − σ x ) / √ [23]. Now, con-sider the following re-writing: I CHSH = 1 0 1 00 1 0 11 0 0 10 1 1 0 = 2 1 1 1 00 1 1 11 0 0 11 1 0 0 − − I − . (5)The Bell expression I is CGLMP in the version ofZohren and Gill [30], which was already known tobe equivalent to CHSH for no-signaling P ’s. Indeed,let’s prove that the rightmost table is a complicatedway of writing the constant k = 3 for no-signaling P ’s. The top left block is P (00 |
00) + 2 P (01 |
00) + P (11 |
00) = P A (0 |
0) + P B (1 | . Treating the two off-diagonal blocks similarly, we find that the table repre-sents [ P A (0 | P B (1 | P A (1 | P B (0 | P A (1 | P B (0 | P (01 | − P (10 |
11) = 2+ P A (1 | P B (0 | P (01 | − P (10 | . Again because of no-signaling,one has P A (1 |
1) = P (10 |
11) + P (11 | and P B (0 |
1) = P (00 |
11) + P (10 | , which proves the claim.It follows from these observations that I Q is obtainedfor the same point on the quantum boundary that gives I QCHSH , which as we said is achievable only with thestates and measurements written above. Then, given theoperators, the bounds h B ( x, a ) i ≤ λ ( x,a ) of I are givenin terms of the P ( b | y, a x ) by P (0 | , ) + P (1 | , ) + P (0 | , ) ≤ λ , ,P (1 | , ) + P (0 | , ) + P (1 | , ) ≤ λ , ,P (0 | , ) + P (1 | , ) ≤ λ , ,P (0 | , ) + P (1 | , ) = 1 , where λ , = λ , = 2 and λ , = 1 + √ [18].But for the state under consideration, h B (0 , a ) i = + √ < λ ,a ) for both a = 0 , . Hence, the CGLMP game I is not an OW-game. This case study shows thatdifferent Bell expressions may behave differently on theOW characterization. C. Transformations that represent equivalenceunder no-signaling
As explained with the example of CHSH andCGLMP , we are going to look for alternative Bell ex-pressions of the same inequality obtained by adding aconstant term k , and check if at least one of them definesan OW-game.In our notation, a Bell expression of a Bell inequality I is represented by the table (4) — we keep the discussionin the (2,2,2,2) Bell scenario, but the generalisation isobvious. First notice that tables of the type k = k k k k k k k k k k k k k k k k , (6)for a real number k , and convex combinations thereof, doindeed represent the constant k due to the normalisationconstraint P a,b P ( a, b | x, y ) = 1 ∀ x, y . This representa-tion of a constant is pretty trivial and indeed one cancheck that the OW character of a Bell expression is notchanged by adding k expressed in this way. For instance,starting from a Bell expression I , one can always con-struct I ′ with the same OW character such that all the V abxy are non-negative.If we now enforce the no-signalling constraint for Bob P ( b | x = 0 , y ) = P ( b | x = 1 , y ) = P ( b | y ) , we obtain less trivial representations of the same constant: k = k k k k k k k k k
00 0 k
00 0 0 k k = 0 0 0 k k k
00 0 k , (7)and convex combinations thereof. Similarly, enforcingthe no-signaling constraint for Alice P ( a | x, y = 0) = P ( a | x, y = 1) = P ( a | x ) , we have the additional repre-sentations k = k k k k k kk k k k k k = 0 0 0 00 0 0 00 0 k kk k , (8)and convex combinations thereof. Such rewritings basedon no-signaling may change the OW character of a Bellexpression, as it happened in the CHSH example above.Having come to terms with this flexibility, it is conve-nient to recast the same information in difference tables D = D D D D (9)with D xy = V xy − V xy V xy − V xy V xy − V xy V xy − V xy . (10)This representation is handy because the transformationsallowed by no-signaling take a very simple form. Indeed,two difference tables are equivalent under no-signalingif and only if there exist α, β, γ, δ ∈ R such that D ′ = D + ∆( α, β, γ, δ ) with ∆( α, β, γ, δ ) = + α + β + γ + γ − γ − γ + α + β − α − β + δ + δ − δ − δ − α − β . (11)In particular, if a difference table D k represents a con-stant k under the no-signaling constraints for Alice andBob, there must exist α, β, γ, δ such that D ′ k = D k + ∆( α, β, γ, δ ) = 0 00 0 0 00 00 00 0 0 00 0 . (12) D. Checking for OW-games
Having introduced the context, we can finally explainhow one can look for an OW-game.First notice that even the verification of the OW crite-rion for a given Bell expression is not trivial a priori . In-deed, while the property of “being at the quantum bound-ary” is determined by the Bell expression alone, checkingthe OW criterion involves finding the states and oper-ators that realize the quantum point P . For a genericBell expression, it is not known how to find such a quan-tum realisation; and even once one is found, there is noguarantee that it is unique. If P could be obtained withinequivalent realisations of the state and the measure-ments, one would have to say whether saturation of (3)holds for all realisations, or it is enough that it holds forone. As it turns out, for all the cases explicitly studiedso far, a P on the quantum boundary can be obtainedby a unique choice of the state and the measurements,up to local isometries (“ self-testing ”). The independentconjecture that self-testing holds on the whole quantumboundary is interesting in its own right, but we don’taddress it here.Having clarified how the OW criterion is going totested, we need to move one step back and explain howone can try and guess a Bell expression that is a candi-date for OW-game, given all the freedom allowed by theequivalence under no-signaling. The heuristic methodwe found consists in enforcing first some necessary con-ditions. Indeed, it is clear that the OW criteria (3) canonly be satisfied if ˆ B ( x, a ) is diagonal in the basis which ρ B | x,a is diagonal. This condition imposes several con-straints on V abxy , that largely restrict the candidate Bellexpressions. The remaining ones can then be tested di-rectly. In Appendix A we describe in greater detail howthese constraints are used in the case of self-testing pointsin the (2 , , , Bell scenario.
III. POINTS WITH OW-GAME
We present now the points on the quantum boundaryfor which we have found an OW-game. For the (2 , , , Bell scenario, we first discuss the two points that al-legedly provided counterexamples (III A), then two wholefamilies of points (III B). Then we present one point in the (2 , , d, d ) Bell scenario, the one that violates maxi-mally the CGLMP d inequality (III C). Finally, one pointin a three-partite scenario, the one that violates maxi-mally the Mermin inequality, together with the suitabledefinition of steering in the multipartite case (III D). A. Alleged counterexamples
Ramanathan and coworkers [25] provided two gamesthat are not OW-games. As we know by now, this isnot sufficient to prove that there is no OW-game for thecorresponding points on the quantum boundary — andas it turns out, there is.For the first point, we use the family of Bell expressions I c (Γ) = + Γ Γ Γ Γ Γ − Γ − Γ − Γ − I c (0) + 2Γ . (13)The fact that the rightmost table is equal to k = 2Γ forno-signaling P ’s can be checked with the tools describedabove. The bounds h B ( x, a ) i ≤ λ ( x,a ) now read Γ P (0 | , ) + (1 − Γ) P (1 | , ) ≤ λ c , , (1 − Γ) P (1 | , ) + Γ P (0 | , ) ≤ λ c , ,P (1 | , ) + P (1 | , ) ≤ λ c , , (2 − Γ) P (0 | , ) + (1 − Γ) P (0 | , ) ≤ λ c , , In Ref. [25] it is proved that I c (0) self-tests a given two-qubit state and suitable measurements; and that the OWcriteria (3) do not hold. However, using those same stateand measurements, for Γ ≃ . we find numer-ically that the criteria hold, with λ c , = λ c , ≃ . , λ c , ≃ . , λ c , ≃ . .For the second point, we use the Bell expressions I c (Γ) = + 0 Γ − Γ 00 Γ − Γ 0Γ 0 0 − ΓΓ 0 0 − Γ = I c (0) + 0 (14)because the table we added is equal to k = 0 for no-signaling P ’s. The corresponding bounds (1 − Γ) P (0 | , ) + (1 + Γ) P (1 | , ) ≤ λ c , , (1 + Γ) P (1 | , ) + (1 − Γ) P (0 | , ) ≤ λ c , ,P (1 | , ) + P (1 | , ) ≤ λ c , , (1 + Γ) P (0 | , ) + Γ P (0 | , ) ≤ λ c , , are not saturated for Γ = 0 , as proved in Ref. [25]; butthey are for Γ ≃ . , in which case λ c , = λ c , = λ c , ≃ . , λ c , ≃ . . B. Families of points
Even for the (2 , , , -scenario, we do not know a com-plete parametrisation of the quantum boundary. Themost famous family of points is the three-parameterfamily that describes the slice with unbiased marginals P ( a | x ) = P ( b | y ) = . The boundary is known tobe given by P x,y ( − xy Arcsin ( E xy ) = π with E xy = P ( a = b | xy ) − P ( a = b | xy ) , or suitable permutationsof the settings and the outcomes [6, 10]. The points onthese boundaries are also those that self-test | Φ + i in the (2 , , , -scenario [28]. Now, for E xy ≡ cos α xy = ± ,the inequality that describes the tangent to each of thesepoints can be cast as the game [14, 28] I ~E = α α α α α − α α − α . (15)This is a weighted XOR game, the non-zero V abxy beingdifferent for different ( x, y ) . We checked numerically thatthe OW criteria (3) hold by sampling 156849 such pointsat random.The other family that we consider is the one-parameter family of the points that violate maximally one of the tilted CHSH inequalities αE A + E + E + E − E ≤ α where E A = P A (0 | − P A (1 | and α ∈ [0 , [1].Each of these points self-tests a corresponding partiallyentangled qubit state | ψ ( θ ) i = cos θ | i + sin θ | i with α = 2 / √ θ , for the measurements A = σ z , A = σ x , B = cos µσ z + sin µσ x and B = cos µσ z − sin µσ x where tan µ = sin 2 θ [2, 29]. For these points, wework with the family of Bell expressions I α (Γ) = 1 + α α − Γ 0 − cos 2 θ − cos 2 θ − cos 2 θ − cos 2 θ sin θ cos θ sin θ cos θ sin θ cos θ sin θ cos θ (16)The rightmost table is k = 2 sin θ for no-signaling P ’s,so the local bound is I Lα (Γ) = 2 + α −
2Γ sin θ . The case Γ = 0 is the game that one would naturally write downfrom the inequality as stated, but it can be checked thatit is not an OW-game for any α ∈ (0 , [19]. However, Γ = 1 provides an OW-game. In this case, the bounds h B ( x, a ) i ≤ λ ( x,a ) are given by P (0 | , ) + P (0 | , ) ≤ λ α (0 , ,P (1 | , ) + P (1 | , ) ≤ λ α (0 , , cos θP (0 | , ) + sin θP (1 | , ) ≤ λ α (1 , , sin θP (1 | , ) + cos θP (0 | , ) ≤ λ α (1 , , with λ α (0 , = 1 + q − cos 4 θ , λ α (0 , = − cos 4 θ − cos 4 θ −√ − θ and λ α (1 , = λ α (1 , = + √ − θ .Interestingly, even if these Bell expressions are asym-metric between Alice and Bob, they can be used to steerin the either direction. We have just presented the steer-ing from Alice to Bob. That from Bob to Alice, theOW criteria is given by h B ′ ( y, b ) i = λ ′ ( y,b ) ∀ y, b where B ′ ( y, b ) = P yb V abxy Π xa and λ ′ ( y,b ) is the largest eigen-value of B ′ ( y, b ) .From the tilted CHSH inequalities, we can write downanother family of Bell expressions, denoted by I ′ α , whichis given by: I ′ α (Γ) = 1 + α α − α − α X − α X − α − X − X X X − X − X (17)where X = − Λ + − Λ − Λ + − Λ − , X = + Λ − − Λ + − Λ − Λ + − Λ − and Λ ± = θ − θ ± √ θ . Similarly, I ′ α (Γ) is an OW-game forthe case Γ = 1 but not
Γ = 0 . In the case
Γ = 1 , thebounds h B ( y, b ) i ≤ λ ( y,b ) are given by P (0 | , ) + ( X − α ) P (1 | , )+ P (0 | , ) + X P (1 | , ) ≤ λ ′ α (0 , , ( X + 1 − α ) P (1 | , ) + ( X + 1) P (1 | , ) ≤ λ ′ α (0 , ,P (0 | , ) − X P (1 | , ) + P (0 | , )+(2 − X ) P (1 | , ) ≤ λ ′ α (1 , , (1 − X ) P (1 | , ) + 2 P (0 | , )+(1 − X ) P (1 | , ) ≤ λ ′ α (1 , . For the steering scenario of Bob to Alice, the probabilitieswritten above are P ( a | x b y ) .A final remark: we have also explored a third family ofpoints, those that violate maximally the “Hardy inequal-ities” introduced by by Mančinska and Wehner [11]. Wehave strong numerical evidence of both the fact that asample of these points are self-testing and that one canconstruct OW-games for each of them. We don’t thinkthat this paper will be significantly improved by a de-tailed presentation of these optimisations as they stand. C. OW-games with more outcomes: maximalviolation of CGLMP d In this section, we leave the (2 , , , -scenario to dis-cuss one point with OW-game in the (2 , , d, d ) -scenario,for any d ≥ . Concretely, we consider the points thatviolate maximally each of the CGLMP d inequalities. Theconclusions of this subsection rely on the conjecture thatthe maximal quantum violation is indeed achieved by thepoints constructed below (proved up to numerical preci-sion for d ≤ , see Table 1 of [16]). We have also to warnthat the self-testing of the states and measurements hasbeen proved so far only for d = 3 [29].We denote x, y ∈ { , } as the measurement settingsand a, b ∈ { , , . . . , d − } as the outcomes for Aliceand Bob respectively. In this scenario, the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequalities [9] are aclass of facets. Maybe the most compact way of writingthe CGLMP inequality is that of Zohren and Gill: I d = P ( a ≤ b ) + P ( a ≥ b )+ P ( a ≥ b ) + P ( a < b ) ≤ , (18)where P ( a x ≤ b y ) = P a ≤ b P ( ab | xy ) [30]. As it hap-pened for d = 2 , this form is not an OW-game(see Appendix B for the explicit proof in the case d = 3 ). An OW-game based on CGLMP was con-structed in Ref. [25]: G = P x,y =0 P ( a x = b y − xy ) + P ( a x = b y + x + y − , where P ( a x = b y + ∆) = P dk =0 P ( a = k, b = ( k − ∆) mod d | xy ) . By inspection,one finds that G = 3 I − .This construction can be generalised to high dimen-sional case. Now consider the non-local game G d = X xy d − X ∆=0 ∆ P (cid:16) a − b = ( − x + y (∆ + 1) − xy | xy (cid:17) = d − d − · · · d − · · · d − d − · · · d − d − ... ...... ... ... ... ... ... ... d − · · · d − · · · d − d − d − · · · d − d − · · · d − d − ... ... ... ...... ... ... ... ... ... d − · · · d − d − d − · · · . (19) G d is also a weighted XOR game. It’s easy to checkthat G is the CHSH-XOR game [20] and G is the gamepresented in Ref. [25]. In fact, G d = dI d − holds for all d , so G d is a CGLMP d game. It can be shown that thisis an OW-game for all d , under the conjecture mentionedabove on the form of the optimal measurements. Weleave the proof of this statement to Appendix C. D. Multipartite example: Maximal Violation ofMermin Inequality
All discussions made on OW criteria in the literaturefocus on bipartite Bell scenarios, which is not surprisingbecause the idea of steering is somehow naturally bipar-tite. However, a multipartite generalisation of steeringhas been introduced. Following the work of Cavalcantiand coworkers [5], in the tripartite Bell scenario, one candistinguish two types of steering: (i) 1 black box thatsteers to 2 characterised devices and (ii) 2 black boxesthat steer to 1 characterised device. Each type of steering would give rise to different sets of OW criteria, namely:(i): (cid:10) ˆ B ( x, a ) (cid:11) ρ BC | x,a = λ ( x,a ) ∀ x, a (20)(ii): (cid:10) ˆ B ( x, y, a, b ) (cid:11) ρ C | x,y,a,b = λ ( x,y,a,b ) ∀ x, y, a, b (21)where ˆ B ( x, a ) := X bcyz V abcxyz Π yb ⊗ Π zc , (22) ˆ B ( x, y, a, b ) := X cz V abcxyz Π zc (23)and λ ( x,a ) and λ ( x,y,a,b ) are the largest eigenvalues of ˆ B ( x, a ) and ˆ B ( x, y, a, b ) respectively.We study the point that violates maximally the Mer-min inequality[13] h A B C i−h A B C i−h A B C i−h A B C i ≤ (24)When rewritten to the form of a Bell expression, we get: I M ( P ) := X abcxyz V abcxyz P ( a, b, c | x, y, z ) ≤ (25)where V abcxyz = δ a ⊕ b ⊕ c,x ∨ y ∨ z δ xyz, δ xyz, δ xyz, δ xyz, (26)The maximal quantum bound of I M ( P ) is given by 4and it self-tests [21] the measured quantum to be theGHZ state | GHZ i = | i + | i√ (27)and the measurements to be ˆ A = ˆ B = ˆ C = σ z , (28) ˆ A = ˆ B = ˆ C = σ y . (29)Hence, one can easily check that: (cid:10) ˆ B M ( x, a ) (cid:11) ρ BC | x,a = λ M ( x,a ) = 2 ∀ x, a (30) (cid:10) ˆ B M ( x, y, a, b ) (cid:11) ρ C | x,y,a,b = λ M ( x,y,a,b ) = 1 ∀ x, y, a, b (31)Thus, this concludes that Mermin inequality is an OW-game for both types of steering. IV. CONCLUSION
The quantum set of correlations is defined by all the P ( ab | xy ) that can be obtained by measuring quantumstates, without any constraint on the Hilbert space di-mension of the underlying system. The characteriza-tion of its boundary in terms of physical or mathemat-ical properties is still elusive. In this paper, we haveshown that the Oppenheim-Wehner criteria are fulfilledby many points on the quantum boundary, includingsome that can’t maximise any XOR game and two thatwere believed to provide counterexamples.No counterexample has been found so far, which mayinspire the conjecture that every point on the quantumboundary, in any scenario, has an associated OW-game.In order to test the truth of this conjecture, one wouldhave to solve long standing problems that are of inter-est in themselves (and even arguably of greater interest).Indeed, in order to state the OW criteria, one needs theknowledge of the state and the measurements that re-alise the probability point on the quantum boundary. Itis not even sure that such a point is unique: this is anopen conjecture on self-testing. Even taking uniquenessfor granted, it would be a breakthrough by itself, if onewere able to provide the quantum realisation of the max-imal violation of a generic Bell expression (even for Bellinequalities this is usually unknown).In the context of nonlocal games, it would be interest-ing to study in which context the choice of a representa-tion that is a OW-game may be an advantage. It mustclearly be a situation in which the equivalence under no-signaling is not important. ACKNOWLEDGMENTS
Y.Z.Z., Y.L.Z. and K.C. acknowledge hospitality fromCQT, Singapore, when this collaboration was started.An early discussion with Antonio Acín was in the backof V.S.’s mind and triggered this project. We also thankFrancesco Buscemi and Ravishankar Ramanathan forvaluable discussions, and Manik Banik for highlightingtypos in a previous version.This research is supported by the Singapore Ministryof Education Academic Research Fund Tier 3 (Grant No.MOE2012-T3-1-009); by the National Research Fund andthe Ministry of Education, Singapore, under the Re-search Centres of Excellence programme; by the ChineseAcademy of Science; and by the National Natural Sci-ence Foundation of China (Grants No. 11175170, andNo. 11575174). V.S. acknowledges further support from aNUS Provost Chair grant, Y.Z.Z. and Y.L.Z. from USTCStudent Scholarship and China Scholarship Council. [1] Acín, A., Durt, T., Gisin, N., and Latorre, J. I.,Phys. Rev. A , 052325 (2002).[2] Bamps, C. and Pironio, S.,Phys. Rev. A , 052111 (2015).[3] Bell, J. S., Physics (Long Island City, N.Y.) , 195 (1964).[4] Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., andWehner, S., Rev. Mod. Phys. , 419 (2014).[5] Cavalcanti, D., Skrzypczyk, P., Aguilar, G., Nery, R.,Ribeiro, P. S., and Walborn, S., Nat. Commun. , 7941(2015).[6] Cirel’son, B. S., Lett. Math. Phys. , 93 (1980).[7] Clauser, J. F. and Horne, M. A.,Phys. Rev. D , 526 (1974).[8] Clauser, J. F., Horne, M. A., Shimony, A., and Holt,R. A., Phys. Rev. Lett. , 880 (1969).[9] Collins, D., Gisin, N., Linden, N., Massar, S., andPopescu, S., Phys. Rev. Lett. , 040404 (2002).[10] Landau, L. J., Found. Phys. , 449 (1988).[11] Mančinska, L. and Wehner, S., J. Phys. A: Math. Theor. , 424027 (2014).[12] Masanes, L., arXiv preprint quant-ph/0512100 (2005).[13] Mermin, N. D., Phys. Rev. Lett. , 1838 (1990).[14] Miller, C. A. and Shi, Y., arXiv , 1207.1819 (2012).[15] Navascués, M., Guryanova, Y., Hoban, M. J., and Acín,A., Nat. Commun. (2015).[16] Navascués, M., Pironio, S., and Acín, A.,New J. Phys. , 073013 (2008).[17] Navascués, M. and Wunderlich, H., in Proceedings ofthe Royal Society of London A: Mathematical, Physicaland Engineering Sciences (The Royal Society, 2009) p.rspa20090453.[18] Notice that conditions like P (0 | , ) + P (1 | , ) = 1 hold by definition, so one could just have written the firstline as P (0 | , ) ≤ λ ′ , where λ ′ , = λ , − ; and similarly for the second line.[19] For the state and the measurements that define thepoint at the quantum boundary, one finds (cid:10) B (1 , b ) (cid:11) =1+(1 − cos 4 θ ) / ( √ − θ ) for b = 0 , . This is strictlysmaller than λ (1 ,b ) = 1 + | sin 2 θ | q − cos 4 θ for α ∈ (0 , .[20] Oppenheim, J. and Wehner, S.,Science , 1072 (2010).[21] Pál, K. F., Vértesi, T., and Navascués, M., Phys. Rev.A , 042340 (2014).[22] Pawlowski, M., Paterek, T., Kaszlikowski, D.,Scarani, V., Winter, A., and Zukowski, M.,Nature , 1101 (2009).[23] Popescu, S. and Rohrlich, D.,Phys. Rev. A , 411 (1992).[24] Popescu, S. and Rohrlich, D.,Found. Phys. , 379 (1994).[25] Ramanathan, R., Goyeneche, D., Mironowicz, P., andHorodecki, P., arXiv , 1506.05100 (2015).[26] Rosset, D., Bancal, J.-D., and Gisin, N.,J. Phys. A: Math. Theor. , 424022 (2014).[27] Scarani, V., Gisin, N., Brunner, N., Masanes, L., Pino,S., and Acín, A., Phys. Rev. A , 042339 (2006).[28] Wang, Y., Wu, X., and Scarani, V., arXiv , 1511.04886(2015).[29] Yang, T. H. and Navascués, M.,Phys. Rev. A , 050102 (2013).[30] Zohren, S. and Gill, R. D.,Phys. Rev. Lett. , 120406 (2008). Appendix A: Enforcing necessary conditions for the (2 , , , Bell scenario under self-testing
In this appendix we show more explicitly how to imple-ment the constraints discussed in section II D in the caseof self-testing probability distributions in the (2 , , , Bell scenario. In this scenario, we are guaranteed thatthe maximal Bell violation by a quantum resource canbe achieved by a pure bipartite qubits state and projec-tive measurements [12]. If the point is self-testing, thenit does self-test a pure two-qubit state and those mea-surements. In particular, the steered state on Bob willbe a pure qubit state.Define now the unitary transformation U x,a such that U x,a ρ B | x,a U † x,a = | ih | . (A1)The projectors written in the basis where the steeredstate is diagonal are given in the following form: U x,a Π y =0 b =0 U † x,a = p ( x, a ) q ( x, a ) q ( x, a ) 1 − p ( x, a ) ! (A2) U x,a Π y =0 b =1 U † x,a = − p ( x, a ) − q ( x, a ) − q ( x, a ) p ( x, a ) ! (A3) U x,a Π y =1 b =0 U † x,a = p ( x, a ) q ( x, a ) q ( x, a ) 1 − p ( x, a ) ! (A4) U x,a Π y =1 b =1 U † x,a = − p ( x, a ) − q ( x, a ) − q ( x, a ) p ( x, a ) ! (A5)where p ( x, a ) , q ( x, a ) , p ( x, a ) and q ( x, a ) are somereal numbers between 0 and 1. The necessary conditionswhich for a Bell expresssion to be an OW-game on V abxy are then q ( x, a )( V a x − V a x )+ q x, a )( V a x − V a x ) = 0 ∀ x, a . (A6)In particular, for cases where the r ( x, a ) := q ( x,a ) q ( x,a ) arewell-defined for all x, a pairs, an OW-game has the formof: OW = A B C r (0 , A − B ) + CD E F r (0 , D − E ) + FG H I r (1 , G − H ) + IJ K L r (1 , J − K ) + L (A7)where the capital Roman alphabet letters are free vari-ables.In order to check whether a point may have an OW-game, we can now take any Bell expression I that is max-imally violated by that point, and check if there exist anOW-game such that I − OW = k . As discussed above,this is going to be simplest by passing in the differencerepresentation and using equation (12). Appendix B: The Zohren-Gill version of CGLMP isnot an OW-game In this section, we show that I as defined in equation(18) is not an OW-game.The bounds h B ( x, a ) i ≤ λ ( x,a ) are given by: P (0 | , ) + P (1 | , ) + P (2 | , ) + P (0 | , ) ≤ λ , P (1 | , ) + P (2 | , ) + P (0 | , ) + P (1 | , ) ≤ λ , P (2 | , ) + P (0 | , ) + P (1 | , ) + P (2 | , ) ≤ λ , P (0 | , ) + P (1 | , ) + P (2 | , ) ≤ λ , P (0 | , ) + P (1 | , ) + P (2 | , ) ≤ λ , P (0 | , ) + P (1 | , ) + P (2 | , ) ≤ λ , Since the maximal violation of the CGLMP inequalityis self-testing [29], the optimal state and measurementsto violate the CGLMP inequality are unique up to lo-cal isometries. The optimal state is given by | ψ γ i =( | i + γ | i + | i ) / p γ where γ = √ −√ [1],while the optimal measurements are described in equa-tion (C2). Hence, it is possible to study the the inequali-ties h B ( x, a ) i ≤ λ ( x,a ) at the point of maximal CGLMP violation. Table I shows the values of h B ( x, a ) i and λ x,a ) ∀ x, a when I ( P ) = I Q : since they are different,the non-local game I does not exhibit the OW property. TABLE I. B ( x, a ) and λ x,a ) of the CGLMP game I ( x, a ) (0 ,
0) (0 ,
1) (0 ,
2) (1 ,
0) (1 ,
1) (1 , λ x,a ) h B ( x, a ) i Appendix C: OW-games for the maximal violationof CGLMP d In this Appendix, we provide the explicit proof thatthe non-local game G d defined in (19) is an OW-gamefor all d .We first write the non-local game G d in a way to showAlice’s steering: G d = d − X a =0 P ( a | d − X ∆=0 ∆ h P ( b = a − − ∆ | a , P ( b = a + 1 + ∆ | a , i + d − X a =0 P ( a | d − X ∆=0 ∆ h P ( b = a + 1 + ∆ | a , P ( b = a − ∆ | a , i . (C1)We assume that the maximal violation of CGLMP d canonly be obtained by a suitable state and the projectivemeasurements E xa = | a x i h a x | and E yb = | b y i h b y | definedby | a x i = 1 √ d d − X k =0 exp (cid:18) i πd ka (cid:19) exp ( ikφ x ) | k i , | b y i = 1 √ d d − X k =0 exp (cid:18) − i πd kb (cid:19) exp ( ikθ y ) | k i , (C2)with φ = 0 , φ = πd , θ = − π d , and θ = π d [9]. Forlarge d , this form of the optimal measurements is conjec-tured based on numerical results [30].It follows that B (0 , a ) = d − X ∆=0 ∆ h E y =0 b = a − − ∆ + E y =1 b = a +1+∆ i = d − X k,k ′ =0 exp (cid:18) − i πd a ( k − k ′ ) (cid:19) f ( k, k ′ ) | k i h k ′ | , where f ( k, k ′ ) = d P d − ∆ cos (cid:2) π d ( k − k ′ ) (4∆ + 3) (cid:3) ; and similarly, B (1 , a ) = d − X ∆=0 ∆ h E y =0 b = a +1+∆ + E y =1 b = a − ∆ i = d − X k,k ′ =0 exp (cid:18) − i πd a ( k − k ′ ) (cid:19) f ( k, k ′ ) | k i h k ′ | , where f ( k, k ′ ) = f ( k, k ′ ) exp (cid:0) − i πd ( k − k ′ ) (cid:1) .The four B ( x, a ) can be transformed into each othersby unitaries. For fixed x , U a ′ a B ( x, a ) U † a ′ a = B ( x, a ′ ) holds for U a ′ a = P k e − i πd ( a ′ − a ) k | k i h k | . Similarly,there exist V = P k e − i πd k | k i h k | such that B (1 , a ) = V B (0 , a ) V † . This implies that all B ( x, a ) share the samemaximal eigenvalue, i.e. λ d ( x,a ) = λ d . We’ll denote by | β ( x,a ) i the eigenstate associated to the maximal eigen-value of B ( x, a ) .Now we need to show that there always exists abipartite pure state | ψ AB i such that the OW crite-rion (3) holds. Let’s set | β (0 , i = P k β k | k i ; by theunitary relationship between different B ( x,a ) , it followsthat | β (0 ,a ) i = U a | β (0 , i and | β (1 ,a ) i = U a V | β (0 , i .Using these relations, one can verify that the bipar-tite state | ψ AB i = P k β k | kk i is such that | ψ ( x,a ) B i = P k exp (cid:0) − i πd ak (cid:1) exp ( − ikφ x ) β k | k i ≡ | β ( x,a ) i . Thisconcludes the proof that G dd