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PHYSICAL REVIEW LETTERS
Strong Monogamies of No-Signaling Violations for Bipartite Correlation Bell Inequalities 1
Ravishankar Ramanathan1,2,* and Paweł Horodecki1,3
National Quantum Information Center of Gdansk, 81-824 Sopot, Poland 2 University of Gdansk, 80-952 Gdansk, Poland 3 Faculty of Applied Physics and Mathematics, Technical University of Gdansk, 80-233 Gdansk, Poland (Received 20 February 2014; revised manuscript received 21 August 2014; published 20 November 2014) The phenomenon of monogamy of Bell inequality violations is interesting both from the fundamental perspective as well as in cryptographic applications such as the extraction of randomness and secret bits. In this article, we derive new and stronger monogamy relations for violations of Bell inequalities in general no-signaling theories. These relations are applicable to the class of binary output correlation inequalities known as XOR games, and to free unique games. In many instances of interest, we show that the derived relation provides a significant strengthening over previously known results. Our result connects, for the first time, the property of monogamy with that crucial part of the Bell expression that is necessary for revealing a contradiction with local realistic predictions, thus shifting the paradigm in the field of monogamy of correlations. DOI: 10.1103/PhysRevLett.113.210403
PACS numbers: 03.65.Ud, 03.65.Ta, 03.67.Dd, 03.67.Mn
Introduction.—The violation of a Bell inequality is a defining feature of quantum correlations that distinguishes them from classical correlations. Apart from the fundamental interest in the absence of a local hidden variable description of nature, this feature of quantum theory has led to numerous applications in communication protocols, including key distribution [1,2] and generation of randomness [3,4]. Many interesting properties of this phenomenon dubbed nonlocality have been discovered leading to a better understanding of the set of quantum correlations [5,6] that is crucial in the development of further applications. It is known that the set of quantum correlations is a convex set sandwiched between the classical polytope and the nosignaling polytope, which is the set of correlations obeying the no-signaling principle (impossibility of faster-than-light communication). The violation of Bell inequalities in general no-signaling theories (which includes quantum theory) displays a very interesting property called monogamy. Strong nonlocal correlations between two parties in extreme cases lead to weak correlations between these parties and any other nosignaling system. Specifically, there are instances where the violation of a Bell inequality by Alice and Bob precludes its violation by Alice and any other party Charlie, when Alice uses the same measurement results in both experiments. This phenomenon is seen to be important in secure communication protocols for key distribution or randomness generation between these parties, due to the fact that any third party such as an eavesdropper is only able to establish weak correlations with their systems [4,7,8]. The monogamy of no-signaling correlations was first discovered for the CHSH inequality in [9] and has since been shown to be a generic feature of all no-signaling theories [5]. A general monogamy relation applicable to 0031-9007=14=113(21)=210403(5)
any no-signaling correlations in the two-party Bell scenario that only involves the number of settings (inputs) for one party was developed in [10,11]. This approach follows the idea of symmetric extensions and shareability of correlations which can also be recast in terms of the existence of joint probability distributions [12]. See also [13] for a novel method to deriving monogamies based on the powerful machinery of de Finetti theorems for no-signaling probability distributions. In this Letter, we adopt a different approach and derive a strengthened version of the monogamy relation for two-party inequalities that holds in many flagship scenarios involving correlation expressions, in particular, to the wide class of binary output correlation scenarios known as XOR games and to so-called free unique games. We also investigate the validity of the monogamy relation and the strengthening it provides in many relevant scenarios. The result highlights the importance in monogamies of the number of settings of one party that lead to a contradiction with local realistic predictions. Monogamy relations for Bell inequality violations from no-signaling constraints.—The general two-party Bell inequality associated to a nonlocality game is written as BAB ≔ BAB :fPða; bjx; yÞg ¼
mA X mB X d X
μðx; yÞVða; bjx; yÞPða; bjx; yÞ
x¼1 y¼1 a;b¼1
≤ RL ðBÞ; where mA ðmB Þ denotes the number of settings for Alice (Bob) and d ¼ dA ; dB denotes the number of outcomes of both Alice and Bob, with redundant outcomes added if necessary to make dA ¼ dB . The probability distribution
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with which the inputs are chosen is denoted by μðx; yÞ (0 ≤ μðx; yÞ ≤ 1). For the commonly considered situation of “free” games, this distribution is taken to be of product form μðx; yÞ ¼ μA ðxÞμB ðyÞ, reflecting the independence of Alice and Bob in choosing their measurement settings. The Vða; bjx; yÞ picks out the probabilities Pða; bjx; yÞ with appropriate coefficients that enter the Bell inequality. The bound RL ðBÞ is the maximum attainable value of the Bell expression by local deterministic boxes and consequently by their mixtures. The Bell vector BAB has entries BAB ða; b; x; yÞ ¼ μðx; yÞVða; bjx; yÞ and the no-signaling box fPða; bjx; yÞg describes the output distributions for different inputs. We note that the input probability distributions μðx; yÞ only appear in the general form above if we consider the Bell inequality as corresponding to a nonlocal game played by Alice and Bob; more general Bell expressions need not refer to input probability distributions at all. All the considerations in this article are confined to two-party Bell inequalities, so we will skip the suffix AB wherever it is not required. All the proofs are deferred to the Supplemental Material [14]. We consider the scenario where Alice performs the Bell experiment simultaneously with many Bobs Bi , with i ∈ ½n for some n. A no-signaling constraint across all parties ~ y~Þ with b~ ¼ fb1 ; …; bn g; is imposed on the box Pða; bjx; 1 n y~ ¼ fy ; …; y g, inputs x ∈ ½mA ; yi ∈ ½mB and outputs a; bi ∈ ½d ∀i, i.e., d X
~ y~Þ ¼ Pða; bjx;
a¼1 d X
d X
ðiÞ
bi ¼1
d X
~ 0 ; y~Þ∀b; ~ y~;x;x0 ; Pða; bjx ðiÞ ðiÞ ðiÞ Pða; b~ jx; y~0 Þ ∀ a;x; y~0 ; y~ðiÞ ;
bi ¼1
ð1Þ ðiÞ where b~ ¼ fb1 ; …; bi ; …; bn g, y~ðiÞ ¼ fy1 ;…;yi ;…;yn g, ðiÞ and y~0 ¼ fy1 ; …; y0i ; …; yn g. Building on the idea of symmetric extensions [10], it was shown in [11] that a monogamy relation holds for arbitrary Bell inequalities in any no-signaling theory in the following form. Theorem 1 [10,11]: Consider the scenario where Alice wants to perform a certain Bell experiment B with K Bobs denoted B1 ; …; BK , simultaneously. The associated Bell expressions BABi for i ∈ ½K satisfy a monogamy relation when K ¼ mB , i.e., for any box fPða; b1 ; …; bmB × jx; y1 ; …; ymB Þg with x ∈ ½mA ; yi ∈ ½mB and a; bi ∈ ½d ∀i, in any no-signaling theory it holds that mB X i¼1
BABi ¼
mB X
BABi :fPða; b1 ; …; bmB jx; y1 ; …; ymB Þg
i¼1
≤ mB RL ðBÞ:
Notice that the above relation depends upon the inequality only in so far as the number of settings of Bob (mB ) is concerned, in particular, it is independent of mA ; d and μðx; yÞ. Quantum theory being a no-signaling theory obeys Eq. (2) and, in particular instances, is also known to satisfy more stringent monogamy relations [9,17] due to its additional structure. From denote Pn here on, we 1 i BAB :fPða; bjx; yÞgP by B and i¼1 BAB :fPða; b ; …; bn × 1 n jx; y ; …; y Þg as ni¼1 BABi for ease of notation. In this Letter, we derive a no-signaling monogamy relation that is applicable to a wide class of bipartite inequalities involving correlation expressions. To this end, we consider a parameter that we call the contradiction number characterizing the Bell expression. This quantity denotes the difference between the number of measurement settings of one party (in this case Bob) and the maximum number of their measurement settings up to which the optimal no-signaling value can be attained by a local deterministic box. Definition 1 [Contradiction number]: For any Bell inequality BAB ≤ RL ≤ RNS, denote by SðBÞ the set of settings of party B of minimum cardinality CB ≔ jSðBÞj whose removal leads to the optimum no-signaling value being achieved by a local deterministic box, i.e., nc nc nc nc Bnc AB ≤ RL ðB Þ ¼ RNS ðB Þ, where Bnc AB
≔
mA mX B −CB X x¼1
a¼1
Pða; b~ jx; y~ðiÞ Þ ¼
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ð2Þ
y¼1
μðx; yÞ
d X
Vða; bjx; yÞPða; bjx; yÞ ð3Þ
a;b¼1
nc and Rnc L ðB Þ is the optimum local value of the expression nc nc BAB while Rnc NS ðB Þ is its optimum no-signaling value. We then call CB as the contradiction number for the Bell inequality. The contradiction number of a Bell inequality is a crucial parameter that represents the number of settings which give rise to a contradiction with the predictions of a local hidden variable model. For instance, when there is restricted free choice in each run of the experiment so that these number of settings have zero probability of being chosen (perhaps due to a faulty random number generator), a local hidden variable explanation emerges for the correlations. The strengthened monogamy relation that we derive is given in terms of the contradiction number of the inequality as CX B þ1 i¼1
BABi ¼
CX B þ1
BABi :fPða; b1 ; …; bCB þ1 jx; y1 ; …; yCB þ1 Þg
i¼1
≤ ðCB þ 1ÞRL ðBÞ:
ð4Þ
Clearly, CB ≤ mB − 1 since a classical winning strategy always exists when considering only a single setting of Bob. As we shall see, in many instances of interest the inequality is strict so that the relation in Eq. (4) provides a significant strengthening to that in Eq. (2). Note that the above definition could be modified to take into account
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either Alice or Bob, in which case the monogamy relation will hold with the common party being the one that does not achieve the minimum cardinality. Monogamy of correlation Bell expressions.—A natural generalization of the CHSH inequality to more inputs mA ; mB is to Bell inequalities involving the correlators Ex;y defined as [18] Ex;y ¼
d−1 X
λk Pða − b ¼ k mod djx; yÞ;
ð5Þ
of choice in many applications of nonlocality. The paradigmatic example of such a XOR game for which Eq. (4) significantly outperforms previously known monogamy relations is the chained Bell inequality of Braunstein and Caves [22]. These inequalities have been applied in randomness amplification and key distribution protocols secure against no-signaling adversaries [4,7,8]. The chained Bell expression Bch;N AB involves two parties with mA ¼ mB ¼ N inputs and d ¼ 2 outcomes, it is written as N X X
k¼0
with real parameters λk . These “correlation Bell expressions” do not depend on the individual values assigned to Alice and Bob’s outcomes, but rather on how these outcomes are correlated with each other. These are the most common Bell inequalities and are written in general as B≑ ≔
mA X mB X
αx;y Ex;y ≤ RL ðB≑ Þ;
ð6Þ
x¼1 y¼1
where αx;y are arbitrary reals. In the scenario of binary outputs (d ¼ 2), the correlation inequalities are also known as XOR games. XOR games (for two parties) consider the scenario when the predicate Vða; bjx; yÞ ∈ f0; 1g only depends upon the xor of the two parties’ outcomes Vða ⊕ bjx; yÞ. The fact that these are equivalent to correlation inequalities for d ¼ 2 is simply seen by noting that for a; b; k ∈ f0; 1g, we have Pða ⊕ b ¼ kjx; yÞ ¼ 12 ½1 þ ð−1Þk Exy , where the correlators Exy are defined using λk ¼ ð−1Þk . XOR games are a widely studied and the most well-understood class of Bell inequalities due to the fact that the maximum quantum values of these inequalities are directly calculable by a semidefinite program [19–21]. Our first main result is the following. Proposition 1: Consider the general two-party XOR Bell expression B⊕ : ⊕ B⊕ AB ≔ BAB :fPða; bjx; yÞg mA X mB X X μðx; yÞ Vða ⊕ bjx; yÞPða; bjx; yÞ ¼ x¼1 y¼1
≤ RL
a;b¼0;1
ðB⊕ Þ
with corresponding number of contradictions CB⊕ . For any mA ; mB ; μðx; yÞ and any no-signaling box fPða; b1 ; …; bCB⊕ þ1 jx; y1 ; …; yCB⊕ þ1 Þg with x ∈ ½mA ; yi ∈ ½mB and a; bi ∈ f0; 1g∀ i, we have that B⊕ satisfies Eq. (4), i.e., CB⊕ þ1
X i¼1
B⊕ ≤ ðCB⊕ þ 1ÞRL ðB⊕ Þ: ABi
ð7Þ
The interest in the above statement is due to the fact that two-party XOR games still provide the Bell inequality
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a;b¼0;1
μðx; yÞPða ⊕ b ¼ 0jx; yÞ
x;y¼1 x¼y∨x¼yþ1
þμð1; NÞPða ⊕ b ¼ 1j1; NÞ ≤ RL ðBch;N Þ;
ð8Þ
where RL ðBch;N Þ ¼ 1 − minx;y μðx; yÞ is the maximum achievable value by a local box. Example: For any no-signaling box fPða; b; cjx; y; zÞg with a; b; c ∈ f0; 1g and x; y; z ∈ ½N, the chain Bell inequality satisfies the strong monogamy relation ch;N ch;N Bch;N Þ: AB þ BAC ≤ 2RL ðB
ð9Þ
The fact that the contradiction number for chain inequalities is one is due to the following observation. If Bob fails to measure even a single setting out of the N, perhaps due to an imperfect random number generator choosing his inputs [4], all the constraints in the remaining part of the expression can be satisfied by a deterministic box. Explicitly, if Bob fails to measure setting k ∈ ½N in an N chain, a classical winning strategy is for Alice to output a ¼ 0 for 1 ≤ x ≤ k and a ¼ 1 for k < x ≤ N while Bob outputs b ¼ 0 for y ∈ ½k − 1 and b ¼ 1 for k < y ≤ N. In this case therefore, CBch;N þ 1 ¼ 2 is much smaller than mB ¼ N, so that Eq. (4) provides a significant strengthening over Eq. (2). Interestingly, the monogamy relation also holds when one considers the same family of XOR games, i.e., where the predicate is defined by the same condition but with the inputs being specified over different domain sizes N and M, an observation we pursue elsewhere. The strong monogamy relation for the chain inequality implies that any violation by Alice and Bob precludes its violation by Alice and Charlie. In a cryptographic application [4,8], this suggests that if Alice and Bob are able to test for strong correlations leading to a violation of the chain inequality, then the correlations that their systems share with an eavesdropper are necessarily weaker. Interestingly though, the very fact that the contradiction number is small provides an attack in this task to the eavesdropper who may, for instance, tamper with the random number generator in the Bell test so that Bob does not measure a single setting and a local model exists. Unique games.—A generalization of xor games to more outputs is a family of Bell inequalities called unique games
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[23], which also involve only certain correlations between Alice and Bob’s outputs with no constraint on the marginals. A two-party game is called unique if for every input pair x; y with x ∈ ½mA and y ∈ ½mB and for every output a ∈ ½d of Alice, Bob is required to produce a unique output b ∈ ½d defined by some permutation of Alice’s output which can be different for different input pairs, i.e., Vða; bjx; yÞ ¼ 1 if and only if b ¼ σ x;y ðaÞ and 0 otherwise. Unique games are an important class of Bell expressions, finding application in the field of computational complexity in the approximation of computationally hard problems. It is easy to see that the maximum value of every unique game with fixed fσ x;y g can be achieved by a no-signaling box fPða; bjx; yÞg with nonzero entries Pða; σ x;y ðaÞjx; yÞ ¼ ð1=dÞ ∀ a ∈ ½d. We now investigate the monogamy relations for this important category of Bell expressions. In order to do this, we introduce a strengthened version of the concept of contradiction number. Definition 2 [Strong contradiction number]: For any Bell inequality BAB ≤ RL ≤ RNS, denote by SðsÞ ðBÞ the set ðsÞ of settings of party B of minimum cardinality CB ≔ jSðsÞ ðBÞj whose removal leads to the optimum no-signaling value being achieved by local deterministic boxes, nc nc nc nc i.e.,Bnc AB ≤ RL ðB Þ ¼ RNS ðB Þ, where ðsÞ
Bnc AB ≔
−CB mA mBX X x¼1
μðx;yÞ
y¼1
d X
Vða;bjx;yÞPða;bjx;yÞ; ð10Þ
a;b¼1
and moreover there exists for every fa; xg a local deternc ministic box achieving value Rnc L ðB Þ that deterministiðsÞ cally outputs a for input x. We then call CB as the strong contradiction number for the Bell inequality. In the important case of the so-called free unique games, i.e., where μðx; yÞ ¼ μA ðxÞμB ðyÞ the strong monogamy relation does hold. For more general unique games, the proofs presented here do not apply and it is unclear whether the strengthened monogamy relation holds; however, it does for the specific case when mA ¼ 2, and arbitrary mB ; d; μðx; yÞ. Proposition 2: Any unique game defining an inequality BU AB ≔
mA X mB X
μðx; yÞ
x¼1 y¼1
d X
Vða; σ x;y ðaÞjx; yÞPða; bjx; yÞ
a;b¼1
≤ RL ðBU Þ
ð11Þ
with the restriction that μðx; yÞ ¼ μA ðxÞμB ðyÞ ∀ x; y and ðsÞ with associated strong contradiction number CBU satisfies Eq. (4) in any no-signaling theory independently of mA ; mB ; d, i.e., ðsÞ C U þ1 B
X i¼1
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PCþ1 FIG. 1 (color online). Sketch of the proof: Rewrite i¼1 BABi PCþ1 as a sum i¼1 Bi , where Bi consists of two parts: one between Alice and the ith Bob for which the local value equals the no-signaling value, and a second where each of the remaining C Bobs has a single input.
Moreover, every unique game with mA ¼ 2 obeys Eq. (4) independent of mB ; d, and μðx; yÞ. Before we proceed, let us briefly sketch the proofs of Propositions 1 and 2; an illustration is provided in Fig. 1. First, we rewrite the sum of Bell expressions BABi in Eq. (4) c as a sum of expressions of the form Bi ¼ Bnc i þ Bi , where nc Bi is an expression involving a single Bob and having i i no contradictions (so that a local box fPnc L ða; b jx; y Þg c achieves its optimal value) and Bi is an expression involving C other Bobs, each of whom measures a single setting (so that ~ ~ again a local box fPcL ða; bj jx; yj Þg with ~j ¼ ½C þ 1nfig achieves its optimum). The crucial part of the proof is to show the existence of a single local box fPL ða; b1 ; …; bCþ1 jx; y1 ; …; yCþ1 Þg that achieves the optimal value of c both expressions Bnc i and Bi simultaneously. In order to do this, we show a useful Lemma that for the inequalities under consideration, the optimal value of Bnc i is achieved by a local box for any marginal distribution on Alice’s side. We now proceed to note an interesting property of unique games, namely, that every nontrivial unique game [with RNS ðBU Þ > RL ðBU Þ] is monogamous at least to a slight degree in the sense that it is not possible for AliceBob and Alice-Charlie to simultaneously achieve the maximum no-signaling value of the game. Observation 1: For any unique game BU with parameters mA ; mB ; d, and RNS ðBU Þ > RL ðBU Þ and any nosignaling box fPða; b; cjx; y; zÞg with a; b; c ∈ ½d and x ∈ ½mA ; y; z ∈ ½mB , we have U U ðBU AB þ BAC Þ⋅fPða; b; cjx; y; zÞg < 2RNS ðB Þ
ðsÞ
BU ≤ ðCBU þ 1ÞRL ðBU Þ: ABi
ð12Þ
ð13Þ
Discussion.—A natural question to investigate is whether the strengthened monogamy relations derived here could
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possibly hold for all inequalities beyond the correlation expressions considered here. Another interesting question is whether all Bell inequality violations are monogamous at least to a slight degree, i.e., is it always the case that when two parties Alice and Bob violate a Bell inequality to its optimal no-signaling value, Alice and Charlie are unable to also optimally violate the inequality? Interestingly, the answer to both questions turns out to be negative due to the following. Claim 1: There exists a two-party Bell expression B with mA ¼ mB ¼ 3; d ¼ 4, associated contradiction numðsÞ bers CB ¼ CB ¼ 1, maximum no-signaling value greater than the maximum local value (RNS > RL ), and a threeparty no-signaling box fPða; b; cjx; y; zÞg with a; b; c ∈ f1; 2; 3; 4g and x; y; z ∈ f1; 2; 3g such that ðBAB þ BAC Þ:fPða; b; cjx; y; zÞg ¼ 2RNS ðBÞ:
ð14Þ
We present in the Supplemental Material [14] the Bell inequality which does not fall within the classes considered in this Letter and which counteracts the hope that the strengthened monogamy relation for the contradiction number can be applied to arbitrary Bell inequalities beyond those involving correlation expressions. The domain of validity and the tightness of the monogamy relations deserves further investigation. Conclusions.—In this Letter, we have presented a strong monogamy relation that applies to a wide class of Bell inequalities such as XOR games and free unique games. These monogamy relations provide a significant improvement over previously known results and suggest why well-known inequalities such as the Braunstein-Caves chain inequalities are useful in many cryptographic tasks. The fact that the monogamy relations depend exactly on the contradiction number of the Bell inequality enables the identification of suitable inequalities for further use in device-independent protocols. The results in our Letter suggest the use of games with a small contradiction number for applications, as these are more likely to guarantee weak correlations with an eavesdropper. The methods presented here can also be applied to derive monogamy relations between different Bell inequalities within the same class, an investigation we defer for the future. Apart from the quantitative results presented here, the shift in the paradigm from the importance of the number of settings to the number of contradictions leads also to questions regarding the exact physical reasons underpinning these monogamies and their multipartite generalization. One might also consider monogamies in scenarios with multiple Alices and Bobs, in which case in addition to the number of contradictions, the placement and relations between different contradictory settings should also be relevant.
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This work is supported by ERC AdG Grant QOLAPS and by the Foundation for Polish Science (FNP) TEAM project co-financed by the EU European Regional Development Fund. We thank M. L. Nowakowski, M. Horodecki, and K. Horodecki for useful discussions. Part of this work was carried out when the authors attended the program “Mathematical Challenges in Quantum Information” at the Isaac Newton Institute for Mathematical Sciences in the University of Cambridge.
*
[email protected] [1] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). [2] J. Barrett, L. Hardy, and A. Kent, Phys. Rev. Lett. 95, 010503 (2005). [3] S. Pironio et al., Nature (London) 464, 1021 (2010). [4] R. Colbeck and R. Renner, Nat. Phys. 8, 450 (2012). [5] Ll. Masanes, A. Acin, and N. Gisin, Phys. Rev. A. 73, 012112 (2006). [6] N. Brunner and P. Skrzypczyk, Phys. Rev. Lett. 102, 160403 (2009). [7] R. Augusiak, M. Demianowicz, M. Pawłowski, J. Tura, and A. Acin, arXiv:1307.6390. [8] J. Barrett, R. Colbeck, and A. Kent, Phys. Rev. A 86, 062326 (2012). [9] B. Toner, Proc. R. Soc. A 465, 59 (2009). [10] B. M. Terhal, A. C. Doherty, and D. Schwab, Phys. Rev. Lett. 90, 157903 (2003). [11] M. Pawłowski and C. Brukner, Phys. Rev. Lett. 102, 030403 (2009). [12] R. Ramanathan, A. Soeda, P. Kurzynski, and D. Kaszlikowski, Phys. Rev. Lett. 109, 050404 (2012). [13] F. G. S. L. Brandao and A. W. Harrow, Proceedings of the 45th ACM Symposium on Theory of Computing (STOC 2013), p. 861 (ACM, New York, 2013). [14] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.113.210403, for the proofs of the Propositions stated in the main text, which includes Refs. [15,16]. [15] N. S. Jones and Ll. Masanes, Phys. Rev. A 72, 052312 (2005). [16] A. Cabello, Phys. Lett. A 377, 64 (2012). [17] P. Kurzynski, T. Paterek, R. Ramanathan, W. Laskowski, and D. Kaszlikowski, Phys. Rev. Lett. 106, 180402 (2011). [18] J.-D. Bancal, C. Branciard, N. Brunner, N. Gisin, and Y-C. Liang, J. Phys. A 45, 125301 (2012). [19] B. S. Cirelson, Lett. Math. Phys. 4, 93 (1980). [20] S. Wehner, Phys. Rev. A 73, 022110 (2006). [21] R. Cleve, P. Hoyer, B. Toner, and J. Watrous, arXiv:quantph/0404076. [22] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 61, 662 (1988). [23] J. Kempe and O. Regev, arXiv:0911.0201.
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PHYSICAL REVIEW LETTERS
Strong Monogamies of No-Signaling Violations for Bipartite Correlation Bell Inequalities 1
Ravishankar Ramanathan1,2,* and Paweł Horodecki1,3
National Quantum Information Center of Gdansk, 81-824 Sopot, Poland 2 University of Gdansk, 80-952 Gdansk, Poland 3 Faculty of Applied Physics and Mathematics, Technical University of Gdansk, 80-233 Gdansk, Poland (Received 20 February 2014; revised manuscript received 21 August 2014; published 20 November 2014) The phenomenon of monogamy of Bell inequality violations is interesting both from the fundamental perspective as well as in cryptographic applications such as the extraction of randomness and secret bits. In this article, we derive new and stronger monogamy relations for violations of Bell inequalities in general no-signaling theories. These relations are applicable to the class of binary output correlation inequalities known as XOR games, and to free unique games. In many instances of interest, we show that the derived relation provides a significant strengthening over previously known results. Our result connects, for the first time, the property of monogamy with that crucial part of the Bell expression that is necessary for revealing a contradiction with local realistic predictions, thus shifting the paradigm in the field of monogamy of correlations. DOI: 10.1103/PhysRevLett.113.210403
PACS numbers: 03.65.Ud, 03.65.Ta, 03.67.Dd, 03.67.Mn
Introduction.—The violation of a Bell inequality is a defining feature of quantum correlations that distinguishes them from classical correlations. Apart from the fundamental interest in the absence of a local hidden variable description of nature, this feature of quantum theory has led to numerous applications in communication protocols, including key distribution [1,2] and generation of randomness [3,4]. Many interesting properties of this phenomenon dubbed nonlocality have been discovered leading to a better understanding of the set of quantum correlations [5,6] that is crucial in the development of further applications. It is known that the set of quantum correlations is a convex set sandwiched between the classical polytope and the nosignaling polytope, which is the set of correlations obeying the no-signaling principle (impossibility of faster-than-light communication). The violation of Bell inequalities in general no-signaling theories (which includes quantum theory) displays a very interesting property called monogamy. Strong nonlocal correlations between two parties in extreme cases lead to weak correlations between these parties and any other nosignaling system. Specifically, there are instances where the violation of a Bell inequality by Alice and Bob precludes its violation by Alice and any other party Charlie, when Alice uses the same measurement results in both experiments. This phenomenon is seen to be important in secure communication protocols for key distribution or randomness generation between these parties, due to the fact that any third party such as an eavesdropper is only able to establish weak correlations with their systems [4,7,8]. The monogamy of no-signaling correlations was first discovered for the CHSH inequality in [9] and has since been shown to be a generic feature of all no-signaling theories [5]. A general monogamy relation applicable to 0031-9007=14=113(21)=210403(5)
any no-signaling correlations in the two-party Bell scenario that only involves the number of settings (inputs) for one party was developed in [10,11]. This approach follows the idea of symmetric extensions and shareability of correlations which can also be recast in terms of the existence of joint probability distributions [12]. See also [13] for a novel method to deriving monogamies based on the powerful machinery of de Finetti theorems for no-signaling probability distributions. In this Letter, we adopt a different approach and derive a strengthened version of the monogamy relation for two-party inequalities that holds in many flagship scenarios involving correlation expressions, in particular, to the wide class of binary output correlation scenarios known as XOR games and to so-called free unique games. We also investigate the validity of the monogamy relation and the strengthening it provides in many relevant scenarios. The result highlights the importance in monogamies of the number of settings of one party that lead to a contradiction with local realistic predictions. Monogamy relations for Bell inequality violations from no-signaling constraints.—The general two-party Bell inequality associated to a nonlocality game is written as BAB ≔ BAB :fPða; bjx; yÞg ¼
mA X mB X d X
μðx; yÞVða; bjx; yÞPða; bjx; yÞ
x¼1 y¼1 a;b¼1
≤ RL ðBÞ; where mA ðmB Þ denotes the number of settings for Alice (Bob) and d ¼ dA ; dB denotes the number of outcomes of both Alice and Bob, with redundant outcomes added if necessary to make dA ¼ dB . The probability distribution
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with which the inputs are chosen is denoted by μðx; yÞ (0 ≤ μðx; yÞ ≤ 1). For the commonly considered situation of “free” games, this distribution is taken to be of product form μðx; yÞ ¼ μA ðxÞμB ðyÞ, reflecting the independence of Alice and Bob in choosing their measurement settings. The Vða; bjx; yÞ picks out the probabilities Pða; bjx; yÞ with appropriate coefficients that enter the Bell inequality. The bound RL ðBÞ is the maximum attainable value of the Bell expression by local deterministic boxes and consequently by their mixtures. The Bell vector BAB has entries BAB ða; b; x; yÞ ¼ μðx; yÞVða; bjx; yÞ and the no-signaling box fPða; bjx; yÞg describes the output distributions for different inputs. We note that the input probability distributions μðx; yÞ only appear in the general form above if we consider the Bell inequality as corresponding to a nonlocal game played by Alice and Bob; more general Bell expressions need not refer to input probability distributions at all. All the considerations in this article are confined to two-party Bell inequalities, so we will skip the suffix AB wherever it is not required. All the proofs are deferred to the Supplemental Material [14]. We consider the scenario where Alice performs the Bell experiment simultaneously with many Bobs Bi , with i ∈ ½n for some n. A no-signaling constraint across all parties ~ y~Þ with b~ ¼ fb1 ; …; bn g; is imposed on the box Pða; bjx; 1 n y~ ¼ fy ; …; y g, inputs x ∈ ½mA ; yi ∈ ½mB and outputs a; bi ∈ ½d ∀i, i.e., d X
~ y~Þ ¼ Pða; bjx;
a¼1 d X
d X
ðiÞ
bi ¼1
d X
~ 0 ; y~Þ∀b; ~ y~;x;x0 ; Pða; bjx ðiÞ ðiÞ ðiÞ Pða; b~ jx; y~0 Þ ∀ a;x; y~0 ; y~ðiÞ ;
bi ¼1
ð1Þ ðiÞ where b~ ¼ fb1 ; …; bi ; …; bn g, y~ðiÞ ¼ fy1 ;…;yi ;…;yn g, ðiÞ and y~0 ¼ fy1 ; …; y0i ; …; yn g. Building on the idea of symmetric extensions [10], it was shown in [11] that a monogamy relation holds for arbitrary Bell inequalities in any no-signaling theory in the following form. Theorem 1 [10,11]: Consider the scenario where Alice wants to perform a certain Bell experiment B with K Bobs denoted B1 ; …; BK , simultaneously. The associated Bell expressions BABi for i ∈ ½K satisfy a monogamy relation when K ¼ mB , i.e., for any box fPða; b1 ; …; bmB × jx; y1 ; …; ymB Þg with x ∈ ½mA ; yi ∈ ½mB and a; bi ∈ ½d ∀i, in any no-signaling theory it holds that mB X i¼1
BABi ¼
mB X
BABi :fPða; b1 ; …; bmB jx; y1 ; …; ymB Þg
i¼1
≤ mB RL ðBÞ:
Notice that the above relation depends upon the inequality only in so far as the number of settings of Bob (mB ) is concerned, in particular, it is independent of mA ; d and μðx; yÞ. Quantum theory being a no-signaling theory obeys Eq. (2) and, in particular instances, is also known to satisfy more stringent monogamy relations [9,17] due to its additional structure. From denote Pn here on, we 1 i BAB :fPða; bjx; yÞgP by B and i¼1 BAB :fPða; b ; …; bn × 1 n jx; y ; …; y Þg as ni¼1 BABi for ease of notation. In this Letter, we derive a no-signaling monogamy relation that is applicable to a wide class of bipartite inequalities involving correlation expressions. To this end, we consider a parameter that we call the contradiction number characterizing the Bell expression. This quantity denotes the difference between the number of measurement settings of one party (in this case Bob) and the maximum number of their measurement settings up to which the optimal no-signaling value can be attained by a local deterministic box. Definition 1 [Contradiction number]: For any Bell inequality BAB ≤ RL ≤ RNS, denote by SðBÞ the set of settings of party B of minimum cardinality CB ≔ jSðBÞj whose removal leads to the optimum no-signaling value being achieved by a local deterministic box, i.e., nc nc nc nc Bnc AB ≤ RL ðB Þ ¼ RNS ðB Þ, where Bnc AB
≔
mA mX B −CB X x¼1
a¼1
Pða; b~ jx; y~ðiÞ Þ ¼
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ð2Þ
y¼1
μðx; yÞ
d X
Vða; bjx; yÞPða; bjx; yÞ ð3Þ
a;b¼1
nc and Rnc L ðB Þ is the optimum local value of the expression nc nc BAB while Rnc NS ðB Þ is its optimum no-signaling value. We then call CB as the contradiction number for the Bell inequality. The contradiction number of a Bell inequality is a crucial parameter that represents the number of settings which give rise to a contradiction with the predictions of a local hidden variable model. For instance, when there is restricted free choice in each run of the experiment so that these number of settings have zero probability of being chosen (perhaps due to a faulty random number generator), a local hidden variable explanation emerges for the correlations. The strengthened monogamy relation that we derive is given in terms of the contradiction number of the inequality as CX B þ1 i¼1
BABi ¼
CX B þ1
BABi :fPða; b1 ; …; bCB þ1 jx; y1 ; …; yCB þ1 Þg
i¼1
≤ ðCB þ 1ÞRL ðBÞ:
ð4Þ
Clearly, CB ≤ mB − 1 since a classical winning strategy always exists when considering only a single setting of Bob. As we shall see, in many instances of interest the inequality is strict so that the relation in Eq. (4) provides a significant strengthening to that in Eq. (2). Note that the above definition could be modified to take into account
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either Alice or Bob, in which case the monogamy relation will hold with the common party being the one that does not achieve the minimum cardinality. Monogamy of correlation Bell expressions.—A natural generalization of the CHSH inequality to more inputs mA ; mB is to Bell inequalities involving the correlators Ex;y defined as [18] Ex;y ¼
d−1 X
λk Pða − b ¼ k mod djx; yÞ;
ð5Þ
of choice in many applications of nonlocality. The paradigmatic example of such a XOR game for which Eq. (4) significantly outperforms previously known monogamy relations is the chained Bell inequality of Braunstein and Caves [22]. These inequalities have been applied in randomness amplification and key distribution protocols secure against no-signaling adversaries [4,7,8]. The chained Bell expression Bch;N AB involves two parties with mA ¼ mB ¼ N inputs and d ¼ 2 outcomes, it is written as N X X
k¼0
with real parameters λk . These “correlation Bell expressions” do not depend on the individual values assigned to Alice and Bob’s outcomes, but rather on how these outcomes are correlated with each other. These are the most common Bell inequalities and are written in general as B≑ ≔
mA X mB X
αx;y Ex;y ≤ RL ðB≑ Þ;
ð6Þ
x¼1 y¼1
where αx;y are arbitrary reals. In the scenario of binary outputs (d ¼ 2), the correlation inequalities are also known as XOR games. XOR games (for two parties) consider the scenario when the predicate Vða; bjx; yÞ ∈ f0; 1g only depends upon the xor of the two parties’ outcomes Vða ⊕ bjx; yÞ. The fact that these are equivalent to correlation inequalities for d ¼ 2 is simply seen by noting that for a; b; k ∈ f0; 1g, we have Pða ⊕ b ¼ kjx; yÞ ¼ 12 ½1 þ ð−1Þk Exy , where the correlators Exy are defined using λk ¼ ð−1Þk . XOR games are a widely studied and the most well-understood class of Bell inequalities due to the fact that the maximum quantum values of these inequalities are directly calculable by a semidefinite program [19–21]. Our first main result is the following. Proposition 1: Consider the general two-party XOR Bell expression B⊕ : ⊕ B⊕ AB ≔ BAB :fPða; bjx; yÞg mA X mB X X μðx; yÞ Vða ⊕ bjx; yÞPða; bjx; yÞ ¼ x¼1 y¼1
≤ RL
a;b¼0;1
ðB⊕ Þ
with corresponding number of contradictions CB⊕ . For any mA ; mB ; μðx; yÞ and any no-signaling box fPða; b1 ; …; bCB⊕ þ1 jx; y1 ; …; yCB⊕ þ1 Þg with x ∈ ½mA ; yi ∈ ½mB and a; bi ∈ f0; 1g∀ i, we have that B⊕ satisfies Eq. (4), i.e., CB⊕ þ1
X i¼1
B⊕ ≤ ðCB⊕ þ 1ÞRL ðB⊕ Þ: ABi
ð7Þ
The interest in the above statement is due to the fact that two-party XOR games still provide the Bell inequality
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a;b¼0;1
μðx; yÞPða ⊕ b ¼ 0jx; yÞ
x;y¼1 x¼y∨x¼yþ1
þμð1; NÞPða ⊕ b ¼ 1j1; NÞ ≤ RL ðBch;N Þ;
ð8Þ
where RL ðBch;N Þ ¼ 1 − minx;y μðx; yÞ is the maximum achievable value by a local box. Example: For any no-signaling box fPða; b; cjx; y; zÞg with a; b; c ∈ f0; 1g and x; y; z ∈ ½N, the chain Bell inequality satisfies the strong monogamy relation ch;N ch;N Bch;N Þ: AB þ BAC ≤ 2RL ðB
ð9Þ
The fact that the contradiction number for chain inequalities is one is due to the following observation. If Bob fails to measure even a single setting out of the N, perhaps due to an imperfect random number generator choosing his inputs [4], all the constraints in the remaining part of the expression can be satisfied by a deterministic box. Explicitly, if Bob fails to measure setting k ∈ ½N in an N chain, a classical winning strategy is for Alice to output a ¼ 0 for 1 ≤ x ≤ k and a ¼ 1 for k < x ≤ N while Bob outputs b ¼ 0 for y ∈ ½k − 1 and b ¼ 1 for k < y ≤ N. In this case therefore, CBch;N þ 1 ¼ 2 is much smaller than mB ¼ N, so that Eq. (4) provides a significant strengthening over Eq. (2). Interestingly, the monogamy relation also holds when one considers the same family of XOR games, i.e., where the predicate is defined by the same condition but with the inputs being specified over different domain sizes N and M, an observation we pursue elsewhere. The strong monogamy relation for the chain inequality implies that any violation by Alice and Bob precludes its violation by Alice and Charlie. In a cryptographic application [4,8], this suggests that if Alice and Bob are able to test for strong correlations leading to a violation of the chain inequality, then the correlations that their systems share with an eavesdropper are necessarily weaker. Interestingly though, the very fact that the contradiction number is small provides an attack in this task to the eavesdropper who may, for instance, tamper with the random number generator in the Bell test so that Bob does not measure a single setting and a local model exists. Unique games.—A generalization of xor games to more outputs is a family of Bell inequalities called unique games
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[23], which also involve only certain correlations between Alice and Bob’s outputs with no constraint on the marginals. A two-party game is called unique if for every input pair x; y with x ∈ ½mA and y ∈ ½mB and for every output a ∈ ½d of Alice, Bob is required to produce a unique output b ∈ ½d defined by some permutation of Alice’s output which can be different for different input pairs, i.e., Vða; bjx; yÞ ¼ 1 if and only if b ¼ σ x;y ðaÞ and 0 otherwise. Unique games are an important class of Bell expressions, finding application in the field of computational complexity in the approximation of computationally hard problems. It is easy to see that the maximum value of every unique game with fixed fσ x;y g can be achieved by a no-signaling box fPða; bjx; yÞg with nonzero entries Pða; σ x;y ðaÞjx; yÞ ¼ ð1=dÞ ∀ a ∈ ½d. We now investigate the monogamy relations for this important category of Bell expressions. In order to do this, we introduce a strengthened version of the concept of contradiction number. Definition 2 [Strong contradiction number]: For any Bell inequality BAB ≤ RL ≤ RNS, denote by SðsÞ ðBÞ the set ðsÞ of settings of party B of minimum cardinality CB ≔ jSðsÞ ðBÞj whose removal leads to the optimum no-signaling value being achieved by local deterministic boxes, nc nc nc nc i.e.,Bnc AB ≤ RL ðB Þ ¼ RNS ðB Þ, where ðsÞ
Bnc AB ≔
−CB mA mBX X x¼1
μðx;yÞ
y¼1
d X
Vða;bjx;yÞPða;bjx;yÞ; ð10Þ
a;b¼1
and moreover there exists for every fa; xg a local deternc ministic box achieving value Rnc L ðB Þ that deterministiðsÞ cally outputs a for input x. We then call CB as the strong contradiction number for the Bell inequality. In the important case of the so-called free unique games, i.e., where μðx; yÞ ¼ μA ðxÞμB ðyÞ the strong monogamy relation does hold. For more general unique games, the proofs presented here do not apply and it is unclear whether the strengthened monogamy relation holds; however, it does for the specific case when mA ¼ 2, and arbitrary mB ; d; μðx; yÞ. Proposition 2: Any unique game defining an inequality BU AB ≔
mA X mB X
μðx; yÞ
x¼1 y¼1
d X
Vða; σ x;y ðaÞjx; yÞPða; bjx; yÞ
a;b¼1
≤ RL ðBU Þ
ð11Þ
with the restriction that μðx; yÞ ¼ μA ðxÞμB ðyÞ ∀ x; y and ðsÞ with associated strong contradiction number CBU satisfies Eq. (4) in any no-signaling theory independently of mA ; mB ; d, i.e., ðsÞ C U þ1 B
X i¼1
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PCþ1 FIG. 1 (color online). Sketch of the proof: Rewrite i¼1 BABi PCþ1 as a sum i¼1 Bi , where Bi consists of two parts: one between Alice and the ith Bob for which the local value equals the no-signaling value, and a second where each of the remaining C Bobs has a single input.
Moreover, every unique game with mA ¼ 2 obeys Eq. (4) independent of mB ; d, and μðx; yÞ. Before we proceed, let us briefly sketch the proofs of Propositions 1 and 2; an illustration is provided in Fig. 1. First, we rewrite the sum of Bell expressions BABi in Eq. (4) c as a sum of expressions of the form Bi ¼ Bnc i þ Bi , where nc Bi is an expression involving a single Bob and having i i no contradictions (so that a local box fPnc L ða; b jx; y Þg c achieves its optimal value) and Bi is an expression involving C other Bobs, each of whom measures a single setting (so that ~ ~ again a local box fPcL ða; bj jx; yj Þg with ~j ¼ ½C þ 1nfig achieves its optimum). The crucial part of the proof is to show the existence of a single local box fPL ða; b1 ; …; bCþ1 jx; y1 ; …; yCþ1 Þg that achieves the optimal value of c both expressions Bnc i and Bi simultaneously. In order to do this, we show a useful Lemma that for the inequalities under consideration, the optimal value of Bnc i is achieved by a local box for any marginal distribution on Alice’s side. We now proceed to note an interesting property of unique games, namely, that every nontrivial unique game [with RNS ðBU Þ > RL ðBU Þ] is monogamous at least to a slight degree in the sense that it is not possible for AliceBob and Alice-Charlie to simultaneously achieve the maximum no-signaling value of the game. Observation 1: For any unique game BU with parameters mA ; mB ; d, and RNS ðBU Þ > RL ðBU Þ and any nosignaling box fPða; b; cjx; y; zÞg with a; b; c ∈ ½d and x ∈ ½mA ; y; z ∈ ½mB , we have U U ðBU AB þ BAC Þ⋅fPða; b; cjx; y; zÞg < 2RNS ðB Þ
ðsÞ
BU ≤ ðCBU þ 1ÞRL ðBU Þ: ABi
ð12Þ
ð13Þ
Discussion.—A natural question to investigate is whether the strengthened monogamy relations derived here could
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possibly hold for all inequalities beyond the correlation expressions considered here. Another interesting question is whether all Bell inequality violations are monogamous at least to a slight degree, i.e., is it always the case that when two parties Alice and Bob violate a Bell inequality to its optimal no-signaling value, Alice and Charlie are unable to also optimally violate the inequality? Interestingly, the answer to both questions turns out to be negative due to the following. Claim 1: There exists a two-party Bell expression B with mA ¼ mB ¼ 3; d ¼ 4, associated contradiction numðsÞ bers CB ¼ CB ¼ 1, maximum no-signaling value greater than the maximum local value (RNS > RL ), and a threeparty no-signaling box fPða; b; cjx; y; zÞg with a; b; c ∈ f1; 2; 3; 4g and x; y; z ∈ f1; 2; 3g such that ðBAB þ BAC Þ:fPða; b; cjx; y; zÞg ¼ 2RNS ðBÞ:
ð14Þ
We present in the Supplemental Material [14] the Bell inequality which does not fall within the classes considered in this Letter and which counteracts the hope that the strengthened monogamy relation for the contradiction number can be applied to arbitrary Bell inequalities beyond those involving correlation expressions. The domain of validity and the tightness of the monogamy relations deserves further investigation. Conclusions.—In this Letter, we have presented a strong monogamy relation that applies to a wide class of Bell inequalities such as XOR games and free unique games. These monogamy relations provide a significant improvement over previously known results and suggest why well-known inequalities such as the Braunstein-Caves chain inequalities are useful in many cryptographic tasks. The fact that the monogamy relations depend exactly on the contradiction number of the Bell inequality enables the identification of suitable inequalities for further use in device-independent protocols. The results in our Letter suggest the use of games with a small contradiction number for applications, as these are more likely to guarantee weak correlations with an eavesdropper. The methods presented here can also be applied to derive monogamy relations between different Bell inequalities within the same class, an investigation we defer for the future. Apart from the quantitative results presented here, the shift in the paradigm from the importance of the number of settings to the number of contradictions leads also to questions regarding the exact physical reasons underpinning these monogamies and their multipartite generalization. One might also consider monogamies in scenarios with multiple Alices and Bobs, in which case in addition to the number of contradictions, the placement and relations between different contradictory settings should also be relevant.
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This work is supported by ERC AdG Grant QOLAPS and by the Foundation for Polish Science (FNP) TEAM project co-financed by the EU European Regional Development Fund. We thank M. L. Nowakowski, M. Horodecki, and K. Horodecki for useful discussions. Part of this work was carried out when the authors attended the program “Mathematical Challenges in Quantum Information” at the Isaac Newton Institute for Mathematical Sciences in the University of Cambridge.
*
[email protected] [1] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). [2] J. Barrett, L. Hardy, and A. Kent, Phys. Rev. Lett. 95, 010503 (2005). [3] S. Pironio et al., Nature (London) 464, 1021 (2010). [4] R. Colbeck and R. Renner, Nat. Phys. 8, 450 (2012). [5] Ll. Masanes, A. Acin, and N. Gisin, Phys. Rev. A. 73, 012112 (2006). [6] N. Brunner and P. Skrzypczyk, Phys. Rev. Lett. 102, 160403 (2009). [7] R. Augusiak, M. Demianowicz, M. Pawłowski, J. Tura, and A. Acin, arXiv:1307.6390. [8] J. Barrett, R. Colbeck, and A. Kent, Phys. Rev. A 86, 062326 (2012). [9] B. Toner, Proc. R. Soc. A 465, 59 (2009). [10] B. M. Terhal, A. C. Doherty, and D. Schwab, Phys. Rev. Lett. 90, 157903 (2003). [11] M. Pawłowski and C. Brukner, Phys. Rev. Lett. 102, 030403 (2009). [12] R. Ramanathan, A. Soeda, P. Kurzynski, and D. Kaszlikowski, Phys. Rev. Lett. 109, 050404 (2012). [13] F. G. S. L. Brandao and A. W. Harrow, Proceedings of the 45th ACM Symposium on Theory of Computing (STOC 2013), p. 861 (ACM, New York, 2013). [14] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.113.210403, for the proofs of the Propositions stated in the main text, which includes Refs. [15,16]. [15] N. S. Jones and Ll. Masanes, Phys. Rev. A 72, 052312 (2005). [16] A. Cabello, Phys. Lett. A 377, 64 (2012). [17] P. Kurzynski, T. Paterek, R. Ramanathan, W. Laskowski, and D. Kaszlikowski, Phys. Rev. Lett. 106, 180402 (2011). [18] J.-D. Bancal, C. Branciard, N. Brunner, N. Gisin, and Y-C. Liang, J. Phys. A 45, 125301 (2012). [19] B. S. Cirelson, Lett. Math. Phys. 4, 93 (1980). [20] S. Wehner, Phys. Rev. A 73, 022110 (2006). [21] R. Cleve, P. Hoyer, B. Toner, and J. Watrous, arXiv:quantph/0404076. [22] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 61, 662 (1988). [23] J. Kempe and O. Regev, arXiv:0911.0201.
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