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AIP conference proceedings

No-signaling, perfect bipartite dichotomic correlations and local randomness

M. P. Seevinck, Andrei Yu. Khrennikov, Gregg S. Jaeger, M. Schlosshauer, Gregor Weihs

Published Jan 1, 2011• 0 citations
Publisher

Abstract

The no‐signaling constraint on bi‐partite correlations is reviewed. It is shown that in order to obtain non‐trivial Bell‐type inequalities that discern no‐signaling correlations from more general ones, one must go beyond considering expectation values of products of observables only. A new set of nontrivial no‐signaling inequalities is derived which have a remarkably close resemblance to the CHSH inequality, yet are fundamentally different. A set of inequalities by Roy and Singh [29] and Avis et al.. [1], which is claimed to be useful for discerning no‐signaling correlations, is shown to be trivially satisfied by any correlation whatsoever. Finally, using the set of newly derived no‐signaling inequalities a result with potential cryptographic consequences is proven: if different parties use identical devices, then, once they have perfect correlations at spacelike separation between dichotomic observables, they know that because of no‐signaling the local marginals cannot but be completely random.

DOI
10.1063/1.3578723

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The no‐signaling constraint on bi‐partite correlations is reviewed. It is shown that in order to obtain non‐trivial Bell‐type inequalities that discern no‐signaling correlations from more general ones, one must go beyond considering expectation values of products of observables only. A new set of nontrivial no‐signaling inequalities is derived which have a remarkably close resemblance to the CHSH inequality, yet are fundamentally different. A set of inequalities by Roy and Singh [29] and Avis et al.. [1], which is claimed to be useful for discerning no‐signaling correlations, is shown to be trivially satisfied by any correlation whatsoever. Finally, using the set of newly derived no‐signaling inequalities a result with potential cryptographic consequences is proven: if different parties use identical devices, then, once they have perfect correlations at spacelike separation between dichotomic observables, they know that because of no‐signaling the local marginals cannot but be completely random.
The no-signaling constraint on bi-partite correlations is reviewed. It is shown that in order to obtain non-trivial Bell-type inequalities that discern no-signaling correlations from more general ones, one must go beyond considering expectation values of products of observables only. A new set of nontrivial no-signaling inequalities is derived which have a remarkably close resemblance to the CHSH inequality, yet are fundamentally different. A set of inequalities by Roy and Singh and Avis et al., which is claimed to be useful for discerning no-signaling correlations, is shown to be trivially satisfied by any correlation whatsoever. Finally, using the set of newly derived no-signaling inequalities a result with potential cryptographic consequences is proven: if different parties use identical devices, then, once they have perfect correlations at spacelike separation between dichotomic observables, they know that because of no-signaling the local marginals cannot but be completely random.
Determinism, no signaling and measurement independence are some of the constraints required for framing Bell inequality. Any model simulating nonlocal correlations must either individually or jointly give up these constraints. Recently M. J. W. Hall (Phys Review A, \textbf{84}, 022102 (2011)) derived different forms of Bell inequalities under the assumption of individual or joint relaxation of those constraints on both(i.e., two) the sides of a bipartite system. In this work we have investigated whether one sided relaxation can also be a useful resource for simulating nonlocal correlations or not. We have derived Bell-type inequalities under the assumption of joint relaxation of these constraints only by one party of a bipartite system. Interestingly we found that any amount of randomness in correlations of one party in absence of signaling between two parties is incapable of showing any sort of Bell-CHSH violation whereas signaling and measurement dependence individually can simulate any nonlocal correlations. We have also completed the proof of a recent conjecture due to Hall (Phys. Rev. A \textbf{82}, 062117 (2010); Phys. Rev. A \textbf{84}, 022102 (2011)) for one sided relaxation scenario only.
Derivation of the full set of Bell inequalities involving correlation functions, for two parties, with binary observables, and N possible local settings is not as easy as it seemed. The proof of v1 is wrong. Additionaly one can find a counterexample, which will be presented soon. Thus our thesis is dead. Still the series of Bell inequalities discussed in the manuscript (v1) form a necessary condition for local realism, and are tight. They are tight and complete (sufficient) only for N=3 settings per observer (as shown in quant-ph/0611086, fortunately using an entirely different approach).
A Bell inequality is a constraint on a set of correlations whose violation can be used to certify nonlocality. They are instrumental for device-independent tasks such as key distribution or randomness expansion. In this work, we consider bipartite Bell inequalities where two parties have mA{m}_{A}mA​ and mB{m}_{B}mB​ possible inputs and give nA{n}_{A}nA​ and nB{n}_{B}nB​ possible outputs, referring to this as the (mA,mB,nA,nB)({m}_{A},{m}_{B},{n}_{A},{n}_{B})(mA​,mB​,nA​,nB​) scenario. By exploiting knowledge of the set of extremal no-signaling distributions, we find all 175 Bell inequality classes in the (4,4,2,2) scenario as well as provide a partial list of 18 277 classes in the (4,5,2,2) scenario. We also use a probabilistic algorithm to obtain 5 classes of inequality in the (2,3,3,2) scenario, which we confirmed to be complete, 25 classes in the (3,3,2,3) scenario, and a partial list of 21 170 classes in the (3,3,3,3) scenario. Our inequalities are given as Supplemental Material. Finally, we discuss the application of these inequalities to the detection loophole problem and provide lower bounds on the detection efficiency threshold for small numbers of inputs and outputs.
We give a set of necessary conditions for locality in bipartite systems, which include and generalize known Bell's inequalities. Each condition corresponds to a specific order of the expansion of random variables defined on graphs, in terms of genuinely n-variable correlation functions. The first non-trivial order leads to known Bell inequalities, while higher orders produce additional, non-equivalent conditions. In particular, in CHSH settings, we obtain at least two additional tight conditions which do not reduce to the CHSH inequality. This shows that tight Bell inequalities are sufficient but not necessary conditions for the non-locality of bipartite quantum correlations.
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.