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arXiv:1507.07514 · Entropy

Tsirelson’s Bound Prohibits Communication through a Disconnected Channel

Avishy Carmi, Daniel Moskovich

Published Feb 27, 2018• 6 citations
arXivHTMLPDF

Abstract

Why does nature only allow nonlocal correlations up to Tsirelson's bound and not beyond? We construct a channel whose input is statistically independent of its output, but through which communication is nevertheless possible if and only if Tsirelson's bound is violated. This provides a statistical justification for Tsirelson's bound on nonlocal correlations in a bipartite setting.

DOI
10.3390/e20030151

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Why does nature only allow nonlocal correlations up to Tsirelson's bound and not beyond? We construct a channel whose input is statistically independent of its output, but through which communication is nevertheless possible if and only if Tsirelson's bound is violated. This provides a statistical justification for Tsirelson's bound on nonlocal correlations in a bipartite setting.
A central theme in quantum mechanics is nonlocality, which states that a pair of quantum systems which are shown not to be physically interacting may nevertheless be impossible to describe as independent entities. Quantum theory, however, forbids nonlocal correlations beyond a certain limit. Why is nature only so nonlocal and not more? Approaching the question from the direction of statistics and statistical inference, we identify a statistical no-signaling principle which states that no information may pass through a disconnected channel. We show this principle to be equivalent to the Tsirelson bound on nonlocality for the Bell--CHSH inequality.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
Measurements on entangled quantum states can produce outcomes that are nonlocally correlated. But according to Tsirelson's theorem, there is a quantitative limit on quantum nonlocality. It is interesting to explore what would happen if Tsirelson's bound were violated. To this end, we consider a model that allows arbitrary nonlocal correlations, colloquially referred to as 'box world'. We show that while box world allows more highly entangled states than quantum theory, measurements in box world are rather limited. As a consequence there is no entanglement swapping, teleportation or dense coding.
We present a simple hidden-variable model for the singlet state of a pair of qubits, characterized by two kinds of hierarchically ordered hidden variables. The model is such that, averaging over both types of variables, one reproduces all the quantum mechanical correlations of the singlet state. In contrast, averaging only over the hidden variables of the lower level, one obtains a general formal theoretical scheme exhibiting correlations stronger than the quantum ones, but with faster-than-light communication forbidden. By varying the parameters of the model, it is possible to obtain an arbitrary violation of the Clauser-Horne-Shimony-Holt and Tsirelson bounds. This result is interesting by itself since it shows that a violation of the quantum bound for nonlocal correlations can be implemented in a precise physical manner and not only mathematically and it suggests that resorting to two levels of nonlocal hidden variables might lead to a deeper understanding of the physical principles at the basis of quantum nonlocality. It is useful to mention that a model having features similar to ours has already been considered in the literature. However, from the point of view of its nonlocal aspects, it is physically much less appealing since averaging over the lower-level hidden variables yields correlations that can take only a discrete set of values for any choice of the upper-level variables and not the continuous set of values characterizing our approach. We discuss this problem in the concluding section of our work.
The study of Bell nonlocality and the bounds of quantum correlations, the so-called Tsirelson bounds, is fundamental to quantum information science and the exploration of the limits of quantum theory. While quantum bounds for qubit systems have been extensively characterized, determining tight quantum bounds for correlations attainable with high-dimensional quantum states and measurements remains a significant challenge. In this work, we propose a systematic derivation of bipartite Tsirelson and local bounds written in terms of sum-of-squares decompositions. Using this method, we discover novel bounds and recover established results for maximally entangled states of qubits and qudddits.