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arXiv:1610.05060 · Quantum science and technology

Quantum Correlations and Synchronization Measures

Fernando Galve, Gian Luca Giorgi, Roberta Zambrini

Published Jan 1, 2017• 50 citations
arXivHTMLPDF
DOI
10.1007/978-3-319-53412-1_18

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Nonlocal games with synchronous correlations are a natural generalization of functions between two finite sets. In this work we examine analogues of Bell's inequalities for such correlations, and derive a synchronous device-independent quantum key distribution protocol. This protocol has the advantage of symmetry between the two users and self-testing while generating shared secret key without requiring a preshared secret. We show that, unlike general correlations and the CHSH inequality, there can be no quantum Bell violation among synchronous correlations with two measurement settings. However we exhibit explicit analogues of Bell's inequalities for synchronous correlations with three measurement settings and two outputs, provide an analogue of Tsirl'son's bound in this setting, and prove existence and rigidity of quantum correlations that saturate this bound. We conclude by posing a security assumption that bypasses the locality, or causality, loophole and examine the protocol's robustness against measurement error and depolarization noise.
The study of quantum correlation sets initiated by Tsirelson in the 1980s and originally motivated by questions in the foundations of quantum mechanics has more recently been tied to questions in quantum cryptography, complexity theory, operator space theory, group theory, and more. Synchronous correlation sets introduced by Paulsen et al. [J. Funct. Anal. 270, 2188–2222 (2016)] are a subclass of correlations that has proven particularly useful to study and arises naturally in applications. We show that any correlation that is almost synchronous, in a natural ℓ1 sense, arises from a state and measurement operators that are well-approximated by a convex combination of projective measurements on a maximally entangled state. This extends a result of Paulsen et al. [J. Funct. Anal. 270, 2188–2222 (2016)] that applies to exactly synchronous correlations. Crucially, the quality of approximation is independent of the dimension of the Hilbert spaces or of the size of the correlation. Our result allows one to reduce the analysis of many classes of nonlocal games, including rigidity properties, to the case of strategies using maximally entangled states that are generally easier to manipulate.