Bounding the Set of Quantum Correlations

TLDR

The paper introduces a hierarchy of necessary conditions for any probability distribution arising from local measurements on a shared quantum state. The hierarchy is constructed as a sequence of semidefinite programs that test whether a given distribution can be realized by quantum measurements. The method yields upper bounds on quantum violations of Bell inequalities, providing tight bounds for the Collins et al.

Abstract

We introduce a hierarchy of conditions necessarily satisfied by any distribution P_{alphabeta} representing the probabilities for two separate observers to obtain outcomes alpha and beta when making local measurements on a shared quantum state. Each condition in this hierarchy is formulated as a semidefinite program. Among other applications, our approach can be used to obtain upper bounds on the quantum violation of an arbitrary Bell inequality. It yields, for instance, tight bounds for the violations of the Collins et al. inequalities.