Structure of the set of quantum correlators via semidefinite programming

TL;DR

This paper explores the geometric structure of quantum correlators using semidefinite programming, providing new analytic descriptions, conditions for extremality, and an operational interpretation related to self-testing.

Contribution

It generalizes existing analytic descriptions, establishes necessary and sufficient conditions for extremality and exposedness, and offers an operational interpretation in the context of self-testing.

Findings

Generalization of Tsirelson-Landau-Masanes description

Necessary and sufficient conditions for extremality and exposedness

Operational interpretation of extremality via self-testing

Abstract

Quantum information leverages properties of quantum behaviors in order to perform useful tasks such as secure communication and randomness certification. Nevertheless, not much is known about the intricate geometric features of the set quantum behaviors. In this paper we study the structure of the set of quantum correlators using semidefinite programming. Our main results are (i) a generalization of the analytic description by Tsirelson-Landau-Masanes, (ii) necessary and sufficient conditions for extremality and exposedness, and (iii) an operational interpretation of extremality in the case of two dichotomic measurements, in terms of self-testing. We illustrate the usefulness of our theoretical findings with many examples and extensive computational work.