Maximal violation of steering inequalities and the matrix cube

TL;DR

This paper links the maximal violation of steering inequalities in quantum theory to the inclusion constants of the matrix cube, providing new bounds and optimality results using convex optimization tools.

Contribution

It establishes a novel connection between quantum steering violations and matrix cube inclusion constants, advancing the understanding of quantum incompatibility and steering.

Findings

Maximal violation of steering inequalities equals matrix cube inclusion constants.

New upper bounds on steering inequality violations are derived.

Inclusion constants of the matrix cube and diamond are shown to be equal.

Abstract

In this work, we characterize the amount of steerability present in quantum theory by connecting the maximal violation of a steering inequality to an inclusion problem of free spectrahedra. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, which is a well-studied object in convex optimization theory. This allows us to find new upper bounds on the maximal violation of steering inequalities and to show that previously obtained violations are optimal. In order to do this, we prove lower bounds on the inclusion constants of the complex matrix cube, which might be of independent interest. Finally, we show that the inclusion constants of the matrix cube and the matrix diamond are the same. This allows us to derive new bounds on the amount of incompatibility available in dichotomic quantum measurements in fixed dimension.