Inclusion constants for free spectrahedra with applications to quantum incompatibility

TL;DR

This paper develops a method to compute inclusion constants for free spectrahedra using non-commutative polynomial optimization, providing new bounds on quantum measurement incompatibility.

Contribution

It introduces a novel approach to calculate inclusion constants for Cartesian products of free simplices with applications to quantum incompatibility.

Findings

Derived closed-form expressions for inclusion constants.

Established bounds on noise tolerance for incompatible measurements.

Analyzed extreme points of free spectrahedra for optimization.

Abstract

Building on the matrix cube problem, inclusions of free spectrahedra have been used successfully to obtain relaxations of hard spectrahedral inclusion problems. The quality of such a relaxation is quantified by the inclusion constant associated with each free spectrahedron. While optimal values of inclusion constants were known in certain highly symmetric cases, no general method for computing them was available. In this work, we show that inclusion constants for Cartesian products of free simplices can be computed using methods from non-commutative polynomial optimization, together with a detailed analysis of the extreme points of the associated free spectrahedra. This analysis also yields new closed-form analytic expressions for these constants. As an application to quantum information theory, we prove new bounds on the amount of white noise that incompatible measurements can tolerate before they become compatible. In particular, we study the case of one dichotomic and one $k$-outcome measurement, as well as the case of four dichotomic qubit measurements.