Compatibility of quantum measurements and inclusion constants for the matrix jewel

TL;DR

This paper links the compatibility of quantum measurements with the inclusion of a novel free spectrahedron called the matrix jewel, providing new bounds on measurement compatibility using quantum information techniques.

Contribution

It generalizes previous work by establishing a connection between free spectrahedra and measurement compatibility for any number of outcomes, introducing the matrix jewel as a key concept.

Findings

Bound the set of inclusion constants for the matrix jewel.

Derived new lower bounds on quantum measurement compatibility.

Utilized quantum cloning and mutually unbiased bases techniques.

Abstract

In this work, we establish the connection between the study of free spectrahedra and the compatibility of quantum measurements with an arbitrary number of outcomes. This generalizes previous results by the authors for measurements with two outcomes. Free spectrahedra arise from matricial relaxations of linear matrix inequalities. A particular free spectrahedron which we define in this work is the matrix jewel. We find that the compatibility of arbitrary measurements corresponds to the inclusion of the matrix jewel into a free spectrahedron defined by the effect operators of the measurements under study. We subsequently use this connection to bound the set of (asymmetric) inclusion constants for the matrix jewel using results from quantum information theory and symmetrization. The latter translate to new lower bounds on the compatibility of quantum measurements. Among the techniques we employ are approximate quantum cloning and mutually unbiased bases.