Joint measurability of quantum effects and the matrix diamond

TL;DR

This paper explores the relationship between joint measurability of quantum effects and free spectrahedra, especially the matrix diamond, providing new insights into measurement incompatibility and spectrahedral inclusion.

Contribution

It establishes an equivalence between joint measurability and matrix diamond inclusion in free spectrahedra, linking quantum measurement incompatibility to spectrahedral geometry.

Findings

Characterizes joint measurability via matrix diamond inclusion.

Quantifies measurement incompatibility using inclusion constants.

Provides new results on spectrahedral inclusion for the matrix diamond.

Abstract

In this work, we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra. Free spectrahedra typically arise as matricial relaxations of linear matrix inequalities. An example of a free spectrahedron is the matrix diamond, which is a matricial relaxation of the $\ell_1$-ball. We find that joint measurability of binary POVMs is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study. This connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements. In particular, we completely characterize the case in which the dimension is exponential in the number of measurements. Conversely, we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond.