Incompatibility in general probabilistic theories, generalized spectrahedra, and tensor norms

TL;DR

This paper explores measurement incompatibility in general probabilistic theories by establishing new characterizations involving spectrahedra and tensor norms, leading to bounds on incompatibility in quantum systems.

Contribution

It introduces novel characterizations of measurement compatibility using spectrahedra and tensor norms, extending free spectrahedra theory to ordered vector spaces.

Findings

Characterizations of compatibility via positivity, spectrahedra inclusion, and tensor norms.

Complete description of compatibility regions for certain GPTs.

New bounds on maximal incompatibility in multi-qubit measurements.

Abstract

In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility present in different GPTs, i.e. their compatibility regions. For centrally symmetric GPTs, we show that the compatibility degree is given as the ratio of the injective and the projective norm of the tensor product of associated Banach spaces. This allows us to completely characterize the compatibility regions of several GPTs, and to obtain optimal universal bounds on the compatibility degree in terms of the 1-summing constants of the associated Banach spaces. Moreover, we find new bounds on the maximal incompatibility present in more than three qubit measurements.