All measurements in a probabilistic theory are compatible if and only if the state space is a simplex

TL;DR

This paper investigates the conditions under which measurements in a general probabilistic framework are compatible, revealing that incompatibility arises unless the state space is a simplex, and provides a linear programming approach for compatibility testing.

Contribution

It establishes necessary and sufficient conditions for measurement compatibility and links incompatibility to the geometric structure of the state space.

Findings

Measurements are compatible if and only if the state space is a simplex.

Incompatibility exists in non-simplex state spaces.

Linear programming can determine measurement compatibility.

Abstract

We study the compatibility of measurements on finite-dimensional compact convex state space in the framework of general probabilistic theory. Our main emphasis is on formulation of necessary and sufficient conditions for two-outcome measurements to be compatible and we use these conditions to show that there exist incompatible measurements whenever the state space is not a simplex. We also formulate the linear programming problem for the compatibility of two-outcome measurements.