A necessary condition for incompatibility of observables in general probabilistic theories

TL;DR

This paper establishes a quantitative condition for the incompatibility of observables in general probabilistic theories, with applications to quantum theory and polytope state spaces, providing a simple formula for noise content.

Contribution

It introduces a noise content inequality for incompatible observables and applies it across various probabilistic frameworks, including quantum and polytope state spaces.

Findings

Derived a noise content inequality for incompatible observables

Applied the inequality to quantum theory, quantum processes, and polytope state spaces

Provided a simple formula for noise content of POVMs

Abstract

We quantify the intrinsic noise content of an observable in a general probabilistic theory and derive a noise content inequality for incompatible observables. We apply the derived inequality to standard quantum theory, the quantum theory of processes, and polytope state spaces. The noise content for positive operator-valued measures takes a particularly simple form and equals the sum of minimal eigenvalues of all the effects. We illustrate our findings with a number of examples including the introduced notion of reverse observables.