No-free-information principle in general probabilistic theories

TL;DR

This paper investigates the validity of no-information-without-disturbance and no-free-information principles in general probabilistic theories, revealing that these principles do not universally hold and depend on specific state space structures.

Contribution

It characterizes when the no-information-without-disturbance and no-free-information principles hold in general probabilistic theories, especially in polygon state spaces.

Findings

No-information-without-disturbance principle always holds in polygon state spaces.

The no-free-information principle's validity depends on the parity of the polygon's vertices.

These principles do not universally apply in general probabilistic theories.

Abstract

In quantum theory, the no-information-without-disturbance and no-free-information theorems express that those observables that do not disturb the measurement of another observable and those that can be measured jointly with any other observable must be trivial, i.e., coin tossing observables. We show that in the framework of general probabilistic theories these statements do not hold in general and continue to completely specify these two classes of observables. In this way, we obtain characterizations of the probabilistic theories where these statements hold. As a particular class of state spaces we consider the polygon state spaces, in which we demonstrate our results and show that while the no-information-without-disturbance principle always holds, the validity of the no-free-information principle depends on the parity of the number of vertices of the polygons.