Symmetries and Wigner representations of operational theories

TL;DR

This paper develops Wigner representations for general probabilistic theories, enabling a unified description of classical and quantum systems through symmetries and observable transformations.

Contribution

It introduces a general framework for Wigner representations in operational theories, including criteria for symmetry definitions and uniqueness.

Findings

Wigner representations can be constructed for a broad class of operational theories.

Symmetries in the Wigner framework can be well-defined or unique under certain conditions.

The approach unifies classical and quantum descriptions via observable-based representations.

Abstract

We develop the theory of Wigner representations for general probabilistic theories (GPTs), a large class of operational theories that include both classical and quantum theory. The Wigner representations that we introduce are a natural way to describe the theory in terms of some fixed observables; these observables are often picked to be position and momentum or spin observables. This allows us to introduce symmetries which transform the outcomes of the observables used to construct the Wigner representation; we obtain several results for when these symmetries are well defined or when they uniquely specify the Wigner representation.