Necessity of negativity in quantum theory

TL;DR

This paper extends the proof that negativity is essential in quasiprobability representations of quantum theory from finite to infinite-dimensional Hilbert spaces, unifying the framework and including the Wigner function as a special case.

Contribution

It generalizes previous finite-dimensional results to infinite-dimensional spaces, establishing negativity as a necessary feature in all quasiprobability representations.

Findings

Negativity is necessary in quasiprobability representations for infinite-dimensional quantum systems.

Unified framework for all quasiprobability representations including the Wigner function.

Extension of finite-dimensional results to infinite-dimensional Hilbert spaces.

Abstract

A unification of the set of quasiprobability representations using the mathematical theory of frames was recently developed for quantum systems with finite-dimensional Hilbert spaces, in which it was proven that such representations require negative probability in either the states or the effects. In this article we extend those results to Hilbert spaces of infinite dimension, for which the celebrated Wigner function is a special case. Hence, this article presents a unified framework for describing the set of possible quasiprobability representations of quantum theory, and a proof that the presence of negativity is a necessary feature of such representations.