Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations

TL;DR

This paper unifies various finite-dimensional quantum quasi-probability representations using frame theory and proves that such representations inherently require negativity or a modified probability calculus.

Contribution

It introduces a unified formalism for all known quasi-probability representations of finite quantum systems using frame theory and establishes the necessity of negativity in these representations.

Findings

All known quasi-probability representations can be described by frame representations.

Any such representation must exhibit negativity or a deformed probability calculus.

The formalism clarifies the physical significance and limitations of quasi-probability representations.

Abstract

Several finite dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted dimensions and their physical significance in contexts such as drawing quantum-classical comparisons is limited by the non-uniqueness of the particular representation. Here we show how the mathematical theory of frames provides a unified formalism which accommodates all known quasi-probability representations of finite dimensional quantum systems. Moreover, we show that any quasi-probability representation satisfying two reasonable properties is equivalent to a frame representation and then prove that any such representation of quantum mechanics must exhibit either negativity or a deformed probability calculus.