Framed Hilbert space: hanging the quasi-probability pictures of quantum theory

TL;DR

This paper advances the mathematical formalism of quantum phase-space representations using frames, clarifies the role of negativity and contextuality in non-classicality, and unifies various quasi-probability representations within this framework.

Contribution

It develops a frame-based formalism for quantum quasi-probability representations, proves their equivalence with classical generalizations, and clarifies the role of negativity and contextuality.

Findings

Established a formalism based on frames for quantum phase-space representations

Proved the equivalence of different approaches to quasi-probability representations

Clarified the relationship between negativity, contextuality, and non-classicality

Abstract

Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an alternate approach to defining a set of quasi-probability representations, based on a more natural generalization of a classical representation, is equivalent to our earlier approach based on frames, and therefore is also subject to our no-go theorem for a non-negative representation. Furthermore, we clarify the relationship between the contextuality of quantum theory and the necessity of negativity in quasi-probability representations and discuss their relevance as criteria for non-classicality. We also provide a comprehensive overview of known quasi-probability representations and their expression within the frame formalism.