On Quantisations, Quasi-probabilities and the Weak Value

TL;DR

This paper introduces a unified framework for quantum and quasi-classical transformations using quasi-joint-spectral distributions, providing new insights into quantum correlations, conditional expectations, and the interpretation of weak values.

Contribution

It develops the concept of QJSDs to systematically derive quantum/quasi-classical transformations and offers a statistical interpretation of weak values within this formalism.

Findings

QJSDs uniquely generate various quantum/classical transformations.

The formalism provides a statistical interpretation of weak values.

Introduces quantum correlations and conditional expectations analogous to classical theory.

Abstract

We propose a general framework of the quantum/quasi-classical transformations by introducing the concept of quasi-joint-spectral distribution (QJSD). Specifically, we show that the QJSDs uniquely yield various pairs of quantum/quasi-classical transformations, including the Wigner-Weyl transform. We also discuss the statistical behaviour of combinations of generally non-commutin quantum observables by introducing the concept of quantum correlations and conditional expectations defined analogously to the classical counterpart. Based on these, Aharonov's weak value is given a statistical interpretation as one realisation of the quantum conditional expectations furnished in our formalism.