The Leggett-Garg Inequalities and No-Signalling in Time: A Quasi-Probability Approach

TL;DR

This paper introduces a quasi-probability framework to analyze Leggett-Garg inequalities, providing a measurable, non-invasive way to assess macrorealism and its relation to no-signalling in time, with implications for quantum measurement theory.

Contribution

It develops a quasi-probability approach that generalizes no-signalling in time and offers a detailed characterization of macrorealism beyond traditional LG inequalities.

Findings

Quasi-probability is measurable and correlates with two-time measurement probabilities.

When non-negative, the quasi-probability aligns with EPRB system properties.

A simple spin model illustrates the approach's key features.

Abstract

The Leggett-Garg (LG) inequalities were proposed in order to assess whether sets of pairs of sequential measurements on a single quantum system can be consistent with an underlying notion of macrorealism. Here, the LG inequalities are explored using a simple quasi-probability linear in the projection operators to describe the properties of the system at two times. We show that this quasi-probability is measurable, has the same correlation function as the usual two-time measurement probability (for the bivalent variables considered here) and has the key property that the probabilities for the later time are independent of whether an earlier measurement was made, a generalization of the no-signalling in time condition of Kofler and Brukner. We argue that this quasi-probability, appropriately measured, provides a non-invasive measure of macrorealism per se at the two time level. This measure, when combined with the LG inequalities, provides a characterization of macrorealism more detailed than that provided by the LG inequalities alone. When the quasi-probability is non-negative, the LG system has a natural parallel with the EPRB system and Fine's theorem. A simple spin model illustrating key features of the approach is exhibited.