Wigner's form of the Leggett-Garg inequality, No-Signalling in Time, and Unsharp Measurements

TL;DR

This paper introduces a new temporal form of the Leggett-Garg inequality based on Wigner's approach, demonstrating its robustness against unsharp measurements and its potential to test macrorealism violations where traditional LGI fails.

Contribution

It derives a novel Wigner-based temporal inequality for two-level systems and shows its effectiveness in detecting quantum violations under unsharp measurement conditions.

Findings

Wigner's form of LGI can be violated even when traditional LGI is satisfied.

Quantum violations of no-signalling in time persist regardless of measurement sharpness.

Wigner's LGI offers a new way to test macrorealism with unsharp measurements.

Abstract

Wigner's form of the local realist inequality is used to derive its temporal version for an oscillating two-level system involving two-time joint probabilities. Such an inequality may be regarded as a novel form of the Leggett-Garg inequality (LGI) constituting a necessary condition for macrorealism. The robustness of its quantum mechanical (QM) violation against unsharpness of measurement is investigated by using a suitable model of unsharp measurements. It is found that there exists a range of values of the sharpness parameter (characterizing precision of the relevant measurements) for which the usual LGI is satisfied by QM, but Wigner's form of the LGI (WLGI) is violated. This implies that for such unsharp measurements, the QM violation of macrorealism cannot be tested using the usual LGI, but can be tested using WLGI. In showing this, we take into account the general form of the usual LGI involving an arbitrary number of pairs of two-time correlation functions. Another recently proposed necessary condition for macrorealism, called `no-signalling in time', is also probed, showing that its QM violation persists for arbitrarily unsharp measurements.