# Exploration of an augmented set of Leggett-Garg inequalities using a noninvasive continuous-in-time velocity measurement

## TL;DR

This paper experimentally investigates Leggett-Garg inequalities using a nuclear magnetic resonance setup, testing necessary and sufficient conditions for macroscopic realism with a novel non-invasive measurement protocol based on velocity measurements.

## Contribution

It implements a comprehensive test of necessary and sufficient Leggett-Garg inequalities and introduces a new non-invasive measurement protocol using velocity to assess temporal correlations.

## Key findings

- Confirmed the validity of the combined LGI tests under various regimes.
- Demonstrated the effectiveness of the velocity-based non-invasive measurement protocol.
- Compared the new protocol with traditional ideal negative measurements.

## Abstract

Macroscopic realism (MR) is the view that a system may possess definite properties at any time independent of past or future measurements, and may be tested experimentally using the Leggett-Garg inequalities (LGIs). In this work we advance the study of LGIs in two ways using experiments carried out on a nuclear magnetic resonance spectrometer. Firstly, we addresses the fact that the LGIs are only necessary conditions for MR but not sufficient ones. We implement a recently-proposed test of necessary and sufficient conditions for MR which consists of a combination of the original four three-time LGIs augmented with a set of twelve two-time LGIs. We explore different regimes in which the two- and three-time LGIs may each be satisfied or violated. Secondly, we implement a recent proposal for a measurement protocol which determines the temporal correlation functions in an approximately non-invasive manner. It employs a measurement of the velocity of a dichotomic variable $Q$, continuous in time, from which a possible sign change of $Q$ may be determined in a single measurement of an ancilla coupled to the velocity. This protocol involves a significantly different set of assumptions to the traditional ideal negative measurement protocol and a comparison with the latter is carried out.