Spatial Leggett-Garg Inequalities

TL;DR

This paper introduces a spatial extension of the Leggett-Garg inequality to analyze quantum perturbation propagation in many-body systems, revealing violations linked to Lieb-Robinson physics and interaction range.

Contribution

It formulates a new spatial Leggett-Garg inequality applicable to many-body systems and demonstrates its violation in spin chains, connecting to Lieb-Robinson bounds.

Findings

Violation time scales with distance between observers.

Violations increase with interaction range.

Observable in current experiments.

Abstract

We formulate a spatial extension of the Leggett-Garg inequality by considering three distant observers locally measuring a many-body system at three subsequent times. The spatial form, in particular, is specially suited to analyze propagation of quantum perturbations through spin chains, by capturing how a measurement at one site can later affect distant sites due to the interactions. We illustrate our proposal for a Heisenberg chain in a magnetic field, showing indeed that the first inequality-violation time scales proportionally to the distance between measuring parties. We attribute this phenomenon to Lieb-Robinson physics and, confirming this connection, we find that violations are anticipated when increasing the interaction range. The inequality violation, readily observable in current experiments, demonstrates the incompatibility between two-point correlation functions and the macrorealistic hypothesis. In outlook, spatial Leggett-Garg inequalities constitute a new tool for analyzing the non-relativistic dynamics of many-body quantum systems.