Temporal Bell inequalities in non-relativistic many-body physics

Abstract: Analyzing the spreading of information in many-body systems is crucial to understanding their quantum dynamics. At the most fundamental level, this task is accomplished by Bell inequalities, whose violation by quantum mechanics implies that information cannot always be stored locally. While Bell-like inequalities, such as the one of Clauser and Horne, envisage a situation in which two parties perform measurements on systems at different positions, one could formulate temporal inequalities, in which the two parties measure at different times. However, for causally-connected measurement events, these extensions are compatible with a local description, so that no intrinsically-quantum information spreading is involved in such temporal correlations. Here we show that a temporal Clauser-Horne inequality for two spins is violated for a nonzero time interval between the measurements if the two measured parties are connected by a spin chain. Since the chain constitutes the sole medium for the spreading of quantum information, it prevents the immediate vanishing of Bell correlations after the first measurement and it induces violation revivals. The dynamics we analyze shows that, as expected in a non-relativistic setup, the spreading of information is fundamentally limited by the Lieb-Robinson bound. New insights on many-body quantum dynamics could emerge through future applications of our temporal Bell inequality to more general systems.