Spread of entanglement and causality

TL;DR

This paper explores the limits of entanglement spread in relativistic quantum systems, proving bounds on the tsunami velocity and comparing free and interacting models across dimensions, with implications for holographic theories.

Contribution

It generalizes entanglement spreading models to higher dimensions, establishes bounds on entanglement growth rates, and introduces a geometric tensor network model for entanglement propagation.

Findings

Tsunami velocity is bounded by the speed of light.

Entanglement spread depends on initial entanglement patterns in higher dimensions.

Interacting models provide upper bounds closer to holographic results.

Abstract

We investigate causality constraints on the time evolution of entanglement entropy after a global quench in relativistic theories. We first provide a general proof that the so-called tsunami velocity is bounded by the speed of light. We then generalize the free particle streaming model of arXiv:cond-mat/0503393 to general dimensions and to an arbitrary entanglement pattern of the initial state. In more than two spacetime dimensions the spread of entanglement in these models is highly sensitive to the initial entanglement pattern, but we are able to prove an upper bound on the normalized rate of growth of entanglement entropy, and hence the tsunami velocity. The bound is smaller than what one gets for quenches in holographic theories, which highlights the importance of interactions in the spread of entanglement in many-body systems. We propose an interacting model which we believe provides an upper bound on the spread of entanglement for interacting relativistic theories. In two spacetime dimensions with multiple intervals, this model and its variations are able to reproduce intricate results exhibited by holographic theories for a significant part of the parameter space. For higher dimensions, the model bounds the tsunami velocity at the speed of light. Finally, we construct a geometric model for entanglement propagation based on a tensor network construction for global quenches.