Spread of entanglement and causality

TL;DR

This work establishes a causality-based bound on the speed of entanglement propagation after a global quench in relativistic theories, showing the tsunami velocity v_E cannot exceed the speed of light. It develops multiple frameworks to understand entanglement spreading: free streaming with light-cone–based measures, interacting models including an infinite-scattering limit, and tensor-network–inspired entropy constructions that reproduce holographic results in 1+1 dimensions. A key finding is that free models alone cannot account for the holographic entanglement growth, indicating interactions are essential in higher dimensions; the maximal free-streaming bound v_E^{free} is strictly below holographic v_E, with interactions able to raise the bound to v_E=1 in most cases. The paper also links v_E to mutual information, provides a rigorous bound on propagation speed, and offers a tensor-network–based continuum model that achieves holographic-like entanglement patterns while respecting strong subadditivity in targeted regimes, illuminating how quantum information spreads in strongly interacting systems.

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.

Figures (30)

• Figure 1: The growth in entanglement entropy can be visualized as occurring via an "entanglement tsunami" wave carrying entanglement inward from $\Sigma$. The region that has been covered by the wave (i.e. the orange region in the plot) is entangled with the region outside $\Sigma$, while the white region is not yet entangled.
• Figure 2: A half space at time $t$ and another at time $t+{{\delta}} t$. The null region $X$ connects the boundaries of these two regions. The horizontal direction is $x_1$ and vertical direction is time. Directions parallel to the boundary are not shown.
• Figure 3: The small strip regions $A$, $B$, and $C$ have all the same width. The null regions $A^\prime$ and $B^{\prime}$ are included in the causal development of regions $A$ and $B$ respectively.
• Figure 4: Regions $A, B, C$ on ${{\mathcal{M}}}$.
• Figure 5: Analysis of the late time behavior. The collection of points ${{\mathcal{N}}}_{\Sigma} (t)$, whose light cones intersect with $\Sigma$ is given by a shell centered around $\Sigma$ (green region). The approximate radius of the shell is $t$. For a fixed $\vec{\eta}$, the intersections of light cones with $\Sigma$ from different values of $y$ provide a foliation of region ${{\mathcal{A}}}$ enclosed by $\Sigma$.