Speed Limits for Entanglement

TL;DR

This work establishes universal bounds on entanglement growth in relativistic quantum field theories by linking entanglement entropy dynamics to thermal relative entropy. The authors derive an instantaneous speed limit $|v_E|\le 1$ using monotonicity of relative entropy with respect to a thermal reference state, valid for translation-invariant states and large region sizes, and show shape-dependent nonlinear bounds via causal volume arguments. In 1+1d CFT, they exploit exact modular Hamiltonians to obtain precise bounds, confirming the general approach and connecting to the tsunami picture of entanglement spreading. The results provide a versatile framework for constraining non-equilibrium entanglement dynamics across dimensions and suggest extensions to inhomogeneous states and lattice systems, with potential ties to hydrodynamics.

Abstract

We show that in any relativistic system, entanglement entropy obeys a speed limit set by the entanglement in thermal equilibrium. The bound is derived from inequalities on relative entropy with respect to a thermal reference state. Thus the thermal state constrains far-from-equilibrium entanglement dynamics whether or not the system actually equilibrates, in a manner reminiscent of fluctuation theorems in classical statistical mechanics. A similar shape-dependent bound constrains the full nonlinear time evolution, supporting a simple physical picture for entanglement propagation that has previously been motivated by holographic calculations in conformal field theory. We discuss general quantum field theories in any spacetime dimension, but also derive some results of independent interest for thermal relative entropy in 1+1d CFT.

Figures (8)

• Figure 1: (a) Subregion at a fixed time, $B \subset A$. (b) Regions at different times with nested causal diamonds, $D(B) \subset D(A)$.
• Figure 2: Setup for the derivation of speed limits. The causal diamonds are nested, $D(B) \subset D(A)$, and the boundary of $B$ is null separated from the boundary of $A$.
• Figure 3: Regions for the derivation of the speed limit for a strip.
• Figure 4: The causal volume $V_{causal}(t)$ is defined as the volume of region $A$ at time $t$ causally connected to the region $\bar{A}$ at $t=0$.
• Figure 5: Vector field on the Euclidean thermal cylinder which, upon continuation to Lorentzian signature, generates the modular evolution of an interval in a thermal state.