# Time evolution of entanglement for holographic steady state formation

## TL;DR

This paper studies the time evolution of entanglement in a holographic model of steady state formation, revealing how shockwaves and entanglement entropy behave during the process, with analytical results at low temperatures.

## Contribution

It introduces a holographic approach to analyze entanglement dynamics during steady state formation, including analytical formulas for low-temperature regimes and verification of entanglement inequalities.

## Key findings

- Shockwaves are spacelike in the bulk but do not carry information.
- Entanglement entropy growth rate obeys the same bound as in entanglement tsunami models.
- Derived an analytical formula for entanglement entropy at low temperatures.

## Abstract

Within gauge/gravity duality, we consider the local quench-like time evolution obtained by joining two 1+1-dimensional heat baths at different temperatures at time t=0. A steady state forms and expands in space. For the 2+1-dimensional gravity dual, we find that the shockwaves expanding the steady-state region are of spacelike nature in the bulk despite being null at the boundary. However, they do not transport information. Moreover, by adapting the time-dependent Hubeny-Rangamani-Takayanagi prescription, we holographically calculate the entanglement entropy and also the mutual information for different entangling regions. For general temperatures, we find that the entanglement entropy increase rate satisfies the same bound as in the "entanglement tsunami" setups. For small temperatures of the two baths, we derive an analytical formula for the time dependence of the entanglement entropy. This replaces the entanglement tsunami-like behaviour seen for high temperatures. Finally, we check that strong subadditivity holds in this time-dependent system, as well as further more general entanglement inequalities for five or more regions recently derived for the static case.