Entanglement of General Subregions in Time-Dependent States

TL;DR

This paper introduces a unified method for calculating entanglement entropy in arbitrary spacetime regions of time-dependent 2D conformal field theories, revealing key differences between spacelike and timelike intervals and demonstrating a universal sum rule.

Contribution

It develops a comprehensive framework combining the density matrix formalism and replica method to analyze spacetime entanglement in dynamic quantum states, including new analytic results.

Findings

Timelike entanglement entropy is time-independent in global quenches.

Entanglement entropy depends only on temporal separation for timelike intervals.

The linear sum rule for spacetime entanglement holds in both global and local quenches.

Abstract

We develop a unified framework for computing R\'enyi and entanglement entropies of arbitrary spacetime intervals in time-dependent states of $(1+1)$-dimensional conformal field theories. By combining the spacetime density matrix formalism with the replica method, we show that entanglement entropy is well defined for both spacelike and timelike separations. Applying this framework to global quenches prepared by boundary states and to local quenches generated by operator insertions, we obtain analytic expressions for the entanglement entropy in general spacetime configurations. The results reveal qualitative differences between spacelike and timelike intervals: the timelike entanglement entropy is time-independent in the global quench model, depends solely on the temporal separation, and universally exhibits a constant imaginary contribution. These features are naturally explained by a generalized quasiparticle picture in which entanglement is produced precisely when one worldline of each quasiparticle pair intersects the interval. Furthermore, we demonstrate that the linear sum rule relating time- and spacelike entanglement persists in both global and local quenches, indicating a broader universality of spacetime entanglement in real-time quantum dynamics.