Entanglement of General Subregions in Time-Dependent States

TL;DR

This paper develops a spacetime density matrix framework to define and compute entanglement and Rényi entropies for arbitrary spacetime subregions in time-dependent 1+1D CFTs, unifying spacelike and timelike EE via the Schwinger–Keldysh real-time replica method. It applies the construction to global and local quenches, deriving analytic expressions and revealing distinct timelike versus spacelike entanglement dynamics governed by a quasiparticle picture. A key finding is the persistence of a linear time–space EE sum rule in both quenches, and the emergence of a universal imaginary piece in timelike EE tied to the twist-operator commutator. The results extend the conventional spacelike EE framework, offering a deeper understanding of spacetime entanglement and its universal features in real-time quantum dynamics.

Abstract

We develop a unified framework for computing Rényi 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.

Figures (11)

• Figure 1: Schwinger–Keldysh representation of the spacetime density matrix $\tilde{T}_{C_0C_1}$ (left) and its Hermitian conjugate $\tilde{T}_{C_0C_1}^\dagger$ (right). The initial state at time $t=0$ is denoted by $\Psi_0$, and $i,j,k,l$ denote the boundary conditions. The arrows indicate the direction of the time evolution operator.
• Figure 2: (Left) Schwinger--Keldysh representation of the spacetime density matrix $\tilde{T}_{C_0C_1}$, including the spatial direction $x$. The arrow along the time axis indicates the direction of time evolution. (Right) Representation of the reduced spacetime density matrix $\tilde{T}_{A_0A_1}$. The degrees of freedom on $\bar{A}_0$ and $\bar{A}_1$ are glued together, while cuts remain along $A_0$ and $A_1$.
• Figure 3: The red line represents the interval $A$, and its complement is denoted by the interval $B$. The slanted lines with arrows indicate the worldlines of an entangled pair of quasiparticles. (a)The evolution time has not yet reached the saturation time, $t_0 < \frac{\Delta x}{2}$.(b)The evolution time reaches the saturation time, $t_0 = \frac{\Delta x}{2}$.
• Figure 4: The red line denotes the interval $A$, while the region $B$ represents its entangling complement. The slanted lines with arrows indicate the worldlines of an entangled quasiparticle pair.
• Figure 5: The red line represents the interval $A$, and the region $B$ denotes its entangling partner. The slanted arrows indicate the worldlines of an entangled quasiparticle pair. We consider two temporal separations satisfying $\Delta t_a>\Delta t_b$. (a) The saturation time is reached at $t_0=\frac{\Delta x+\Delta t_a}{2}$, and eight quasiparticle worldlines intersect the interval. (b) The saturation time shifts to $t_0=\frac{\Delta x+\Delta t_b}{2}$, while the number of intersecting worldlines remains eight.