Holographic entanglement of purification for thermofield double states and thermal quench

TL;DR

The paper investigates holographic entanglement of purification (EoP) in Schwarzschild-AdS and Vaidya-AdS spacetimes to understand entanglement structures in thermal states and quenches. It analyzes EoP for two disjoint strips on the same boundary and across two boundaries, revealing a dimension- and width-dependent critical separation for EoP to exist, and characterizes the connectedness transitions of entanglement wedges. The study shows distinct evolution patterns for EoP and holographic mutual information under thermal quenches, with EoP growth largely governed by strip width while MI depends on both width and separation, and finds that EoP can persist even when MI has decayed. These results illuminate how EoP encodes purification-related correlations beyond MI in holographic thermal systems, linking entanglement wedge geometry to purification efficiency and its dynamical behavior.

Abstract

We explore the properties of holographic entanglement of purification (EoP) for two disjoint strips in the Schwarzschild-AdS black brane and the Vaidya-AdS black brane spacetimes. For two given strips on the same boundary of Schwarzschild-AdS spacetime, there is an upper bound of the separation beyond which the holographic EoP will always vanish no matter how wide the strips are. In the case that two strips are in the two boundaries of the spacetime respectively, we find that the holographic EoP exists only when the strips are wide enough. If the width is finite, the EoP can be nonzero in a finite time region. For thermal quench case, we find that the equilibrium time of holographic EoP is only sensitive to the width of strips, while that of the holographic mutual information is sensitive not only to the width of strips but also to their separation.

Figures (12)

• Figure 1: The finite strips on the same boundary of a time slice of the Schwarzschild-AdS black brane spacetime. $m$ and $m'$ are two turning points of minimal surface connecting $ad$ and $bc$. $\Gamma$ is the cross section of entanglement wedge when the entanglement wedge is connected.
• Figure 2: Left panel: The regions below the lines are allowed to have non-vanishing holographic EoP in different dimensional spacetimes. Right panel: The critical length $D_{c}$ of separation when $l\to\infty$ in different dimensions.
• Figure 3: Left and middle panels: The EoP (in unit of $4/V_{d-2}$) for different $l$ and $D$ when $d=2$. Right panel: EoP for strip with very large $l$ in different dimensional spacetimes.
• Figure 4: Extremal surfaces in the AdS black brane. The two time slices at left and right boundary are given by $t_L=t_R=t$ and $\rho=\infty$. The half infinite subregions $A$ and $B$ locate at left and right boundaries, respectively. The entanglement wedge cross section denoted by $\Gamma(t_B)$ hides in the inner region of the black brane.
• Figure 5: Relationship between entanglement of purification and time $t_B$ when $d=3,4,5$. Here we set $z_h=1$ and $\Delta E(t_B)=E(t_B)-E(0)$.