Entanglement Wedge Cross Section from CFT: Dynamics of Local Operator Quench

TL;DR

The paper investigates the real-time dynamics of the reflected entropy for local operator quenches in holographic CFTs and its bulk dual, the entanglement wedge cross section, establishing a precise dynamical correspondence in non-equilibrium AdS/CFT. By leveraging the replica trick with orbifold blocks and analyzing Regge-limit behavior, it derives time-dependent S_R and demonstrates agreement with EWCS in falling-particle BTZ geometries, while also exploring odd entanglement entropy and quantum corrections. It analyzes heavy states, integrable versus chaotic theories, and non-perturbative effects, highlighting how S_R probes both quantum and classical correlations and how replica-limit ordering affects results. The work provides a framework to distinguish holographic from RCFT dynamics, advance understanding of information spreading, and connect subregion duality with non-equilibrium phenomena in AdS/CFT.

Abstract

We derive dynamics of the entanglement wedge cross section from the reflected entropy for local operator quench states in the holographic CFT. By comparing between the reflected entropy and the mutual information in this dynamical setup, we argue that (1) the reflected entropy can diagnose a new perspective of the chaotic nature for given mixed states and (2) it can also characterize classical correlations in the subregion/subregion duality. Moreover, we point out that we must improve the bulk interpretation of a heavy state even in the case of well-studied entanglement entropy. Finally, we show that we can derive the same results from the odd entanglement entropy. The present paper is an extended version of our earlier report arXiv:1907.06646 and includes many new results: non-perturbative quantum correction to the reflected/odd entropy, detailed analysis in both CFT and bulk sides, many technical aspects of replica trick for reflected entropy which turn out to be important for general setup, and explicit forms of multi-point semi-classical conformal blocks under consideration.

Figures (16)

• Figure 2: The path integral representation of the Renyi reflected entropy. Edges labeled with the same number get glued together. We can instead view it as a correlator with four twist operators $\Braket{\sigma_{g_A}(u_1)\sigma_{g_A^{-1}}(v_1) \sigma_{g_B}(u_2) \sigma_{g_B^{-1}}(v_2) }_{\text{CFT}^{\otimes mn}}$.
• Figure 3: To reproduce the entanglement wedge cross section, we first take the large $c$ limit and approximate the correlator by the maximal single conformal block. However, we have to take care of the fact that this maximization is done by two maximization processes. First, we maximize the propagations between external operators (lines colored by blue) and second, we maximize the internal line (colored by red). This order of processes corresponds to the minimizations in bulk side as shown in the upper of this figure. As mentioned in the main text, this order of maximizations can be accomplished by the large $c$ limit under the assumption $2h_n \ll n h_m$.
• Figure 4: Reflected entropy (blue) and mutual information (yellow) for a state locally quenched outside two intervals. Here we have set $(u_1,v_1,u_2,v_2)=(-10,-3,1,20)$, ${\epsilon}=10^{-3}$, ${\gamma}=2$ and we remove the prefactor $\frac{c}{6}$. We check that this parameter set satisfies the connected condition $0<\frac{(v_1-u_2)(u_1-v_2)}{(v_1-v_2)(u_1-u_2)}<\frac{1}{2}$. Each blue dot shows a transition of itself or its first derivative.
• Figure 5: Reflected entropy (blue) and mutual information (yellow) for a state locally quenched inside two intervals. Here we have set $(u_1,v_1,u_2,v_2)=(-20,1,3,10)$, ${\epsilon}=10^{-3}$, ${\gamma}=2$ and we remove the prefactor $\frac{c}{6}$. We check that this parameter set satisfies the connected condition $0<\frac{(v_1-u_2)(u_1-v_2)}{(v_1-v_2)(u_1-u_2)}<\frac{1}{2}$. Each blue dot shows a transition of itself or its first derivative.
• Figure 6: The non-trivial entanglement wedge cross section in the BTZ background has two candidates. One is the connected codimension-2 surface and the other is the disconnected codimension-2 surfaces which have endpoints on the black hole horizon. The correct choice is the minimal one. If we could observe this transition in the CFT side, it should come from a change of the dominant channel in the large $c$ limit as shown in the lower of this figure.