Dynamics of Entanglement Wedge Cross Section from Conformal Field Theories

TL;DR

This work analyzes the dynamics of the entanglement wedge cross section in two-dimensional holographic CFTs after a local operator quench by computing the reflected entropy $S_R(A:B)$ and comparing it with mutual information and RCFT behavior. Using a replica approach and the holographic dictionary, it shows that $S_R(A:B)$ tracks EWCS dynamics and reveals a significant role for classical correlations in chaotic theories, contrasting with RCFTs. The analysis of heavy primary states reveals a two-phase EWCS in BTZ and a violation of subsystem eigenstate thermalization for certain subsystems. Overall, the results provide a bulk interpretation of reflected entropy dynamics and suggest that subregion duality encodes classical correlations, offering a refined diagnostic of chaos.

Abstract

We derive dynamics of the entanglement wedge cross section directly from the two-dimensional holographic CFTs with a local operator quench. This derivation is based on the reflected entropy, a correlation measure for mixed states. We further compare these results with the mutual information and ones for RCFTs. Our results directly suggest the classical correlation also plays an important role in the subregion/subregion duality even for dynamical setup. Besides a local operator quench, we study the reflected entropy in a heavy state and provide improved bulk interpretation. We checked the above results also hold for the odd entanglement entropy, which is another measure for mixed states related to the entanglement wedge cross section.

Figures (4)

• Figure 1: The path integral representation of the Renyi reflected entropy (for vacuum). 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 2: We study the setup $0< u_2<-v_1<-u_1<v_2$. We excite the vacuum by acting an local operator on $x=0$ at $t=0$.
• Figure 3: 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 $\frac{c}{6}$. Each blue dot shows a transition of itself or its first derivative.
• Figure 4: The non-trivial entanglement wedge cross section in the BTZ black hole has two phases. We can also see this transition from the evaluation of pure state in \ref{['eq:Renyi2']}. An important point is that if we evaluate the usual EE for thermal state, the similar phase as right panel never appear, whereas the pure state does. The existence of second phase in our pure state result means that after this transition, we cannot approximate pure state as thermal one.