Towards Entanglement of Purification for Conformal Field Theories

TL;DR

This work derives a boundary-based formula for entanglement of purification in two-dimensional holographic CFTs using the replica trick with internal twist operators, yielding $E_P(A:B) = -\left.\dfrac{\partial}{\partial n}\mathcal{F}_{\Delta_n}\right|_{n\rightarrow 1}$. It connects the CFT computation to bulk geometry by showing that, in the large-$c$ limit, the conformal blocks reproduce the entanglement wedge cross section $E_W$, supporting the $E_P=E_W$ conjecture in AdS$_3$/CFT$_2$. The paper establishes this link through holographic code models, bulk two-point twist operator correlators, and geodesic Witten diagrams, and extends the analysis to BTZ black holes with distinct geometric phases. These results provide a boundary-accessible route to E_P via conformal blocks and clarify how bulk cross-sections emerge from boundary correlation data, with implications for higher dimensions and time dependence.

Abstract

We argue that the entanglement of purification for two dimensional holographic CFT can be obtained from conformal blocks with internal twist operators. First, we explain our formula from the view point of tensor network model of holography. Then, we apply it to bipartite mixed states dual to subregion of AdS${}_3$ and the static BTZ blackhole geometries. The formula in CFT agrees with the entanglement wedge cross section in the bulk, which has been recently conjectured to be equivalent to the entanglement of purification.

Figures (4)

• Figure 1: Figure(a) represents a part around the boundary within the tensor network(TN) of $|0\rangle$. Figure(b) is the TN representation of $|\Psi\rangle_{AB(C)_{1}}$ which is the state cut a perfect tensor from $|0\rangle$. Figure(c) represents the whole TN of $|0\rangle$ and Figure(d) is the optimally purified state $|\Psi_{opt}\rangle$ constructed by the iteratively cutting the perfect tensors.
• Figure 2: The boundary space along $\gamma^{\star}_{C}$ are divided into $A'_{min}$(red line) and $B'_{min}$(orange line). The $y_{1}$ and $y_{2}$ are the two boundary points between $A'_{min}$ and $B'_{min}$. This division is determined so that the length of the geodesic(blue line) between $y_{1}$ and $y_{2}$ become shortest. The $E_W$ is given by that length times $1/4G_{N}$.
• Figure 3: Left figure: a entanglement wedge (shaded region) for BTZ blackhole. In this situation, the wedge does not cover blackhole horizon (black point on center). Here we consider a timeslice $t=0$. Blue solid line represents the minimal cross section. Right figure: related OPE channel for the left diagram. Since we take the semi-classical heavy-light limit, the identity exchange becomes factorized. Hence, it reduces to the 4pt global CB with coordinates transformation $\phi^\prime=\alpha\phi$.
• Figure 4: For sufficiently large subsystems, a entanglement wedge covers the blackhole horizon. In this case, the entanglement wedge cross section becomes blue solid lines $\sigma_{min.}(\phi_{14})+\sigma_{min.}(\phi_{23})$ in the left figure. Right figure shows the related OPE channel. Under the appropriate limit, the corresponding 6pt heavy-light CB becomes equivalent to product of two 4pt heavy-light CBs. Each of CBs produces the one of two cross section, $\sigma_{min.}(\phi_{14})$ or $\sigma_{min.}(\phi_{23})$.