Holographic Entanglement of Purification

TL;DR

The paper introduces the entanglement wedge cross section $E_W$, a geometric proxy in AdS/CFT for the entanglement of purification $E_P$, and shows that $E_W$ satisfies key information-theoretic inequalities such as strong superadditivity. Through tensor-network intuition and explicit calculations in pure AdS$_3$ and BTZ spacetimes, it demonstrates that $E_W$ behaves like $E_P$ for holographic CFTs at large $N$, including an operational interpretation via purifications and regularized quantities like $E_{LOq}$. A covariant generalization to time-dependent backgrounds is provided, along with appendices detailing entanglement measures, a proof of strong superadditivity, and explicit AdS$_3$/CFT$_2$ computations. The work suggests a deep, geometrical holographic realization of correlation measures for mixed states and points to future CFT verifications and operational insights into AdS/CFT.

Abstract

We study properties of the minimal cross section of entanglement wedge which connects two disconnected subsystems in holography. In particular we focus on various inequalities which are satisfied by this quantity. They suggest that it is a holographic counterpart of the quantity called entanglement of purification, which measures a bipartite correlation in a given mixed state. We give a heuristic argument which supports this identification based on a tensor network interpretation of holography. This implies that the entanglement of purification satisfies the strong superadditivity for holographic conformal field theories.

Figures (8)

• Figure 1: The gray regions are the entanglement wedges $M_{AB}$ dual to $\rho_{AB}$. The left one is for subsystems $A$ and $B$ in a pure state e.g. a vacuum state in a CFT. The right one is for subsystems for a thermal state of a CFT dual to a AdS black hole. The surface which divides $M_{AB}$ into two parts each of which ends on $A$ and $B$ is defined as $\Sigma_{AB}$, which is depicted as the dotted surface. Equally, $\Sigma_{AB}$ is the minimal surface which computes the entanglement entropy between $A\cup \Gamma^{(A)}_{AB}$ and $B\cup \Gamma^{(B)}_{AB}$. The surface $\Sigma^{min}_{AB}$ is obtained by minimizing the area of $\Sigma_{AB}$ by varying the choice of $\Gamma_A$. Note also that when $A$ and $B$ gets smaller and more separated, the entanglement wedge gets disconnected into two parts in which case $\Sigma^{min}_{AB}$ becomes empty and we have $E_W=0$.
• Figure 2: The proof of a bound for entanglement wedge cross section. The left picture corresponds to the case where the total system is a pure state, while the right one to the thermal state. It is geometrically clear that we have $S(\rho_A)+S(\rho_B)\leq 2 E_W(\rho_{AB}) +S(\rho_{AB})$. To see this, e.g. in the right picture for a thermal state, we find $E_W(\rho_{AB})=A(\Sigma_{AB})$, $S(\rho_{A,B})=A(\Gamma_{A,B})$, $S(\rho_{AB})=A(\Gamma_{A1})+A(\Gamma_{A2}) +A(\Gamma_{B1})+A(\Gamma_{B2})+A(\Gamma_{BH})$, where we set $4G_N=1$. The bound follows from the inequality $A(\Gamma_A)\leq A(\Gamma_{A1})+A(\Gamma_{A2})+A(\Sigma^{min}_{AB})+A(\Gamma^{(A)}_{BH})$ and a similar one for $B$. Note that $\Gamma^{(A)}_{BH}\cup \Gamma^{(B)}_{BH}$ is the black hole horizon.
• Figure 3: The proof of the strong superadditivity (\ref{['sadd']}). It is obvious that the area of $\Sigma^{min}_{A\tilde{A}B\tilde{B}}$ is larger than the sum of area of $\Sigma^{min}_{AB}$ and $\Sigma^{min}_{\tilde{A}\tilde{B}}$. $\Sigma^{min}_{AB}$ and $\Sigma^{min}_{\tilde{A}\tilde{B}}$ are depicted by the thick surfaces. $\Sigma^{min}_{A\tilde{A}B\tilde{B}}$ is depicted as the dotted surface.
• Figure 4: The computation of $E_W$ for BTZ geometry.
• Figure 5: Three phases of the entanglement wedge for a symmetric setup in a global BTZ black hole.