Does connected wedge imply distillable entanglement?

TL;DR

This work resolves whether a connected entanglement wedge guarantees distillable bipartite entanglement by showing that at leading order in $1/G_N$ there is no LO-distillable entanglement when the wedge surfaces are bulk-separated, even though the boundary mutual information can be large. It introduces and leverages holographic counterparts $E^W$, $E_F\approx E^W$, and $J^W(A|C)$ to map bulk geometry to distillation capabilities, demonstrating that one-shot LOCC distillable entanglement is governed by $J^W(A|C)$ and that a pretty-good bound via the Petz map bounds $E_D^{LO}$ by $\min(J^W(A|C),J^W(C|A))$. The paper also analyzes Haar-random tripartite states as a toy model, proving the proposed bounds in that setting and discussing subleading effects such as traversable wormholes and Planck-scale proximity. It proposes a holographic protocol using holographic measurements to distill approximately $J^W(A|C)$ EPR pairs, and argues for optimality under the holographic measurement constraint. The results illuminate the subtle interplay between bulk causality, entanglement structure, and distillability, and suggest a bound-entangled-like regime in holography where the entanglement wedge is connected yet one-way distillable entanglement vanishes at leading order, with implications for bulk reconstruction and the shadow of the entanglement wedge.

Abstract

The Ryu-Takayanagi formula predicts that two boundary subsystems $A$ and $C$ can exhibit large mutual information $I(A:C)$ even when they are spatially disconnected on the boundary and separated by a buffer subsystem $B$, as long as $A$ and $C$ have connected entanglement wedge in the bulk. However, whether the reduced state $ρ_{AC}$ contains distillable EPR pairs has remained a longstanding open problem. In this work, we resolve this problem by showing that: i) there is no LO-distillable entanglement at leading order in $G_N$, suggesting the absence of bipartite entanglement in a holographic mixed state $ρ_{AC}$, and ii) one-shot, one-way LOCC-distillable entanglement is given at leading order by locally accessible information $J^W(A|C)$, which is related to the entanglement wedge cross section $E^W$ involving the (third) purifying system $B$ via $J^W(A|C) = S_A - E^W(A:B)$. Namely, we demonstrate that a connected entanglement wedge does not necessarily imply nonzero distillable entanglement in one-shot, one-way LOCC. We also show that entanglement of formation $E_{F}(A:C)$ is given by $E^W(A:C)$ at leading order in holography.

Figures (22)

• Figure 1: Connected entanglement wedge. How are two subsystems $A$ and $C$ entangled?
• Figure 2: Examples of the minimal cross section $\Sigma_{A:C}$ (shown in red lines). Thick lines represent the boundaries of the entanglement wedge $\mathcal{E}_{AC}$. a) Pure AdS$_3$ divided into three segments $A,B,C$. b) Pure AdS$_3$ where $C$ (instead of $B$) is a buffer region. c) Two-sided BTZ black hole where $A,C$ are placed on opposite sides. d) Another example for two-sided BTZ black hole. Note that b) and d) have two candidate cross sections.
• Figure 3: EPR pairs cannot be LO-distilled when minimal surfaces are separated in the bulk.
• Figure 4: Examples of (nearly) overlapping minimal surfaces.
• Figure 5: Distillable entanglement and connected wedge as sizes of $A,C$ increase from left to right. The transition point between II and III is obtained in Appendix \ref{['app:hol-calc']}.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Claim 1
• Theorem 4: informal
• Theorem 5: informal
• Corollary 1
• Claim 2
• Claim 3: informal