Entanglement Wedge Reconstruction via Universal Recovery Channels

TL;DR

This work extends entanglement wedge reconstruction to a robust framework based on universal recovery channels, relaxing the need for exact bulk–boundary relative entropy equality. By constructing a boundary channel from the bulk entanglement wedge and applying a universal recovery map, the authors derive an explicit boundary operator formula that reconstructs bulk operators via the boundary modular Hamiltonian, and they prove approximate multiplicativity of recovered operators. The approach applies to finite-dimensional von Neumann algebras and yields a concrete, state-independent reconstruction procedure that remains compatible with correlation functions across multiple wedges. The results connect to JLMS and HKLL concepts, clarify limitations in the infinite-dimensional limit, and offer a clear path for extensions and comparisons with other holographic reconstruction frameworks.

Abstract

We apply and extend the theory of universal recovery channels from quantum information theory to address the problem of entanglement wedge reconstruction in AdS/CFT. It has recently been proposed that any low-energy local bulk operators in a CFT boundary region's entanglement wedge can be reconstructed on that boundary region itself. Existing work arguing for this proposal relies on algebraic consequences of the exact equivalence between bulk and boundary relative entropies, namely the theory of operator algebra quantum error correction. However, bulk and boundary relative entropies are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect. The framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture in addition to new physical insights. Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator. This formula can be interpreted as a noncommutative version of Bayes' rule that attempts to undo the noise induced by restricting to only a portion of the boundary, and has an integral representation in terms of modular flows. To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra.

Figures (3)

• Figure 1: (a) The HKLL procedure provides a way of writing bulk operators in terms of boundary operators living on a strip in the boundary consisting of all points that are spacelike separated from the bulk point. (b) The causal wedge HKLL procedure provides a way of expressing bulk operators in terms of boundary operators living only in the domain of dependence of a boundary region whose associated causal wedge contains the bulk point.
• Figure 2: (a) A bipartition of the boundary into a connected piece $A$ and its complement $\bar{A}$. In this case, the entanglement wedge of $A$ coincides with the causal wedge of $A$, labeled by $a$ in the figure. $\bar{a}$ is then the complement of $a$. (b) A bipartition of the boundary into $A$ and $\bar{A}$ such that $A$ consists of two disconnected components. In the figure above, $A$ spans just more than half of the boundary, and in this case the entanglement wedge of $A$ is not simply the union of the causal wedges of each piece of $A$. In the bulk, $a$ represents the entanglement wedge of $A$ and $\bar{a}$ is the complement of $a$.
• Figure 3: We show in \ref{['multientwedgecorrelation']} that correlation functions of bulk operators can be computed by pushing each bulk operator to the boundary separately, and computing the expectation value in the boundary theory. The bulk operators need not live in the same entanglement wedge. In this figure, the boundary is decomposed into four regions: $A$, $B$, $C$, and $D$. Regions $AB$ and $BC$ have a non-trivial intersection, and the bulk operators $\phi$ are localized to the regions as shown. We can use the recovery maps $\mathcal{R}_{AB}^*$ and $\mathcal{R}_{BC}^*$ to push the operators to $AB$ and $BC$, respectively.

Theorems & Definitions (10)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Corollary 5
• proof