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.
Highlights
The AdS/CFT correspondence is a duality or equivalence between a gravitational theory in (d þ 1)-dimensional asymptotically AdS spacetime and a conformal field theory with one less spatial dimension [1,2,3,4,5]
We show that the framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture as well as new physical insights
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
Summary
The AdS/CFT correspondence is a duality or equivalence between a gravitational theory in (d þ 1)-dimensional asymptotically AdS spacetime and a conformal field theory with one less spatial dimension [1,2,3,4,5]. It has been proposed that for a given subregion A of the conformal boundary, any low-energy bulk operator acting on a spacetime region called the entanglement wedge of A can be reconstructed using only information in A [6,10,11]. For a channel that is not exactly reversible, it is natural to ask whether or not there exists a recovery channel that works approximately in the above sense [24] This question has spurred a flurry of research and was answered only recently in Ref. We apply the theory of universal recovery channels to the problem of entanglement wedge reconstruction in AdS/CFT, and we arrive at an explicit expression for a bulk operator recovered on the boundary. We prove that approximate recovery channels automatically approximately preserve the multiplicative structure of the original bulk algebra, which ensures that correlation functions of boundary reconstructions of the individual operators, even if each operator is reconstructed using a different entanglement wedge
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