Abstract
In the context of quantum theories of spacetime, one overarching question is how quantum information in the bulk spacetime is encoded holographically in boundary degrees of freedom. It is particularly interesting to understand the correspondence between bulk subregions and boundary subregions in order to address the emergence of locality in the bulk quantum spacetime. For the AdS/CFT correspondence, it is known that this bulk information is encoded redundantly on the boundary in the form of an error-correcting code. Having access only to a subregion of the boundary is as if part of the holographic code has been damaged by noise and rendered inaccessible. In quantum-information science, the problem of recovering information from a damaged code is addressed by the theory of universal recovery channels. We apply and extend this theory to address the problem of relating bulk and boundary subregions in AdS/CFT, focusing on a conjecture known as entanglement wedge reconstruction. Existing work relies on the exact equivalence between bulk and boundary relative entropies, but these 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. 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. 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’s rule that attempts to undo the noise induced by restricting to only a portion of the boundary. 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.
- Received 15 October 2018
- Revised 11 February 2019
DOI:https://doi.org/10.1103/PhysRevX.9.031011
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
There has recently been a major shift in the way many physicists think about quantum gravity. This shift arose from the discovery that a wide range of quantum-mechanical systems without gravity are secretly dual to theories of quantum gravity, like a cryptogram with an encoded message. Moreover, one can tease out the hidden quantum-gravity theory by studying how information—in the form of entropy, entanglement, and chaos—is encoded in the quantum-mechanical system. It is natural to ask how objects in the quantum-gravity theory can be rewritten in their masked form in the purely quantum-mechanical theory. We provide a mathematical construction for this rewriting, creating a “communication channel” that inputs objects on the quantum-gravity side and outputs corresponding objects on the purely quantum-mechanical side.
A unique challenge is that quantum-gravitational objects are redundantly encoded in the purely quantum-mechanical system, so that if part of the purely quantum-mechanical theory is erased, the encoded gravitational objects can still be recovered. We construct a “recovery mapping” that leverages a noncommutative analog of Bayesian inference to tell us how to recover quantum-gravitational information when some of its redundant description is erased.
The ability to translate between different yet equivalent descriptions of quantum-gravitational objects lies at the heart of the latest developments in quantum-gravity research.
