Learning the Alpha-bits of Black Holes

TL;DR

This paper explores how boundary operators in AdS/CFT can reconstruct bulk black hole regions with state dependence, introducing the concept of alpha-bits, and demonstrates their implications for quantum error correction and black hole information.

Contribution

It refines the understanding of state dependence in bulk reconstruction, introduces alpha-bits as a measure of reconstructable information, and connects these ideas to quantum error correction and tensor network models.

Findings

Boundary operators can reconstruct large subspaces of black hole microstates.

Black holes serve as capacity-achieving alpha-bit codes.

Entanglement wedge reconstruction can be exact to all orders in 1/N.

Abstract

When the bulk geometry in AdS/CFT contains a black hole, the boundary reconstruction of a given bulk operator will often necessarily depend on the choice of black hole microstate, an example of state dependence. As a result, whether a given bulk operator can be reconstructed on the boundary at all can depend on whether the black hole is described by a pure state or thermal ensemble. We refine this dichotomy, demonstrating that the same boundary operator can often be used for large subspaces of black hole microstates, corresponding to a constant fraction $\alpha$ of the black hole entropy. In the Schrodinger picture, the boundary subregion encodes the $\alpha$-bits (a concept from quantum information) of a bulk region containing the black hole and bounded by extremal surfaces. These results have important consequences for the structure of AdS/CFT and for quantum information. Firstly, they imply that the bulk reconstruction is necessarily only approximate and allow us to place non-perturbative lower bounds on the error when doing so. Second, they provide a simple and tractable limit in which the entanglement wedge is state-dependent, but in a highly controlled way. Although the state dependence of operators comes from ordinary quantum error correction, there are clear connections to the Papadodimas-Raju proposal for understanding operators behind black hole horizons. In tensor network toy models of AdS/CFT, we see how state dependence arises from the bulk operator being `pushed' through the black hole itself. Finally, we show that black holes provide the first `explicit' examples of capacity-achieving $\alpha$-bit codes. Unintuitively, Hawking radiation always reveals the $\alpha$-bits of a black hole as soon as possible. In an appendix, we apply a result from the quantum information literature to prove that entanglement wedge reconstruction can be made exact to all orders in $1/N$.