Quantum Error Correction in the Black Hole Interior

Abstract

We study the quantum error correction properties of the black hole interior in a toy model for an evaporating black hole: Jackiw-Teitelboim gravity entangled with a nongravitational bath. After the Page time, the black hole interior degrees of freedom in this system are encoded in the bath Hilbert space. We use the gravitational path integral to show that the interior density matrix is correctable against the action of quantum operations on the bath which (i) do not have prior access to details of the black hole microstates, and (ii) do not have a large, negative coherent information with respect to the maximally mixed state on the bath, with the lower bound controlled by the black hole entropy and code subspace dimension. Thus, the encoding of the black hole interior in the radiation is robust against generic, low-rank quantum operations. For erasure errors, gravity comes within an O(1) distance of saturating the Singleton bound on the tolerance of error correcting codes. For typical errors in the bath to corrupt the interior, they must have a rank that is a large multiple of the bath Hilbert space dimension, with the precise coefficient set by the black hole entropy and code subspace dimension.

Non-isometric quantum error correction in gravity

“…In some toy models and many holographic scenarios, the dictionary V really is an isometry and leads to an approximate quantum error correcting code between the bulk and boundary where all states in H b look roughly the same from a semiclassical perspective. 4 However, extensions of the standard error correction structure are necessary and indeed sufficient to understand more precise notions of information in bulk subregions [53,54], choices of H b containing states with highly dissimilar bulk entanglement structure [17], and even certain extreme situations motivated by the interiors of evaporating or old black holes where V can be arbitrarily far from an isometry [51,55,70,71]. 5 Of these extensions to standard error correction, the non-isometric extension is the least understood and the most relevant for situations with strong gravitational effects.…”

“…However, unlike the perturbative non-isometry studied in [16,71] by considering small excitations around such evaporating states, we will consider a much larger portion of H b ⊗ H R as a code space. In doing so, we can try to reconstruct bulk physics in both completely disentangled states and also highly entangled states which cause an information problem.…”

“…As we reviewed in section 2, in gravitational models HR can be thought of as a radiation bath into which an AdS black hole can evaporate[16,71].…”

Delicate windows into evaporating black holes

“…A related point was brought up in a recent work [76], where it was shown in the setting of the PSSY (west coast) model [5] that the Page curve phase structure of a portion of radiation is robust to quantum errors. One can think that if slight parameter changes of the JHEP02(2023)080 semiclassical state could be interpreted as quantum errors, our observations of parameter dependence would be at odds with those results.…”

“…One can think that if slight parameter changes of the JHEP02(2023)080 semiclassical state could be interpreted as quantum errors, our observations of parameter dependence would be at odds with those results. However, it is important to note that the quantum error correction results of [76] only involve very large radiation regions at very late times. This regime corresponds to the semi-infinite segments at very late times in our study, which are consistent with robustness of the Page curve.…”