Quantum Error Correction in the Black Hole Interior

TL;DR

This paper studies quantum error correction behind the horizon in a JT-gravity–bath toy model (PSSY) of evaporating black holes. By computing gravitational path integrals, it derives a coherent-information–based criterion that determines when bath operations can corrupt the black hole interior encodings, showing robustness to generic low-rank errors after Page time and near-Singleton-bound behavior for erasures. The interior encoding is shown to be an approximate isometry for small code subspaces relative to the black hole entropy, with explicit bounds from the Frobenius norm. For erasures, gravity nearly saturates the Singleton bound, while typical random errors require a large Kraus-rank on the bath to disrupt the interior, revealing a sharp separation between correctable and non-correctable errors governed by $S_{ ext{BH}}$, the code dimension $d_i$, and the bath size. The work connects the island program and Python's Lunch picture to gravitational QEC, discusses pseudorandomness and complexity, and analyzes implications for semi-classical causality in evaporating spacetimes.

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 non-gravitational 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.

Figures (10)

• Figure 1: A cartoon of an evaporating black hole. (Left) The UV description consists of a holographic quantum mechanical system $B$ (green dot) in a highly excited state coupled to a bath $R$ (blue line). (Right) The dual semi-classical description is that of an evaporating black hole coupled to a bath. Past the Page time, there is a new Quantum Extremal Surface (black dot) relevant for the entropy of the bath, and the region in the black hole interior enclosed by it to its left is called the "island".
• Figure 2: (Left) The state $|\psi^{\alpha}_{i,i'}\rangle_B$ can be constructed in the CFT from a Euclidean path integral with appropriate superpositions of source configurations (indicated by a cross) and an EOW brane boundary condition $\alpha$ at $\tau = - \beta/2$ (red dot). (Right) The bulk description has a black hole with an EOW brane (shown in red) and bulk matter fields in the state $|i,i'\rangle_{\text{code}}$. The black dot is the black hole bifurcation point.
• Figure 3: The Euclidean black hole geometry which enters in determining the normalization factor $\mathcal{N}$ defined in \ref{['eq:normalization-defn']}, specifically contributing to the overlap $\langle \psi_{i,i'}^\alpha \vert \psi_{i,i'}^\alpha \rangle_B$. The red line is the EOW brane with index $\alpha$, the blue line is an interior code subspace excitation with label $i$, and the green line is an exterior code subspace excitation with label $i'$.
• Figure 4: (Left) A tensor network representation of the index structure $f_{\alpha_1,\beta_1,\cdots,\alpha_n,\beta_n}$ with $n=3$. The light grey boxes denote $U_\mathcal{E}$ while dark grey boxes denote $U_\mathcal{E}^{\dagger}$. Dashed blue and red internal legs denote environment and radiation indices respectively. External legs are outgoing (incoming) to denote uncontracted indices in a bra (ket). (Right) The contraction of the $f$-index structure with the gravitational overlaps (i.e., asymptotic boundaries), denoted with solid black lines, required to compute the sums in (\ref{['tr1']}, \ref{['tr2']}). Each term in these sums involves a product of overlaps between $n$ sets of asymptotic gravitational states. The states are defined in terms of operators acting at the asymptotic boundaries which are illustrated here as black lines. Below we will see that the gravitational path integral can fill in these boundaries in various ways that may connect them or not through a bulk geometry.
• Figure 5: The disconnected gravitational contribution to the Rényi entropy of $\mathfrak{R}_i\cup \mathfrak{R}_e \cup \text{env}$. The grey region is the gravitational geometry filling in the boundary conditions, while the red lines are EOW branes. Green and blue lines denote bulk contractions of the exterior and interior factors in the code subspace respectively. Dotted lines indicate identifying and summing over the code-subspace indices.