Black Hole Interior and Quantum Error Correction with Dynamical Gravity

According to the island formula, information in the code subspace defined in the black hole interior is embedded in the Hawking radiation after the Page time. At first sight, this embedding suggests that operations acting on the Hawking radiation could modify the information in the code subspace, potentially leading to an apparent violation of causality. Indeed, in previous studies based on the PSSY model, which incorporates only the topological degrees of freedom of gravity, it was shown that when the error is sufficiently large, a violation of causality can arise, as indicated by a nonvanishing mutual information. In this paper, we investigate the situation in which dynamical gravity also acts on the Hawking radiation. In this case, operations on the Hawking radiation induce nontrivial backreaction on the bulk spacetime appearing in the gravitational path integral for the mutual information – an effect that is absent when the Hawking radiation is non-gravitating. We find that this backreaction renders the relevant mutual information vanishing. This result implies that, in theories with dynamical gravity, the apparent violation of causality is resolved.

💡 Research Summary

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The paper investigates whether the interior of an evaporating black hole can be regarded as a quantum error‑correcting code (QEC) when the Hawking radiation is itself gravitating. Using two two‑dimensional gravity models, the authors compute the Renyi‑2 mutual information (RMI) between a reference system that stores the logical information and the environment that encodes the error. The RMI directly tests the decoupling condition: if it vanishes, the interior code is protected against the error; if it is non‑zero, the code can be disturbed, potentially leading to a causality violation.

First, the authors study the “West Coast model”, a topological gravity theory in which only the Euler characteristic contributes to the action. In this setting the bulk is populated by end‑of‑the‑world (EoW) branes that represent both black‑hole microstates and Hawking quanta. Errors acting on the radiation are modelled as massive branes (Kraus operators) inserted in the gravitational path integral. Because the model lacks dynamical backreaction, the dominant saddle after the Page time is a fully‑connected wormhole linking all replica copies of the two universes (the black hole and its bath). When the error rank is sufficiently large, this wormhole dominates the RMI, giving a positive value. Consequently the decoupling condition fails for a short interval after the Page time, reproducing earlier results from the PSSY model that high‑complexity operations on the radiation can modify the interior and seemingly violate causality.

Next, the authors turn to Jackiw‑Teitelboim (JT) gravity, which includes a dynamical dilaton and allows genuine backreaction of the error brane on the geometry. The full Euclidean action contains a topological term (-S_{0}\chi) and a dynamical term (-\frac{1}{2}\int \phi(R+2)) together with brane actions. When an error acts on the Hawking radiation, the associated Kraus operator is again represented by a massive brane, but now it sources the dilaton equation and changes the bulk geometry. This backreaction raises the action of the fully‑connected wormhole dramatically, making it subdominant for all values of the error rank. Instead, replica geometries in which the wormhole is either absent or only partially connected dominate, and the computed Renyi‑2 mutual information vanishes exactly. Therefore the decoupling condition holds irrespective of the error’s complexity, and the interior code remains intact.

The paper provides detailed derivations of the Renyi‑2 entropies, discusses the scaling dimension of the error operator, and shows how the Kraus operators map onto massive branes. It also analyses the “recoverability + non‑commutativity” condition that signals a causality violation, demonstrating that in JT gravity the backreaction prevents both conditions from being satisfied simultaneously.

The main conclusion is that when gravity acts on the Hawking radiation, the gravitational backreaction generated by any error automatically protects the black‑hole interior from being altered. This resolves the apparent causality paradox that arises in non‑gravitating bath models. The work thus bridges quantum information theory and dynamical gravity, suggesting that the very dynamics of spacetime provide a natural safeguard for holographic quantum error‑correcting codes. Future directions include extending the analysis to higher dimensions, exploring non‑perturbative effects, and constructing explicit recovery protocols in the presence of dynamical backreaction.