Holographic Quantum Error Correction and the Projected Black Hole Interior

TL;DR

This paper develops a holographic quantum error correction framework to reconstruct operators behind the horizons of pure black holes, using SYK microstates produced by projecting out one side of the thermofield double. It introduces both a toy random-tensor model and a rigorous projected subsystem code to show how interior operators can be mapped to the remaining boundary, with the dictionary becoming state- and projection-dependent. The work connects this fluid dictionary to entanglement wedge reconstruction, ER=EPR, and Hayden-Preskill-type information transfer, and discusses how this framework can mediate information flow between black holes connected by a wormhole. It also addresses objections to state dependence, reframing interior reconstruction as a projection-induced OAQEC process and exploring complexity and Page-time implications of the evolving dictionary.

Abstract

The quantum error correction interpretation of AdS/CFT establishes a sense of fluidity to the bulk/boundary dictionary. We show how this property can be utilized to construct a dictionary for operators behind horizons of pure black holes. We demonstrate this within the context of the SYK model with pure black hole microstates obtained via projecting out a single side of the thermofield double (and perturbed versions thereof). Assuming an erasure subsystem code for the duality between the eternal black hole and the thermofield double, this projection results in a rewiring of the dictionary so as to map the interior operators to the remaining boundary in a determinable way. We find this dictionary to be sensitive to the implemented projection in a manner reminiscent of previous state-dependent constructions of the black hole interior. We also comment on how the fluidity of the dictionary can be used to transfer information between two black holes connected by a wormhole, relating the ideas of entanglement wedge reconstruction and the Hayden-Preskill decoding criterion.

Figures (14)

• Figure 1: The diagram on the left is the standard eternal black hole spacetime dual to the thermofield double state. The projected state on the right is dual to a black hole in a pure state with an end-of-the-world (EWB) brane cutting off the spacetime in the interior. The EWB can be viewed as a UV insertion on the left boundary which then proceeds to fall into the black hole.
• Figure 2: Long wormholes supported by out-of-time-ordered (OTO) shockwaves also project into pure states with long throats capped off by an EWB.
• Figure 3: Gravitationally dressing a bulk particle (green) to either boundary pushes the boundary trajectories away from the center of AdS$_2$. The solid boundary lines correspond to dressing entirely to the right, and the dotted trajectories are for dressing entirely to the left. The same story holds for the case of the EWB on the right diagram.
• Figure 4: The naive expectation (left figure) is that a complete projection on the left boundary would distangle the quantum fields across the horizon forming a firewall. This is inconsistent with the motivated picture from the SYK analysis (right) that this projection generates a pure black hole with an interior and a smooth horizon.
• Figure 5: A single tensor can be viewed as the encoding of the state of a pure black hole's horizon degrees of freedom $H_L$ and a set of external modes $a$ into the boundary degrees of freedom $L$.

Theorems & Definitions (4)

• theorem 3.3.1
• proof
• theorem 3.4.1
• proof