Quantum minimal surfaces from quantum error correction

TL;DR

This paper establishes a link between quantum error correction and minimal surface prescriptions for entropies in holographic theories, generalizing previous results and introducing a new bulk geometry framework.

Contribution

It introduces a generalized quantum minimal surface concept for boundary entropies, extending holographic code analysis and bulk reconstruction methods.

Findings

Established equivalence between state-specific reconstruction and quantum minimal surfaces.

Constructed an emergent bulk geometry for general quantum codes.

Extended results to approximate and non-isometric codes, including black hole interiors.

Abstract

We show that complementary state-specific reconstruction of logical (bulk) operators is equivalent to the existence of a quantum minimal surface prescription for physical (boundary) entropies. This significantly generalizes both sides of an equivalence previously shown by Harlow; in particular, we do not require the entanglement wedge to be the same for all states in the code space. In developing this theorem, we construct an emergent bulk geometry for general quantum codes, defining "areas" associated to arbitrary logical subsystems, and argue that this definition is "functionally unique." We also formalize a definition of bulk reconstruction that we call "state-specific product unitary" reconstruction. This definition captures the quantum error correction (QEC) properties present in holographic codes and has potential independent interest as a very broad generalization of QEC; it includes most traditional versions of QEC as special cases. Our results extend to approximate codes, and even to the "non-isometric codes" that seem to describe the interior of a black hole at late times.