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Quantum minimal surfaces from quantum error correction

by Chris Akers, Geoff Penington

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Submission summary

Authors (as registered SciPost users): Christopher Akers
Submission information
Preprint Link: https://arxiv.org/abs/2109.14618v3  (pdf)
Date accepted: 2022-04-26
Date submitted: 2022-04-18 16:10
Submitted by: Akers, Christopher
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical

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.

Published as SciPost Phys. 12, 157 (2022)



Author comments upon resubmission

We thank the reviewers, and we have modified the paper based on their suggestions.

List of changes

1. We fixed typos, including a minus sign in the third paragraph of section 6.1 and in page 6.
2. We clarified Remark 2.5.
3. We added a section 6.4 about extremality, in which we discuss a natural next step towards understanding holographic dynamics in these codes.

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