The black hole interior from non-isometric codes and complexity

TL;DR

This paper proposes that non-isometric quantum error correction codes, protected by computational complexity, can explain the emergence of the black hole interior despite the limitations of isometric encodings.

Contribution

It introduces a novel framework using non-isometric codes and complexity protection to model black hole interiors, unifying several prior ideas in quantum gravity.

Findings

Non-isometric codes can encode black hole interiors.

Complexity constraints explain effective field theory breakdown.

Unified model incorporates null states, Page curve, and state-dependent operators.

Abstract

Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of "non-isometric codes protected by computational complexity". We show that many previous ideas, such as the existence of a large number of "null states", a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, "state-dependent/state-specific" operator reconstruction, and the "simple entropy" approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.