State-dependent geometries from magic-enriched quantum codes

TL;DR

This paper introduces a new framework for quantum error-correcting codes that models state-dependent geometries and gravitational backreaction, extending holographic codes to include approximate recovery and matter-geometry interactions.

Contribution

It develops a Ryu-Takayanagi-like entropy decomposition for approximate codes, linking bulk matter entropy and proto-area, and identifies magic as key to matter-geometry coupling.

Findings

Proto-area increases with bulk entropy in skewed codes.

Magic in the Choi state controls matter-geometry coupling.

Approximate codes can model state-dependent geometries and backreaction.

Abstract

Quantum error-correcting codes provide a powerful framework for emergent spacetime, yet existing holographic code models describe only quantum fields on a fixed background: in exact erasure-correcting codes, the entropic area term is state independent and cannot capture gravitational backreaction. We argue that this limitation is intrinsic to exact subsystem recovery and that incorporating backreaction instead requires approximate quantum error correction. We introduce a Ryu-Takayanagi-like entropy decomposition for approximate subsystem erasure-correcting codes, defining bulk matter entropy via optimal recovery and a complementary proto-area entropy as the difference between boundary entropy and recoverable bulk entropy. For a broad class of skewed quantum codes obtained by small nonlocal perturbations of exact codes, the proto-area increases monotonically with bulk entropy, closely aligning with the behavior of quantum extremal surfaces. We identify the origin of this response as a form of tripartite non-local magic in the Choi state of the encoding map, which vanishes in stabilizer codes and controls the leading matter-geometry coupling in approximate subsystem erasure-correcting codes.