# Shrinking of Operators in Quantum Error Correction and AdS/CFT

## TL;DR

This paper explores how certain operators in quantum error correction can be simplified to act on smaller subsystems, with implications for the AdS/CFT correspondence and entanglement wedge reconstruction.

## Contribution

It introduces a method to shrink operator supports in bipartite states and applies this to construct decoders and analyze operator structures in AdS/CFT-related quantum codes.

## Key findings

- Operators can be supported on smaller subsystems without changing their action.
- Systematic construction of decoders for erasure errors in quantum codes.
- Proof of the validity and converse of entanglement wedge reconstruction.

## Abstract

We first show that a class of operators acting on a given bipartite pure state on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ can shrink its supports on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ to only $\mathcal{H}_{A}$ or $\mathcal{H}_{B}$ while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes against erasure errors. The implications of the results for the operator dictionary in the AdS/CFT correspondence are also discussed. The "subalgebra code with complementary recovery" introduced in the recent work of Harlow is a quantum error-correcting code that shares many common features with the AdS/CFT correspondence. We consider it under the restriction of the bulk (logical) Hilbert space to a subspace that generally has no tensor factorization into subsystems. In this code, the central operators of the reconstructed algebra on the boundary subregion can emerge as a consequence of the restriction of the bulk Hilbert space. Finally, we show a theorem in this code which implies the validity of not only the entanglement wedge reconstruction but also its converse statement with the central operators.