Quantum Error Correction of Observables

TL;DR

This paper develops a formalism for quantum error correction using operator algebras, enabling correction of hybrid quantum-classical information without strict subspace encoding, and provides detailed proofs, new results, and examples.

Contribution

It extends quantum error correction theory to operator algebras, allowing correction of hybrid information and broadening applicability beyond subspace codes.

Findings

Provides detailed proofs for operator algebra-based error correction

Derives new results in quantum error correction formalism

Includes examples and discusses extensions to operator spaces

Abstract

A formalism for quantum error correction based on operator algebras was introduced in [1] via consideration of the Heisenberg picture for quantum dynamics. The resulting theory allows for the correction of hybrid quantum-classical information and does not require an encoded state to be entirely in one of the corresponding subspaces or subsystems. Here, we provide detailed proofs for the results of [1], derive a number of new results, and we elucidate key points with expanded discussions. We also present several examples and indicate how the theory can be extended to operator spaces and general positive operator-valued measures.