Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction

TL;DR

This paper presents a unified algebraic framework for entanglement-assisted quantum error correction, generalizing previous models and introducing new classes of codes with improved theoretical understanding.

Contribution

It unifies existing entanglement-assisted quantum error correction frameworks under a single operator algebra approach and introduces new code classes.

Findings

Proves a general error correction theorem for the unified framework.

Defines a natural notion of distance for these codes.

Constructs new classes of entanglement-assisted subsystem and hybrid codes.

Abstract

We introduce a framework for entanglement-assisted quantum error correcting codes that unifies the three original frameworks for such codes called EAQEC, EAOQEC, and EACQ under a single umbrella. The unification is arrived at by viewing entanglement-assisted codes from the operator algebra quantum error correction perspective, and it is built upon a recently established extension of the stabilizer formalism to that setting. We denote the framework by EAOAQEC, and we prove a general error correction theorem for such codes, derived from the algebraic perspective, that generalizes each of the earlier results. This leads us to a natural notion of distance for such codes, and we derive a number of distance results for subclasses of the codes. We show how EACQ codes form a proper subclass of the entanglement-assisted subspace codes defined by EAOAQEC. We identify and construct new classes of entanglement-assisted subsystem codes and entanglement-assisted hybrid classical-quantum codes that are found outside of the earlier approaches.