Theory of Quantum Error Correction for General Noise

TLDR

A measure of quality of an error‑correcting code is the maximum number of errors that it can correct. The authors introduce a generalized notion of error count that applies to any quantum or classical system with arbitrary interactions and prove the existence of large codes for both quantum and classical information. They model error‑correcting codes as subsystems, linking them to irreducible representations of operator algebras and demonstrating that noiseless subsystems constitute infinite‑distance error‑correcting codes. The study shows that e‑error‑correcting codes protect information without independence assumptions, establishes the existence of large quantum and classical codes, and demonstrates that noiseless subsystems are infinite‑distance error‑correcting codes.

Abstract

A measure of quality of an error-correcting code is the maximum number of errors that it is able to correct. We show that a suitable notion of "number of errors" e makes sense for any quantum or classical system in the presence of arbitrary interactions. Thus, e-error-correcting codes protect information without requiring the usual assumptions of independence. We prove the existence of large codes for both quantum and classical information. By viewing error-correcting codes as subsystems, we relate codes to irreducible representations of operator algebras and show that noiseless subsystems are infinite-distance error-correcting codes.