Coherent information for CSS codes under decoherence

TL;DR

This paper analytically calculates the coherent information for CSS quantum error-correcting codes under local errors, linking decoding transitions to phase transitions in classical statistical models, and confirms the optimality of maximum-likelihood decoding.

Contribution

It provides a rigorous analytical framework connecting CSS code decoding thresholds with phase transitions in classical models, enhancing understanding of quantum error correction.

Findings

Coherent information for CSS codes can be computed via diagonalization and classical model mapping.

Decoding transition corresponds to a phase transition in a classical statistical mechanical model.

Maximum-likelihood decoding achieves the fundamental error correction threshold in the thermodynamic limit.

Abstract

Stabilizer codes lie at the heart of modern quantum-error-correcting codes (QECC). Of particular importance is a class called Calderbank-Shor-Steane (CSS) codes, which includes many important examples such as toric codes, color codes, and fractons. Recent studies have revealed that the decoding transition for these QECCs could be intrinsically captured by calculating information-theoretic quantities from the mixed state. Here we perform a simple analytic calculation of the coherent information for general CSS codes under local incoherent Pauli errors via diagonalization of the density matrices and mapping to classical statistical mechanical (SM) models. Our result establishes a rigorous connection between the decoding transition of the quantum code and the phase transition in the random classical SM model. It is also directly confirmed for CSS codes that exact error correction is possible if and only if the maximum-likelihood (ML) decoder always succeeds in the thermodynamic limit. Thus, the fundamental threshold is saturated by the optimal decoder.