Entropy of a quantum error correction code

TL;DR

This paper introduces a new concept of entropy for quantum error correcting codes, providing multiple equivalent definitions and analyzing its implications for code optimality and structure.

Contribution

It defines and explores a novel entropy measure for quantum codes, linking it to code properties and extending it to subsystem codes.

Findings

Entropy relates to code optimality and error correction capability.

Binary unitary channels analyzed in detail.

Extension of entropy concept to subsystem codes.

Abstract

We define and investigate a notion of entropy for quantum error correcting codes. The entropy of a code for a given quantum channel has a number of equivalent realisations, such as through the coefficients associated with the Knill-Laflamme conditions and the entropy exchange computed with respect to any initial state supported on the code. In general the entropy of a code can be viewed as a measure of how close it is to the minimal entropy case, which is given by unitarily correctable codes (including decoherence-free subspaces), or the maximal entropy case, which from dynamical Choi matrix considerations corresponds to non-degenerate codes. We consider several examples, including a detailed analysis in the case of binary unitary channels, and we discuss an extension of the entropy to operator quantum error correcting subsystem codes.