Assessing quantum error correction: fidelity and entanglement measures with application to photonic codes

TL;DR

This paper introduces a quantitative figure of merit based on fidelity to evaluate quantum error correction schemes, demonstrating its effectiveness with photonic codes and implications for continuous-variable QEC.

Contribution

It proposes a new fidelity-based measure for assessing quantum error correction performance, applicable to photonic and continuous-variable codes, improving over previous qualitative criteria.

Findings

The new measure effectively quantifies QEC performance including uncorrectable errors.

Photonic qubit codes show advantages when evaluated with the new fidelity measure.

The measure reproduces known limitations in Gaussian quantum channels without entanglement calculations.

Abstract

By interpreting the well-known, qualitative criteria for the existence of quantum error correction (QEC) codes by Knill and Laflamme from a quantitative perspective, we propose a figure of merit for assessing a QEC scheme based on the average fidelity between codewords. This enables us to quantify the performance of a protocol as a whole, including errors beyond the correctable set. Various examples are calculated for photonic qubit codes dealing with the experimentally relevant case of photon loss, demonstrating the advantages of the new measure. In the context of continuous-variable QEC, our codeword-overlap measure can be used to reproduce, in a different way with no need for calculating entanglement measures, a previous result concerning the impossibility of improving transmission of Gaussian states through Gaussian channels via Gaussian operations alone.