On the Error Exponent Distribution of Code Ensembles over Classical-Quantum Channels

TL;DR

This paper analyzes the distribution of error exponents in classical-quantum channel code ensembles, revealing how it concentrates around the random coding exponent at high rates and exceeds it at low rates, with implications for understanding quantum communication reliability.

Contribution

It establishes the distribution behavior of error exponents over classical-quantum channels, connecting it with the random coding and typical random coding exponents across different rate regimes.

Findings

Error exponent distribution exceeds the RCE at low rates.

Distribution concentrates around the RCE at high rates.

Threshold matches the TRC exponent in high rate regime.

Abstract

We show that the probability distribution of the error exponent in i.i.d. code ensembles over classical-quantum (CQ) channels with arbitrary output states accumulates above a threshold that is strictly larger than the CQ random coding exponent (RCE) at low rates, while coinciding with it at rates close to the mutual information of the channel. This result, combined with the work by Dalai [1] and the recent ones by Renes [2] and Li and Yang [3], implies that the ensemble distribution of error exponents concentrates around the CQ RCE in the high rate regime. Moreover, in the same rate regime the threshold we derive coincides with the ensemble-average of the exponent, that is, the typical random coding (TRC) exponent [4].