On the Concavity of Auxiliary Function in Classical-Quantum Channels

TL;DR

This paper proves that the auxiliary function of classical-quantum channels is concave, extending known results from classical channels and enabling better analysis of error probabilities in quantum information theory.

Contribution

It establishes the concavity of the auxiliary function for classical-quantum channels, generalizing previous partial results and utilizing geometric means of operators.

Findings

Auxiliary function is concave for classical-quantum channels.

Concavity leads to improved bounds on error probabilities.

Extends classical channel results to quantum channels.

Abstract

The auxiliary function of a classical channel appears in two fundamental quantities that upper and lower bound the error probability, respectively. A crucial property of the auxiliary function is its concavity, which leads to several important results in finite block length analysis. In this paper, we prove that the auxiliary function of a classical-quantum channel also enjoys the same concave property, extending an earlier partial result to its full generality. The key component in our proof is a beautiful result of geometric means of operators.