Gallager Bound of Classical-Quantum Channel Coding

Gallager bound is known as the minimum upper bound of average error probability of classical channel coding[5]. Burnashev and Holevo[8] derived its quantum version in the pure states case, and proposed its quantum version in the mixed states case. The attainability of their quantum version has been an open problem, We prove that the bound proposed by them can be attained by a POVM proposed by Holevo[9]. That is, it is shown that the average error probability is smaller than this bound. As a byproduct, we obtain a simple proof of the achievable part of the channel coding theorem capacity for a stationary memoryless classical-quantum channel.