A Blahut-Arimoto Type Algorithm for Computing Classical-Quantum Channel Capacity

Based on Arimoto's work in 1978 [1], we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut-Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that to reach epsilon accuracy, the iteration complexity of the algorithm is up bounded by log n vertical bar log epsilon/epsilon where n is the size of the input alphabet. In particular, when the output state {rho(x)}x is an element of x is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an epsilon accurate solution with a complexityto O(m(3) log n log epsilon/epsilon), and O(m(3) log epsilon log((1-delta)) epsilon/D(p* vertical bar vertical bar p(N0)) in the special case, where m is the output dimension and D(p* vertical bar vertical bar p(N0)) is the relative entropy of two distributions and delta is a positive number.