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 ε accuracy, the iteration complexity of the algorithm is up bounded by log n log ε ε where n is the size of the input alphabet. In particular, when the output state {ρ_x}_x∈X is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an ε accurate solution with a complexity of O(m³ log n log ε/ε), and O(m³ log ε log_(1-δ) D(ε/p*||p^N(0))) in the special case, where m is the output dimension and D(p*||p^N(0)) is the relative entropy of two distributions and δ is a positive number.