Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis †

Based on Arimoto’s work in 1972, 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 [Formula: see text] accuracy, the iteration complexity of the algorithm is upper bounded by [Formula: see text] where n is the size of the input alphabet. In particular, when the output state [Formula: see text] is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an [Formula: see text] accurate solution with a complexity of [Formula: see text] , and [Formula: see text] in the special case, where m is the output dimension, [Formula: see text] is the relative entropy of two distributions, and [Formula: see text] is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.