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 $\frac{{\log n\log \varepsilon }}{\varepsilon }$ where n is the size of the input alphabet. In particular, when the output state ${\{ {\rho _x}\} _{x \in \mathcal{X}}}$ 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 $O\left( {\frac{{{m^3}\log n\log \varepsilon }}{\varepsilon }} \right)$, and $O\left( {{m^3}\log \varepsilon {{\log }_{(1 - \delta )}}\frac{\varepsilon }{{D\left( {{p^{\ast}}\left\| {{p^{{N_0}}}} \right.} \right)}}} \right)$ in the special case, where m is the output dimension and $D\left( {{p^{\ast}}\left\| {{p^{{N_0}}}} \right.} \right)$ is the relative entropy of two distributions and δ is a positive number.