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

TL;DR

This paper introduces a Blahut-Arimoto type iterative algorithm for efficiently computing the capacity of classical-quantum channels with finite input and output dimensions, including convergence and complexity analysis.

Contribution

It extends the classical Blahut-Arimoto algorithm to classical-quantum channels, providing convergence guarantees and complexity bounds for capacity computation.

Findings

Algorithm achieves $rac{ ext{log} n ext{log} \varepsilon}{\varepsilon}$ iteration complexity.

When output states are linearly independent, the algorithm converges geometrically.

Complexity bounds are provided for general and special cases, involving output dimension and relative entropy.

Abstract

Based on Arimoto's work in 1978, 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 $\varepsilon$ 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 $\varepsilon$ accurate solution with a complexity of $O(\frac{m^3\log n\log\varepsilon}{\varepsilon})$, and $O(m^3\log\varepsilon\log_{(1-\delta)}\frac{\varepsilon}{D(p^*||p^{N_0})})$ in the special case, where $m$ is the output dimension and $D(p^*||p^{N_0})$ is the relative entropy of two distributions and $\delta$ is a positive number.