Non-Commutative Blahut-Arimoto Algorithms
Mario Berta, Navneeth Ramakrishnan, Raban Iten, Volkher B. Scholz · arXiv (Cornell University) · 2019
We generalize alternating optimization algorithms of Blahut-Arimoto type to the quantum setting. In particular, we give iterative algorithms to compute the mutual information of quantum channels, the Holevo quantity of classical-quantum channels, the coherent information of less noisy quantum channels, and the thermodynamic capacity of quantum channels. Our convergence analysis is based on quantum entropy inequalities and leads to a priori additive $\varepsilon$-approximations after $\mathcal{O}\left(\varepsilon^{-1}\log N\right)$ iterations, where $N$ denotes the input dimension of the channel. We complement our analysis with an a posteriori stopping criterion which allows us to terminate the algorithm after significantly fewer iterations compared to the a priori criterion in numerical examples. Finally, we discuss heuristics to accelerate the convergence.