Mirror Descent and Quantum Blahut-Arimoto Algorithms
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Convex optimization problems arise naturally when studying the fundamental limits of communication systems, such as computing quantities known as channel capacities. The Blahut-Arimoto algorithm is a well known method to solve for classical channel capacities, and recent works have extended this algorithm to compute various quantum channel capacities. We show that the classical and quantum Blahut-Arimoto algorithms can be interpreted as mirror descent, which is a well studied generalisation of projected gradient descent for non-Euclidean geometries. Using this perspective, we show how existing tools in convex optimization can be used to guarantee bounds on convergence rates, and how the algorithm can be modified to solve a wider range of related problems in quantum information theory.