Abstract
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 accuracy, the iteration complexity of the algorithm is upper bounded by where n is the size of the input alphabet. In particular, when the output state is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an accurate solution with a complexity of , and in the special case, where m is the output dimension, is the relative entropy of two distributions, and is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.
Highlights
IntroductionThe computation of channel capacity has always been a core problem in information theory
The computation of channel capacity has always been a core problem in information theory.The very well-known Blahut-Arimoto algorithm [1,2] was proposed in 1972 to compute the discrete memoryless classical channel
We propose an algorithm of Blahut-Arimoto type to compute the capacity of discrete memoryless classical-quantum channel
Summary
The computation of channel capacity has always been a core problem in information theory. In 1998, Holevo showed [4] that the classical capacity of the classical-quantum channel is the maximization of a quantity called the Holevo information over all input distributions. Despite the NP-completeness, in [11], an example is given of a qubit quantum channel which requires four inputs to maximize the Holevo capacity. We show that, with proper manipulations, the BA algorithm can be applied to computing the capacity of classical-quantum channel with an input constraint efficiently. Each letter x ∈ X is mapped to a density matrix ρ x , the classical-quantum channel can be represented as a set of density matrices {ρ x } x∈X. The von Neumann entropy of a density matrix ρ is denoted by H (ρ) = − Tr[ρ log ρ].
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