Li Haobo, Cai Ning
School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China.
Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 201210, China.
Entropy (Basel). 2020 Feb 16;22(2):222. doi: 10.3390/e22020222.
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 ε accuracy, the iteration complexity of the algorithm is upper bounded by log n log ε ε where is the size of the input alphabet. In particular, when the output state { ρ x } x ∈ 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 ( m 3 log n log ε ε ) , and O ( m 3 log ε log ( 1 - δ ) ε D ( p * | | p N 0 ) ) in the special case, where is the output dimension, D ( p * | | p N 0 ) is the relative entropy of two distributions, and δ is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.
基于有本1972年的研究成果，我们提出了一种迭代算法，用于计算具有有限输入字母表和有限维输出的离散无记忆经典 - 量子信道的容量，我们将其称为经典 - 量子信道的布莱胡特 - 有本算法，并考虑了输入成本约束。我们证明，为达到ε精度，该算法的迭代复杂度上限为log n log ε ε ，其中 是输入字母表的大小。特别地，当输出态{ ρ x } x ∈ X 在复矩阵空间中线性独立时，该算法具有几何收敛性。我们还证明，该算法以O ( m 3 log n log ε ε ) 的复杂度达到ε精度解，在特殊情况下复杂度为O ( m 3 log ε log ( 1 - δ ) ε D ( p * | | p N 0 ) ) ，其中 是输出维度，D ( p * | | p N 0 ) 是两个分布的相对熵，且δ是一个正数。我们进行了数值实验，并分析了二维二进制情况的近似解。
