  We generalize the quantum Arimoto-Blahut algorithm by Ramakrishnan et al.
(IEEE Trans. IT, 67, 946 (2021)) to a function defined over a set of density
matrices with linear constraints so that our algorithm can be applied to
optimizations of quantum operations. This algorithm has wider applicability.
Hence, we apply our algorithm to the quantum information bottleneck with three
quantum systems, which can be used for quantum learning. We numerically compare
our obtained algorithm with the existing algorithm by Grimsmo and Still (Phys.
Rev. A, 94, 012338 (2016)). Our numerical analysis shows that our algorithm is
better than their algorithm.
