Lax-type equation unifying gradient systems in quantum and classical statistical models

In a previous paper, Y. Uwanoet al.: nlin/SI:0512004, the author made a short remark on a similarity between gradient systems on a quantum information space and on a classical one: One is on the space of regular relative-configurations of multi-qubit states of ordered tuples and the other is on the space of multinomial distributions studied by Y. Nakamura: Japan J. Indust. Appl. Math.10 (1993) 179. This paper aims to report very briefly a Lax-type equation unifying the gradient systems on the information spaces with quantum and with classical features.