A gradient system on the quantum information space realizing the averaged learning equation of Hebb type

The averaged learning equation (ALEH) applicable to the principal component analyzer is studied from both quantum information geometry and dynamical system viewpoints. On the quantum information space (QIS), the space of regular density matrices endowed with the quantum SLD-Fisher metric, a gradient system is given as an extension of the ALEH; on the submanifold, consisting of the diagonal matrices, of the QIS, the gradient flow coincides with the ALEH up to a local diffeomorphism.