Differential Geometrical Aspects of Quantum State Estimation and Relative Entropy

In this article, we treat the space S of finite-dimensional positive density operators (quantum states) in a differential geometrical viewpoint. We suppose that a generalized covariance for arbitrary two observables (Hermitian operators) is specified at each state in S, which includes the symmetrized inner product and the Bogoliubov inner product as special (but important) cases, and introduce a triplet structure (g, ∇(e), ∇(m) on S via the specified covariance, where g is a Riemannian metric and ∇(e) and ∇(m) are affine connections. The structure (g, ∇(e), ∇(m) is regarded as a quantum analogue of the triplet of Fisher metric, exponential connection and mixture connection on a space of probability densities introduced in the information geometry by S. Amari ([1]). Some aspects relating to the quantum state estimation and the relative entropy are treated in terms of the differential geometry, where the theory of dual connections developed by Nagaoka and Amari ([4] [1]) plays an essential role.