The Curvature of the Bogoliubov-kubo-mori Scalar Product on Matrices

The state space of a finite quantum system is identified with the set of positive semidefinite matrices of trace 1. The set of all strictly positive definite matrices of trace 1 becomes naturally a differentiable manifold and the Bogoliubov-Kubo-Mori scalar product defines a Riemannian structure on it. Reference [4] tells about the relation of this metric to the von Neumann entropy functional. Shortly speaking, the von Neumann entropy is a concave functional on the above space of matrices and its negative Hessian is a positive definite inner product knowns as Bogoliubov-Kubo-Mori scalar product (or canonical correlation). For the physical background of the Bogoliubov-Kubo-Mori inner product, [2] is a good source. The objective of the paper is to compute the scalar curvature in the Riemannian geometry of the Bogoliubov-Kubo-Mori scalar product. Earlier this was obtained in [7] for the 2 × 2 matrices and some sectional curvatures were computed in [4] for larger matrices. In this paper, we consider the space of real density matrices which is a geodetic submanifold in the space of complex density matrices. Our study is strongly motivated by the conjectures formulated in [4] and [5]. It was conjectured that the scalar curvature takes its maximum when all eigenvalues of the density matrix are equal, and more generally the scalar curvature is monotone with respect to the majorization relation of matrices. Although we obtain an explicit formula for the scalar curvature, the conjecture remains unproven. (Nevertheless, a huge number of numerical examples are still supporting the conjecture.) The method of computation of the Ricci and scalar curvature is inspired by [3]. First we use a basis in the tangent space to express the scalar curvature and then we get rid of the basis by means of linear algebra. When this paper was nearly finished we received the preprint [1] where the scalar curvature is computed for arbitrary monoton metrics in the complex case by a different method. Our aim is to find a formula for the scalar curvature which depends only on easily computable quantities of matrices. The scalar curvature turns out to be a rather complicated function of the eigenvalues and we express it in terms of some symmetric functions of pairs and triplets of the eigenvalues.