On the Curvature of a Certain Riemannian Space of Matrices
نویسنده
چکیده
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. In short, the von Neumann entropy is a concave functional on the above space of matrices and its negative Hessian is a positive definite inner product known as Bogoliubov–Kubo– Mori scalar product (or canonical correlation). For the physical background of the Bogoliubov–Kubo–Mori inner product, Ref. 2 is a good source. The objective of this paper is to compute the scalar curvature in the Riemannian geometry of the Bogoliubov–Kubo–Mori scalar product. Earlier this was obtained in Ref. 7 for the 2 × 2 matrices and some sectional curvatures were computed in
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