Quantum Statistical Manifolds: The Finite-Dimensional Case

Abstract

Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed. The emphasis is shifted from a manifold of strictly positive density matrices to a manifold M of faithful quantum states on a von Neumann algebra of bounded linear operators working on a Hilbert space. In order to avoid technicalities the theory is developed for the algebra of n-by-n matrices. A chart is introduced which is centered at a given faithful state ωρ. It maps the manifold M onto a real Banach space of self-adjoint operators belonging to the commutant algebra. The operator labeling any state ωσ of M also determines a tangent vector in the point ωρ along the exponential geodesic in the direction of ωσ. A link with the theory of the modular automorphism group is worked out. Explicit expressions for the chart can be derived in terms of the modular conjugation and the relative modular operators.