Information geometry in functional spaces of classical and quantum finite statistical systems

Abstract

Statistical states and bounded random variables (observables) of finite physical systems can be represented in real Banach spaces L s 1 and L s ∞, respectively. Since both norms are Krein-weak, the solution of the estimation problem in these spaces is not necessarily unique. The latter property occurs on the Hilbert-Schmidt space L s 2 which is connected with the Onicescu information energy and the method of least squares. The square information is only an approximation of the “true” logarithmic Shannon information which induces a “logarithmic” asymmetric geometry by means of the concept of relative information (gain of information). This geometry was known in the classical case as the asymmetric Pythagorean geometry (Chentsov [10]) and is approximated by the Riemannian geometry of Fisher's information (Kullback and Leibler [45]). This paper shows that a similar geometric construction is also possible in the quantum case. The fundamental formulae of the quantum case are given (they differ in some details from the classical ones), and possible physical applications are shortly sketched.