Remarks on the statistical origin of the geometrical formulation of quantum mechanics

Abstract

A quantum system can be entirely described by the K\"ahler structure of the projective space P(H) associated to the Hilbert space H of possible states; this is the so-called geometrical formulation of quantum mechanics. In this paper, we give an explicit link between the geometrical formulation (of finite dimensional quantum systems) and statistics through the natural geometry of the space Pn of non-vanishing probabilities p defined on a finite set En:={x1,...,xn}. More precisely, we use the Fisher metric and the exponential connection (both being natural statistical objects living on Pn) to construct, via the Dombrowski splitting Theorem, a K\"ahler structure on TPn (the tangent bundle of Pn) which has the property that it induces the natural K\"ahler structure of a suitably chosen open dense subset of the finite dimensional complex projective space.