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

TL;DR

This paper explores the connection between the geometrical formulation of quantum mechanics and statistical geometry, using Fisher metrics and exponential connections to relate probability spaces to quantum state spaces.

Contribution

It explicitly links finite-dimensional quantum systems' geometry with statistical structures on probability simplices, providing a new geometric interpretation.

Findings

Constructs a K"ahler structure on the tangent bundle of probability spaces.

Shows this structure induces the natural quantum geometric structure.

Establishes a link between statistical and quantum geometries.

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.