**Classical and Quantum Information Geometry **

by Matheus R. Grasselli.

Department of Mathematics, King's College London, 2001.

Supervisor: Raymond F. Streater.

Examiners: Boguslaw Zegarlinski (Imperial College London) and Yuri Suhov (Cambridge
University).
**Abstract**

We begin with the construction of an infinite dimensional Banach manifold of
probability measures using the completion of the set of bounded random variables
in the appropriate Orlicz norm as coordinate spaces. The infinite dimensional
version of the Fisher metric as well as the exponential and mixture connections
are introduced. It is then proved that they form a dualistic structure in the sense
of Amari. The interpolating alpha-connections are defined, at the level of covariant
derivatives, via embeddings into L_r-spaces and then found to be convex mixtures
of the +1 and -1-connections. Several well known parametric results are obtained as finite
dimensional restrictions of the nonparametric case.

Next, for finite dimensional quantum systems, we study a manifold
of density matrices and explore the concepts of monotone metrics and duality
in order to establish that the only monotone metrics with respect to which the
exponential and mixture connections are mutually dual are the scalar multiples
of the Bogoliubov- Kubo-Mori inner product of quantum statistical mechanics.

For infinite dimensional quantum systems, we present a general
construction of a Banach manifold of density operators using the technique of
epsilon-bounded perturbations, which contains small perturbations of forms and
operators in the sense of Kato as special cases. We then describe how to obtain
an affine structure in such a manifold, together with the corresponding exponential
connection. The free energy functional is proved to be analytic on small neighbourhoods
in the manifold.

We conclude with an application of the methods of Information Geometry and
Statistical Dynamics to a concrete problem in fluid dynamics: the derivation of
the time evolution equations for the density, energy and momentum fields of
a fluid under an external field.