Quantum Statistical Manifolds

TL;DR

This paper introduces a new geometric framework for quantum states using differential geometry on faithful states in finite-dimensional C*-algebras, aiming to simplify and generalize quantum information geometry.

Contribution

It shifts focus from positive density matrices to faithful quantum states and adopts a parameter-free approach, establishing Banach manifold structures and affine coordinates for quantum states.

Findings

Established a Banach manifold structure for quantum states.

Introduced affine coordinates compatible with the Bogoliubov inner product.

Provided a foundation for more general quantum information geometry theories.

Abstract

Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection.