
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…
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