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.
Highlights
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
The basic example of a quantum statistical system starts with a self-adjoint operator H on a finite-dimensional or separable Hilbert space H, with the property that the operator exp(− βH ) is trace-class for all β in an open interval D of the real line R
The Hilbert space is assumed to be finite-dimensional to avoid the technicalities coming with unbounded operators
Summary
The basic example of a quantum statistical system starts with a self-adjoint operator H on a finite-dimensional or separable Hilbert space H, with the property that the operator exp(− βH ) is trace-class for all β in an open interval D of the real line R. An alternative approach to parameter-free quantum information geometry is described in [7]. Which approach eventually will lead to a fully developed theory is hard to predict Such a theory is expected to affect several domains of research, including Quantum Information Theory, Statistical Physics, in particular the study of phase transitions, and Complexity Theory. Both the classical case and the quantum case need a regularizing condition on the allowed density functions, respectively density operators Under this condition, they form a Banach manifold. Quantum states are labeled with operators belonging to the commutant of the GNS-representation rather than with density matrices. An Appendix about the GNS-representation and the modular operator is added for convenience of the reader
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