Quantum Statistical Manifolds

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.

submitted to the proceedings of Entropy 2018

References in corpus (4)

• Information geometric methods for complexity
• A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the $q$-$z$ Family
• Quantum Statistical Manifolds
• Quantum statistical manifolds: the linear growth case

Cited by in corpus (9)

• Quantum Statistical Manifolds
• Manifolds of classical probability distributions and quantum density operators in infinite dimensions
• Quantum States, Groups and Monotone Metric Tensors
• Quantum statistical manifolds: the linear growth case
• Exponential arcs in the manifold of vector states on a sigma-finite von Neumann algebra
• Parameter-free description of the manifold of non-degenerate density matrices
• Group actions and Monotone Quantum Metric Tensors
• Exponential arcs in manifolds of quantum states
• Log-affine geodesics in the manifold of vector states on a von Neumann algebra