Exponential arcs in the manifold of vector states on a sigma-finite von Neumann algebra

Abstract

This paper introduces the notion of exponential arcs in Hilbert space and of exponential arcs connecting vector states on a sigma-finite von Neumann algebra in its standard representation. Results from Tomita-Takesaki theory form an essential ingredient. Starting point is a non-commutative Radon-Nikodym theorem that involves positive operators affiliated with the commutant algebra. It is shown that exponential arcs are differentiable and that parts of an exponential arc are again exponential arcs. Special cases of probability theory and of quantum probability are used to illustrate the approach.

31 pages A4, extends and replaces arXiv:1901.06267

References in corpus (2)

• From the Jordan product to Riemannian geometries on classical and quantum states
• Differential geometric aspects of parametric estimation theory for states on finite-dimensional C*-algebras

Cited by in corpus (3)

• Parameter-free description of the manifold of non-degenerate density matrices
• Exponential arcs in manifolds of quantum states
• Group actions and Monotone Quantum Metric Tensors