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

2021/01/19 by Jan Naudts
Computer Science · Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebra over a field #Centralizer and normalizer #Hilbert space #Mathematics #Pure mathematics #Quantum Information and Cryptography #Quantum Mechanics and Applications #Vector space #Von Neumann algebra #Von Neumann architecture #math.OA #msc:53B12 #msc:62B12

paper · pdf · doi:10.1007/s41884-021-00064-4

published as Inf0. Geo. (2022) · 31 pages A4, extends and replaces arXiv:1901.06267

arxiv created 2021/01/19 · openalex publication_date 2022/01/05 · arxiv updated 2022/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Log-affine geodesics in the manifold of vector states on a von Neumann algebra
• The quantum information manifold for ε-bounded forms
• A bounded operator approach to Tomita-Takesaki theory
• On the equilibrium states in quantum statistical mechanics
• Information Geometry and Its Applications
• Tomita's Theory of Modular Hilbert Algebras and its Applications