2003/10/29 by Attila Andai, Andai, Attila
Computer Science · Mathematics · Physics and Astronomy · #53C20 #81Q99 #FOS: Physical sciences #Mathematical Inequalities and Applications #Mathematical Physics (math-ph) #Quantum Information and Cryptography #Statistical Mechanics and Entropy #math-ph #math.MP #msc:53C20 #msc:81Q99
paper · pdf · doi:10.48550/arxiv.math-ph/0310064
20 pages
arxiv created 2003/10/29 · openalex publication_date 2003/10/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The canonical correlation or Kubo-Mori scalar product on the state space of a finite quantum system is a natural generalization of the classical Fisher metric. This metric is induced by the von Neumann entropy or the relative entropy of the quantum mechanical states. An important conjecture of Petz that the scalar curvature of the state space with Kubo-Mori scalar product as Riemannian metric is monotone with respect to the majorisation relation of states: the scalar curvature is increases if one goes to more mixed states. We give an appropriate grouping for the summands in the expression for the scalar curvature. The conjecture will follows from the monotonicity of the summands. We prove the monotonicity for some of these summands and we give numerical evidences that the remaining terms are monotone too. Note that the real density matrices form a submanifold of the complex density matrices. We prove that if Petz's conjecture true for complex density matrices then it is true for real density matrices too.