On the monotonicity of scalar curvature in classical and quantum information geometry

Abstract

We study the statistical monotonicity of the scalar curvature for the alpha-geometries on the simplex of probability vectors. From the results obtained and from numerical data we are led to some conjectures about quantum alpha-geometries and Wigner-Yanase-Dyson information. Finally we show that this last conjecture implies the truth of the Petz conjecture about the monotonicity of the scalar curvature of the Bogoliubov-Kubo-Mori monotone metric.

arXiv comment20 pages, 2 .eps figures; (v2) section 2 rewritten, typos corrected

References in corpus (6)

• Covariance and Fisher information in quantum mechanics
• Wigner-Yanase information on quantum state space:the geometric approach
• The Information Geometry of the Spherical Model
• The Scalar Curvature of the Bures Metric on the Space of Density Matrices
• Monotone Riemannian metrics on density matrices with non-monotone scalar curvature
• On the monotonicity conjecture for the curvature of the Kubo-Mori metric

Cited by in corpus (6)

• Uncertainty Principle and Quantum Fisher Information - II
• Information geometric methods for complexity
• Volume of the quantum mechanical state space
• A volume inequality for quantum Fisher information and the uncertainty principle
• Tomographic Reconstruction of Quantum Metrics
• Monotonicity of the Scalar Curvature of the Quantum Exponential Family for Transverse-Field Ising Chains