aqora

A characterization of extended monotone metrics

Wataru Kumagai

Linear Algebra and its ApplicationsSep 23, 201013 citations

On the monotonicity conjecture for the curvature of the Kubo-Mori metric

Attila Andai

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.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.

arXiv:math-ph/0310064Oct 29, 20035 citations

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

Paolo Gibilisco, Tommaso Isola

We study the monotonicity under mixing of the scalar curvature for the α-geometries on the simplex of probability vectors. From the results obtained and from numerical data, we are led to some conjectures about quantum α-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:math-ph/0407007Journal of Mathematical PhysicsJan 14, 200520 citations

A theorem concerning the positive metric

Derek W. Robinson

Communications in Mathematical PhysicsMar 1, 196530 citations

Curvature of Gaussian quantum states

Harry J. D. Miller

Abstract The space of quantum states can be endowed with a metric structure using the second order derivatives of the relative entropy, giving rise to the so-called Kubo–Mori–Bogoliubov inner product. We explore its geometric properties on the submanifold of faithful, zero-displacement Gaussian states parameterised by their covariance matrices, deriving expressions for the geodesic equations, curvature tensors and scalar curvature. Our analysis suggests that the curvature of the manifold is strictly monotonic with respect to the von Neumann entropy, and thus can be interpreted as a measure of state uncertainty. This provides supporting evidence for the Petz conjecture in continuous variable systems.Abstract The space of quantum states can be endowed with a metric structure using the second order derivatives of the relative entropy, giving rise to the so-called Kubo–Mori–Bogoliubov inner product. We explore its geometric properties on the submanifold of faithful, zero-displacement Gaussian states parameterised by their covariance matrices, deriving expressions for the geodesic equations, curvature tensors and scalar curvature. Our analysis suggests that the curvature of the manifold is strictly monotonic with respect to the von Neumann entropy, and thus can be interpreted as a measure of state uncertainty. This provides supporting evidence for the Petz conjecture in continuous variable systems.

arXiv:2404.09600Journal of Physics A Mathematical and TheoreticalDec 19, 20245 citations

Non-monotone metric on the quantum parametric model

Jun Suzuki

The European Physical Journal PlusJan 17, 20213 citations

On the curvature of monotone metrics and a conjecture concerning the Kubo-Mori metric

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