Geometric phases and related structures
type: Item
instance of: scholarly article
title: Geometric phases and related structures
zbMATH Open document ID: 0891.46051
DOI: 10.1016/0034-4877(96)83640-8
author: Armin Uhlmann
published in: Reports on Mathematical Physics
publication date: 1998-07-20T00:00:00Z
review text: This expository lecture explains the occurrence of the Berry phase within a geometrical setting. The parallel transport of amplitudes and phases are defined in the framework of the state space equipped with the metric of Bures. The behavior of the metric under stochastic mappings and its Reimann form are discussed. These considerations are explicited in the case of density operators, and of two-level systems.
Mathematics Subject Classification ID: 46N50
Mathematics Subject Classification ID: 81P05
zbMATH DE Number: 1087291
zbMATH Keywords: Berry phase
zbMATH Keywords: parallel transport of amplitudes and phases
zbMATH Keywords: metric of Bures
zbMATH Keywords: stochastic mappings
zbMATH Keywords: Reimann form
zbMATH Keywords: density operators
zbMATH Keywords: two-level systems
reviewed by: G. Loupias
MaRDI profile type: Publication
full work available at URL: 0034-4877(96)83640-8
OpenAlex ID: W2101926530
cites work: On equivalence of infinite product measures
cites work: A Radon-Nikodym theorem in 𝑊*-algebras
cites work: Q5624220
cites work: Bures distance function and a generalization o f Sakai's non-commutative Radon-Nikodym theorem
cites work: One-parameter family of Radon-Nikodym theorems for states of a von Neumann algebra
cites work: Functional calculus for sesquilinear forms and the purification map
cites work: The ''transition probability'' in the state space of a \(^*\)-algebra
cites work: Bures distance and relative entropy
cites work: Q4748635
cites work: Means of positive linear operators
cites work: Q3962015
cites work: Q3943797
cites work: A remark on transition probability
cites work: Comparison of Uhlmann's transition probability with the one induced by the natural positive cone of von Neumann algebras in standard form
cites work: On Cantoni's generalized transition probability
cites work: A note on the transition probability over \(C^*\)-algebras
cites work: Stochastic linear maps and transition probability
cites work: On the Bures Distance and the Uhlmann Transition Probability of States on a von Neumann Algebra
cites work: Quantal phase factors accompanying adiabatic changes
cites work: The Transition Probability for States of *‐Algebras
cites work: Parallel transport and ``quantum holonomy'' along density operators
cites work: An Extended Cencov Characterization of the Information Metric
cites work: Q3480859
cites work: Quantum statistical holonomy
cites work: On quantum holonomy for mixed states
cites work: A gauge field governing parallel transport along mixed states
cites work: A study on the geometry of pairs of positive linear forms, algebraic transition probability and geometrical phase over non-commutative operator algebras. II
cites work: A class of connections governing parallel transport along density matrices
cites work: On a connection governing parallel transport along \(2 \times{}2\) density matrices
cites work: Q4859205
cites work: Q4201378
cites work: Density operators as an arena for differential geometry
cites work: Q4877073
cites work: Statistical distance and the geometry of quantum states
cites work: Monotone metrics on matrix spaces
cites work: Geometries of quantum states
description: scientific article; zbMATH DE number 1087291