# GEOMETRIC PHASES AND RELATED STRUCTURES

@article{Uhlmann1995GEOMETRICPA, title={GEOMETRIC PHASES AND RELATED STRUCTURES}, author={Armin Uhlmann}, journal={Reports on Mathematical Physics}, year={1995}, volume={36}, pages={461-481} }

Abstract The parallel transport responsible for the geometric phase is reviewed emphasizing the role of transition probabilities and of the metric of Bures.

## 63 Citations

Geometry of the Rabi Problem and Duality of Loops

- Physics
- 2020

Abstract We investigate the motion of a classical spin processing around a periodic magnetic field using Floquet theory, as well as elementary differential geometry and considering a couple of…

Quantum information geometry and standard purification

- Mathematics
- 2002

We investigate relations between Uhlmann’s parallelism, monotone Riemannian metrics and dual affine connections on the space of density matrices.

Geometric Phases for Three State Systems

- Physics
- 1999

The adiabatic geometric phases for general three state systems are discussed. An explicit parameterization for space of states of these systems is given. The abelian and non-abelian connection…

Contraction coefficients for noisy quantum channels

- Mathematics, Physics
- 2015

Generalized relative entropy, monotone Riemannian metrics, geodesic distance, and trace distance are all known to decrease under the action of quantum channels. We give some new bounds on, and…

Geodesic distances on density matrices

- Physics, Mathematics
- 2004

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

Off-diagonal quantum holonomy along density operators

- Physics
- 2005

Uhlmann's concept of quantum holonomy for paths of density operators is generalised to the off-diagonal case providing insight into the geometry of state space when the Uhlmann holonomy is undefined.…

Matrix-valued optimal mass transportation and its applications

- Engineering
- 2013

University of Minnesota Ph.D. dissertation. November 2013. Major: Electrical Engineering. Advisor: Tryphon T. Georgiou. 1 computer file (PDF); viii, 93 pages.

Holonomy in Quantum Information Geometry

- Physics, Mathematics
- 2019

In this thesis we provide a uniform treatment of two non-adiabatic geometric phases for dynamical systems of mixed quantum states, namely those of Uhlmann and of Sjoqvist et al. We develop a holonomy…

A new kind of geometric phases in open quantum systems and higher gauge theory

- Physics, Mathematics
- 2011

A new approach is proposed, extending the concept of geometric phases to adiabatic open quantum systems described by density matrices (mixed states). This new approach is based on an analogy between…

Purification of Lindblad dynamics, geometry of mixed states and geometric phases

- Mathematics, PhysicsJournal of Geometry and Physics
- 2018

We propose a nonlinear Schrodinger equation in a Hilbert space enlarged with an ancilla such that the partial trace of its solution obeys to the Lindblad equation of an open quantum system. The…

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At first, a short account is given of some basic notations and results on parallel transport along mixed states. A new connection form (gauge field) is introduced to give a geometric meaning to the…

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A class of connections governing parallel transport along nondegenerate density matrices is discussed. These connections are given by certain analytic functions. We develop a calculus for…

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Abstract We investigate a connection governing parallel transport along mixed states recently defined by Uhlmann for the case of 2 × 2 matrices. We discuss the underlying bundle structure including…

Phase change during a cyclic quantum evolution.

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A new geometric phase factor is defined for any cyclic evolution of a quantum system. This is independent of the phase factor relating the initial and final state vectors and the Hamiltonian, for a…