author={Armin Uhlmann},
  journal={Reports on Mathematical Physics},
  • A. Uhlmann
  • Published 1 October 1995
  • Mathematics
  • Reports on Mathematical Physics
Abstract The parallel transport responsible for the geometric phase is reviewed emphasizing the role of transition probabilities and of the metric of Bures. 
Geometry of the Rabi Problem and Duality of Loops
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
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
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
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
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
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
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
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
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
  • D. Viennot
  • Mathematics, Physics
    Journal 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


Explicit computation of the Bures distance for density matrices
Abstract We compute the Bures metric for two-dimensional density matrices explicity and derive a general formula for the n -dimensional case.
A gauge field governing parallel transport along mixed states
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
Comment on "Observation of Berry's topological phase by use of an optical fiber"
A Comment on the Letter by Akira Tomita and Raymond Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
On Cantoni's generalized transition probability
We obtain simple expressions of the “generalized transition probability” proposed by V. Cantoni, for both classical and quantum mechanics. We compare the result with the ordinary quantum mechanical
Appearance of Gauge Structure in Simple Dynamical Systems
Generalizing a construction of Berry and Simon, we show that non-Abelian gauge fields arise in the adiabatic development of simple quantum mechanical systems. Characteristics of the gauge fields are
A class of connections governing parallel transport along density matrices
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
A note on the transition probability over C*-algebras
The algebraic structure of Uhlmann's transition probability between mixed states on unital C*-algebras (see [2]) is analyzed. Several improvements of methods to calculate the transition probability
Statistical distance and the geometry of quantum states.
By finding measurements that optimally resolve neighboring quantum states, we use statistical distinguishability to define a natural Riemannian metric on the space of quantum-mechanical density
On a connection governing parallel transport along 2 × 2 density matrices
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.
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