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}
}
  • 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. 
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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.
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  • 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
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References

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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).
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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
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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
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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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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
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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.
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
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