© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:quant-ph/0304068 · Physical Review A

Geometric phases for nondegenerate and degenerate mixed states

Kuldip Singh, D. M. Tong, Kasturi Basu, J. L. Chen, Jiangfeng Du

Published Mar 18, 2003• 128 citations

Abstract

This paper focuses on the geometric phase of general mixed states under unitary evolution. Here we analyze both nondegenerate as well as degenerate states. Starting with the nondegenerate case, we show that the usual procedure of subtracting the dynamical phase from the total phase to yield the geometric phase for pure states, does not hold for mixed states. To this end, we furnish an expression for the geometric phase that is gauge invariant. The parallelity conditions are shown to be easily derivable from this expression. We also extend our formalism to states that exhibit degeneracies. Here with the holonomy taking on a non-Abelian character, we provide an expression for the geometric phase that is manifestly gauge invariant. As in the case of the nondegenerate case, the form also displays the parallelity conditions clearly. Finally, we furnish explicit examples of the geometric phases for both the nondegenerate as well as degenerate mixed states.

DOI: 10.1103/physreva.67.032106

Related papers

Geometric Phase for Mixed States

Tong Dian-Min, Jing-Ling Chen, Du Jiang-Feng

The geometric phase of mixed states with non-degenerate eigenvalues is investigated. A general formula of geometric phase for mixed state under unitary evolution is given. In particular, we also furnish an expression of Hamiltonians for equivalent evolutions, by which one can understand what kind of evolutional operator U(t) (or Hamiltonian) is related to zero instantaneous dynamic phase. Moreover, the geometric phase and related Hamiltonians in the spin-half case are provided as an explicit example.

Chinese Physics Letters•May 16, 2003•5 citations

Geometric phases for mixed states during cyclic evolutions

Fu, L. B., Chen, J. L.

The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.

•Jan 1, 2004•4 citations

Geometric phases for mixed states during cyclic evolutions

Li-Bin Fu, Jing-Ling Chen

The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.

arXiv:quant-ph/0403070•Journal of Physics A Mathematical and General•Mar 2, 2004•4 citations

A general treatment of geometric phases and dynamical invariants

Eduardo I. Duzzioni, R. M. Serra, M. H. Y. Moussa

Based only on the parallel-transport condition, we present a general method to compute Abelian or non-Abelian geometric phases acquired by the basis states of pure or mixed density operators, which also holds for nonadiabatic and noncyclic evolution. Two interesting features of the non-Abelian geometric phase obtained by our method stand out: i) it is a generalization of Wilczek and Zee's non-Abelian holonomy, in that it describes nonadiabatic evolution where the basis states are parallelly transported between distinct degenerate subspaces, and ii) the non-Abelian character of our geometric phase relies on the transitional evolution of the basis states, even in the nondegenerate case. We apply our formalism to a two-level system evolving nonadiabatically under spontaneous decay to emphasize the non-Abelian nature of the geometric phase induced by the reservoir. We also show, through the generalized invariant theory, that our general approach encompasses previous results in the literature.

arXiv:0706.2448•Europhysics Letters (EPL)•Apr 1, 2008•12 citations

Spectral weighted geometric phase for mixed quantum states

Ole Andersson, Hoshang Heydari

Geometric phase has found a broad spectrum of applications in both classical and quantum physics. In this work we discuss a geometric phase for mixed quantum states based on traces of spectral weighted holonomies. Our approach applies to general unitary evolutions of both nondegenerate and degenerate mixed states, and it generalizes the standard definition of geometric phase for mixed states, which is based on quantum interferometry. We provide an explicit formula for the geometric phase that can be easily implemented for computations in quantum physics, and we discuss higher order analogs of the geometric phase that might be defined at points where the ordinary geometric phase is undefined.

Physica Scripta•Dec 1, 2014•1 citation

Operational geometric phase for mixed quantum states

O Andersson, Hoshang Heydari

The geometric phase has found a broad spectrum of applications in both classical and quantum physics, such as condensed matter and quantum computation. In this paper, we introduce an operational geometric phase for mixed quantum states, based on spectral weighted traces of holonomies, and we prove that it generalizes the standard definition of the geometric phase for mixed states, which is based on quantum interferometry. We also introduce higher order geometric phases, and prove that under a fairly weak, generically satisfied, requirement, there is always a well-defined geometric phase of some order. Our approach applies to general unitary evolutions of both non-degenerate and degenerate mixed states. Moreover, since we provide an explicit formula for the geometric phase that can be easily implemented, it is particularly well suited for computations in quantum physics.

arXiv:1302.1838•New Journal of Physics•May 8, 2013•18 citations

This paper focuses on the geometric phase of general mixed states under unitary evolution. Here we analyze both nondegenerate as well as degenerate states. Starting with the nondegenerate case, we show that the usual procedure of subtracting the dynamical phase from the total phase to yield the geometric phase for pure states, does not hold for mixed states. To this end, we furnish an expression for the geometric phase that is gauge invariant. The parallelity conditions are shown to be easily derivable from this expression. We also extend our formalism to states that exhibit degeneracies. Here with the holonomy taking on a non-Abelian character, we provide an expression for the geometric phase that is manifestly gauge invariant. As in the case of the nondegenerate case, the form also displays the parallelity conditions clearly. Finally, we furnish explicit examples of the geometric phases for both the nondegenerate as well as degenerate mixed states.

The geometric phase of mixed states with non-degenerate eigenvalues is investigated. A general formula of geometric phase for mixed state under unitary evolution is given. In particular, we also furnish an expression of Hamiltonians for equivalent evolutions, by which one can understand what kind of evolutional operator U(t) (or Hamiltonian) is related to zero instantaneous dynamic phase. Moreover, the geometric phase and related Hamiltonians in the spin-half case are provided as an explicit example.

The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.

The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.

Based only on the parallel-transport condition, we present a general method to compute Abelian or non-Abelian geometric phases acquired by the basis states of pure or mixed density operators, which also holds for nonadiabatic and noncyclic evolution. Two interesting features of the non-Abelian geometric phase obtained by our method stand out: i) it is a generalization of Wilczek and Zee's non-Abelian holonomy, in that it describes nonadiabatic evolution where the basis states are parallelly transported between distinct degenerate subspaces, and ii) the non-Abelian character of our geometric phase relies on the transitional evolution of the basis states, even in the nondegenerate case. We apply our formalism to a two-level system evolving nonadiabatically under spontaneous decay to emphasize the non-Abelian nature of the geometric phase induced by the reservoir. We also show, through the generalized invariant theory, that our general approach encompasses previous results in the literature.

Geometric phase has found a broad spectrum of applications in both classical and quantum physics. In this work we discuss a geometric phase for mixed quantum states based on traces of spectral weighted holonomies. Our approach applies to general unitary evolutions of both nondegenerate and degenerate mixed states, and it generalizes the standard definition of geometric phase for mixed states, which is based on quantum interferometry. We provide an explicit formula for the geometric phase that can be easily implemented for computations in quantum physics, and we discuss higher order analogs of the geometric phase that might be defined at points where the ordinary geometric phase is undefined.

The geometric phase has found a broad spectrum of applications in both classical and quantum physics, such as condensed matter and quantum computation. In this paper, we introduce an operational geometric phase for mixed quantum states, based on spectral weighted traces of holonomies, and we prove that it generalizes the standard definition of the geometric phase for mixed states, which is based on quantum interferometry. We also introduce higher order geometric phases, and prove that under a fairly weak, generically satisfied, requirement, there is always a well-defined geometric phase of some order. Our approach applies to general unitary evolutions of both non-degenerate and degenerate mixed states. Moreover, since we provide an explicit formula for the geometric phase that can be easily implemented, it is particularly well suited for computations in quantum physics.