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Chinese Physics Letters

Geometric Phase for Mixed States

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

Published May 16, 2003• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/20/6/304

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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.
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.
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.
Mixed states typically arise when quantum systems interact with the outside world. Evolution of open quantum systems in general are described by quantum operations which are represented by completely positive maps. We elucidate the notion of geometric phase for a quantum system described by a mixed state undergoing unitary evolution and non-unitary evolutions. We discuss parallel transport condition for mixed states both in the case of unitary maps and completely positive maps. We find that the relative phase shift of a system not only depends on the state of the system, but also depends on the initial state of the ancilla with which it might have interacted in the past. The geometric phase change during a sequence of quantum operations is shown to be non-additive in nature. This property can attribute a "memory" to a quantum channel. We explore these ideas and illustrate them with simple examples.
In the paper a problem of geometric phase decomposition for general evolutions in the Hilbert space is addressed. The decomposition of total phase into dynamical and geometrical parts employs a state representation in the noninteraction picture. The noninteraction picture is introduced with the help of decomposition of the system evolution operator into two parts, the one pertaining to free evolution and the other to interaction. The procedure requires the problem to be an exactly solvable case of the Schrödinger equation. The most common class of such problems includes dipole Hamiltonians, for which the evolution operator can be decomposed into a combination of unitary operators. Geometric phase decomposition in the noninteraction picture can be applied to general noncyclic evolutions, but for the cyclic states it reduces to the Floquet decomposition. Defined this way, geometric phase possesses characteristic features of geometric phase for the free system, but extends some stationary properties to temporal dependences. As a case, the time-dependent relationship between geometric phase and the nonstationarity of a quantum state is illustrated by an example of a spin-½ in a rotating magnetic field. It is shown that geometric phase reaches a maximum when the spin state becomes completely nonstationary.