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arXiv:quant-ph/0405052 · Physical Review Letters

Geometric Phase Distributions for Open Quantum Systems

Karl-Peter Marzlin, Shohini Ghose, Barry C. Sanders

Published Dec 20, 2004• 65 citations
arXivHTMLPDF

Abstract

In an open system, the geometric phase should be described by a distribution. We show that a geometric phase distribution for open system dynamics is in general ambiguous, but the imposition of reasonable physical constraints on the environment and its coupling with the system yields a unique geometric phase distribution that applies even for mixed states, nonunitary dynamics, and noncyclic evolutions.

DOI
10.1103/physrevlett.93.260402

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In an open system, the geometric phase should be described by a distribution. We show that a geometric phase distribution for open system dynamics is in general ambiguous, but the imposition of reasonable physical constraints on the environment and its coupling with the system yields a unique geometric phase distribution that applies even for mixed states, nonunitary dynamics, and noncyclic evolutions.
The geometric phase is of fundamental interest and plays an important role in quantum information processing. However, the definition and calculation of this phase for open systems remain problematic due to the lack of agreement on generalizations of the parallel transport condition to mixed-state nonunity evolutions. Here we tackle this problem by associating the open-system geometric phase with the parallel transport of the joint system-reservoir state and find that the associated geometric phase can be calculated when the total Hamiltonian is time independent. Our approach not only provides a way around the nonunitary evolution obstacle, but also sheds light on the relation between the geometric phase and the system-reservoir entanglement. Based on this approach, we calculate the geometric phase of different quantum systems subject to energy decay, showing that it is robust against decoherence, which is in distinct contrast with previous results.
We calculate the geometric phase for an open system (spin-boson model) which interacts with an environment (ohmic or nonohmic) at arbitrary temperature. However there have been many assumptions about the time scale at which the geometric phase can be measured, there has been no reported observation of geometric phases for mixed states under nonunitary evolution yet. We study not only how they are corrected by the presence of the different type of environments but estimate the corresponding times at which decoherence becomes effective as well. These estimations should be taken into account when planning experimental setups to study the geometric phase in the nonunitary regime, particularly important for the application of fault-tolerant quantum computation.
The problem of a geometric phase for an open quantum system is reinvestigated in a unifying approach. Two of the existing methods to define geometric phase, one by Uhlmann's approach and the other by a kinematic approach, which have been considered to be distinct, are shown to be related in this framework. The method is based upon purification of a density matrix by its uniform decomposition and a generalization of the parallel transport condition obtained from this decomposition. It is shown that the generalized parallel transport condition can be satisfied when Uhlmann's condition holds. However, it does not mean that all solutions of the generalized parallel transport condition are compatible with those of Uhlmann's. It is also shown how to recover the earlier known definitions of geometric phase as well as how to generalize them when degeneracy exists and varies in time.
Various types of unravelings of Lindblad master equation have been used to define the geometric phase for an open quantum system. Approaches of this type were criticized for lacking in unitary symmetry of the Lindblad equation [A. Bassi and E. Ippoliti, Phys. Rev. A 73, 062104 (2006)]. We utilize quantum state diffusion (QSD) approach to demonstrate that a geometric phase invariant on the symmetries of the Lindblad equation can be defined. It is then shown that such a definition of the geometric phase could be either invariant on the decomposition of the initial mixed state or gauge invariant, but not both. This alternative is inherent to the definitions based on quantum trajectories. The QSD geometric phase is computed for a qubit in different types of environments.