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arXiv:quant-ph/0205146 · Europhysics Letters (EPL)

Phases of quantum states in completely positive non-unitary evolution

J. G. Peixoto de Faria, A. F. R. de Toledo Piza, Maria Carolina Nemes

Published Jun 1, 2003• 49 citations
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Abstract

We define an operational notion of phases in interferometry for a quantum system undergoing a completely positive non-unitary evolution. This definition is based on the concepts of quantum measurement theory. The suitable generalization of the Pancharatnan connection allows us to determine the dynamical and geometrical parts of the total phase between two states linked by a completely positive map. These results reduce to the known expressions of the total, dynamical and geometrical phases for pure and mixed states evolving unitarily.

DOI
10.1209/epl/i2003-00440-4

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We define an operational notion of phases in interferometry for a quantum system undergoing a completely positive non-unitary evolution. This definition is based on the concepts of quantum measurement theory. The suitable generalization of the Pancharatnan connection allows us to determine the dynamical and geometrical parts of the total phase between two states linked by a completely positive map. These results reduce to the known expressions of the total, dynamical and geometrical phases for pure and mixed states evolving unitarily.
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.
We generalize the notion of relative phase to completely positive (CP) maps with known unitary representation, based on interferometry. Parallel transport conditions that define the geometric phase for such maps are introduced. The interference effect is embodied in a set of interference patterns defined by flipping the environment state in one of the two paths. We show for the qubit that this structure gives rise to interesting additional information about the geometry of the evolution defined by the CP map.
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.
We extend the concepts of Pancharatnam’s connection and phase difference of optical beams to cover random, stationary, partially polarized quasimonochromatic light. The geometric and dynamic structures of the phase are assessed within the framework of classical statistical polarization theory when the beam undergoes a cyclic, unitary polarization-state evolution. The existence of the two phase types is confirmed experimentally by conducting an interferometric measurement where the phase changes are found from the variations in the position of an intensity fringe pattern. The formalism also leads to a definition of the geometric phase for partially polarized beams, which is consistent with the results on mixed (qubit) quantum states.