We generalize the notion of relative phase to completely positive 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.Berry's [1] discovery of a geometric phase accompanying cyclic adiabatic evolution has triggered an immense interest in holonomy effects in quantum mechanics and has led to many generalizations. The restriction of adiabaticity was removed by Aharonov and Anandan [2], who pointed out that the geometric phase is due to the curvature of the projective Hilbert space. It was extended to noncyclic evolution by Samuel and Bhandari The geometric phase has shown to be useful in the context of quantum computing as a tool to achieve faulttolerance [8]. For practical implementations of geometric quantum computing, it is important to understand the behavior of the geometric phase in the presence of decoherence. For this, we generalize in this Letter the idea in [7] to completely positive (CP) maps, i.e. we define the relative (Pancharatnam) phase and introduce a notion of parallel transport with concomitant geometric phase for such maps. These generalized concepts reduces to that of [7] in the case of unitary evolutions.Let us first consider a Mach-Zehnder interferometer with a variable relative U (1) phase χ in one of the interferometer beams (the reference beam) and assume that the interfering particles carry an additional internal degree of freedom, such as spin or polarization, in a pure state |k . The other beam (the target beam) is exposed to the unitary operator U i , yielding the output interference pattern I ∝ 1 + ν cos(χ − α), which is completely determined by the complex quantityThus, by varying χ, the relative phase α and visibility ν can be distinguished experimentally. We note that α is a shift in the maximum of the interference pattern, a fact that motivated Pancharatnam [5] to define it as the relative phase between the internal states |k and U i |k of the two beams. Pancharatnam's analysis was generalized in [7] to mixed states undergoing unitary evolution as follows. Assume that the incoming particle is in a mixed internal state ρ = N k=1 w k |k k|, where N is the dimension of the internal Hilbert space. Each pure component |k of this mixture contributes an interference profile given by k|U i |k = ν k e iα k weighted by the its probability w kNoting that Tr(U i ρ) = w k k|U i |k , this can also be written asThe key result is that the interference fringes, produced by varying the phase χ, is shifted by α = arg Tr(U i ρ) and that this shift reduces to Pancharatnam's original prescription for pure states. These two facts are the central properties for α being a natural generalization of Pancharatnam's relative ...

The geometric phase for a pure quantal state undergoing an arbitrary evolution is a "memory" of the geometry of the path in the projective Hilbert space of the system. We find that Uhlmann's geometric phase for a mixed quantal state undergoing unitary evolution not only depends on the geometry of the path of the system alone but also on a constrained bi-local unitary evolution of the purified entangled state. We analyze this in general, illustrate it for the qubit case, and propose an experiment to test this effect. We also show that the mixed state geometric phase proposed recently in the context of interferometry requires uni-local transformations and is therefore essentially a property of the system alone. In a general context, the geometric phase was defined for nonunitary and non-Schrödinger [6] evolutions. Since the geometric phase for a pure state is a nonintegrable quantity and depends only on the geometry of the path traced in the projective Hilbert space, it acts as a memory of a quantum system.Another important development in this field was initiated by Uhlmann [7] (see also [8]), who introduced a notion of geometric phase for mixed quantal states. More recently, using ideas of interferometry, another definition of mixed state phase was introduced in [9] (see also [10]) and experimentally verified in [11]. A renewed interest in geometric phases for mixed states is due to its potential relevance to geometric quantum computation [12].Mixed states naturally arise when we ignore the ancilla subsystem of a composite object (system+ancilla) that is described by a pure entangled state. In this Letter we wish to consider the mixed state geometric phases in [7,9] in terms of such purifications, and to investigate whether they should be regarded as properties of the system alone or not. More precisely, we would like to address the following question: do mixed state geometric phases depend only on the evolution of the system of interest, or do they also depend on the evolution of the ancilla part with which the system is entangled? By examining, in detail, the case of mixed states undergoing local unitary evolutions, we find that the Uhlmann phase [7] indeed contains a memory of the ancilla part, while the mixed state phase proposed in [9] does not. In particular, we propose an experiment to test the Uhlmann phase using a Franson set up [13] with polarization entangled photons [14,15] that would verify this new memory effect. More importantly, we show that the phase holonomies given in [7] and in [9] are generically different.Consider first the unitary path η : t ∈ [0, τ ] → |ψ t ψ t | of normalized pure state projectors with ψ 0 |ψ τ = 0. The geometric phase associated with η is defined as(1) β is a property only of the path η as it is independent of the lift η −→η : t ∈ [0, τ ] → |ψ t . A parallel lift is defined by requiring that each ψ [(j+1)τ /N ] |ψ [jτ /N ] be real and positive (i.e. ψ|ψ = 0 when N → ∞), so that β takes the formOne may measure β in interferometry as a relative phase shift in the interfere...

Abstract. Supersymmetric solutions of supergravity theories, and consequently metrics with special holonomy, have played an important role in the development of string theory. We describe how a Lorentzian manifold is either completely reducible, and thus essentially known, or not completely reducible so that there exists a degenerate holonomy invariant lightlike subspace and consequently admits a covariantly constant or a recurrent null vector and belongs to the higher-dimensional Kundt class of spacetimes. These Kundt spacetimes (which contain the vanishing and constant curvature invariant spacetimes as special cases) are genuinely Lorentzian and have a number of interesting and unusual properties, which may lead to novel and fundamental physics.

Using the improved quantization technique to the minisuperspace approximation of loop quantum gravity, we study the evolution of black holes supported by a cosmological constant. The addition of a cosmological constant allows for classical solutions with planar, cylindrical, toroidal, and higher-genus black holes. Here we study the quantum analog of these space-times. In all scenarios studied, the singularity present in the classical counterpart is avoided in the quantized version and is replaced by a bounce, and in the late evolution, a series of less severe bounces. Interestingly, although there are differences during the evolution between the various symmetries and topologies, the evolution on the other side of the bounce asymptotes to space-times of Nariai-type, with the exception of the planar black hole analyzed here, whose T-R ¼ constant subspaces seem to continue expanding in the long-term evolution. For the other cases, Nariai-type universes are attractors in the quantum evolution, albeit with different parameters. We study here the quantum evolution of each symmetry in detail.

We show that many topological and geometrical properties of complex projective space can be understood just by looking at a suitably constructed picture. The idea is to view CP n as a set of flat tori parametrized by the positive octant of a round sphere. We pay particular attention to submanifolds of constant entanglement in CP 3 and give a few new results concerning them.