<p>Quantum phenomena related to geometric and topological phases are investigated. The first results presented are theoretical extensions of these phases and related effects. Also experimental proposals to measure some of the described effects are outlined. Thereafter, applications of geometric and topological phases in quantum computation are discussed.</p><p>The notion of geometric phases is extended to cover mixed states undergoing unitary evolutions in interferometry. A comparison with a previously proposed definition of a mixed state geometric phase is made. In addition, an experimental test distinguishing these two phase concepts is proposed. Furthermore, an interferometry based geometric phase is introduced for systems undergoing evolutions described by completely positive maps.</p><p>The dynamics of an Aharonov-Bohm system is investigated within the adiabatic approximation. It is shown that the time-reversal symmetry for a semi-fluxon, a particle with an associated magnetic flux which carries half a flux unit, is unexpectedly broken due to the Aharonov-Casher modification in the adiabatic approximation.</p><p>The Aharonov-Casher Hamiltonian is used to determine the energy quantisation of neutral magnetic dipoles in electric fields. It is shown that for specific electric field configurations, one may acquire energy quantisation similar to the Landau effect for a charged particle in a homogeneous magnetic field.</p><p>We furthermore show how the geometric phase can be used to implement fault tolerant quantum computations. Such computations are robust to area preserving perturbations from the environment. Topological fault-tolerant quantum computations based on the Aharonov-Casher set up are also investigated.</p>

Quantum phenomena related to geometric and topological phases are investigated. The first results presented are theoretical extensions of these phases and related effects. Also experimental proposals to measure some of the described effects are outlined. Thereafter, applications of geometric and topological phases in quantum computation are discussed. The notion of geometric phases is extended to cover mixed states undergoing unitary evolutions in interferometry. A comparison with a previously proposed definition of a mixed state geometric phase is made. In addition, an experimental test distinguishing these two phase concepts is proposed. Furthermore, an interferometry based geometric phase is introduced for systems undergoing evolutions described by completely positive maps. The dynamics of an Aharonov-Bohm system is investigated within the adiabatic approximation. It is shown that the time-reversal symmetry for a semi-fluxon, a particle with an associated magnetic flux which carries half a flux unit, is unexpectedly broken due to the Aharonov-Casher modification in the adiabatic approximation. The Aharonov-Casher Hamiltonian is used to determine the energy quantisation of neutral magnetic dipoles in electric fields. It is shown that for specific electric field configurations, one may acquire energy quantisation similar to the Landau effect for a charged particle in a homogeneous magnetic field. We furthermore show how the geometric phase can be used to implement fault tolerant quantum computations. Such computations are robust to area preserving perturbations from the environment. Topological fault-tolerant quantum computations based on the Aharonov-Casher set up are also investigated.

Descrevemos propriedades espectrais de aplicações positivas em C*- álgebras de dimensão finita, seguindo o trabalho clássico de Evans e Hoegh-Krohn [EH-K]. Conjuntamente, estudamos os pontos extremais do conjunto das aplicações duplamente estocásticas completamente positivas sobre Mn(C), seguindo Landau e Streater [LS]. / We describe spectral properties of positive maps over nite dimensional C* -algebras, following the classical work of Evans and H egh-Krohn [EH-K]. We also study the extremal points of the set of completely positive doubly-stochastic maps over Mn(C), following Landau and Streater [LS].

Descrevemos propriedades espectrais de aplicações positivas em C*- álgebras de dimensão finita, seguindo o trabalho clássico de Evans e Hoegh-Krohn [EH-K]. Conjuntamente, estudamos os pontos extremais do conjunto das aplicações duplamente estocásticas completamente positivas sobre Mn(C), seguindo Landau e Streater [LS]. / We describe spectral properties of positive maps over nite dimensional C* -algebras, following the classical work of Evans and H egh-Krohn [EH-K]. We also study the extremal points of the set of completely positive doubly-stochastic maps over Mn(C), following Landau and Streater [LS].

Descrevemos propriedades espectrais de aplicações positivas em C*- álgebras de dimensão finita, seguindo o trabalho clássico de Evans e Hoegh-Krohn [EH-K]. Conjuntamente, estudamos os pontos extremais do conjunto das aplicações duplamente estocásticas completamente positivas sobre Mn(C), seguindo Landau e Streater [LS]. / We describe spectral properties of positive maps over nite dimensional C* -algebras, following the classical work of Evans and H egh-Krohn [EH-K]. We also study the extremal points of the set of completely positive doubly-stochastic maps over Mn(C), following Landau and Streater [LS].

<p>The effects of open system evolution on single particle interferometry, quantum computation, and the adiabatic approximation are investigated.</p><p>Single particle interferometry: Three concepts concerning completely positive maps (CPMs) and trace preserving CPMs (channels), named subspace preserving (SP) CPMs, subspace local channels, and gluing of CPMs, are introduced. SP channels preserve probability weights on given orthogonal sum decompositions of the Hilbert space of a quantum system. Subspace locality determines what channels act locally with respect to such decompositions. Gluings are the possible total channels obtainable if two evolution devices, characterized by channels, act jointly on a superposition of a particle in their inputs. It is shown that gluings are not uniquely determined by the two channels. We determine all possible interference patterns in single particle interferometry for given channels acting in the interferometer paths. It is shown that the standard interferometric setup cannot distinguish all gluings, but a generalized setup can.</p><p>Quantum computing: The robustness of local and global adiabatic quantum search subject to decoherence in the instantaneous eigenbasis of the search Hamiltonian, is examined. In both the global and local search case the asymptotic time-complexity of the ideal closed case is preserved, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, it is shown that the local adiabatic quantum search performs as the classical search with scaling N, and that the global search scales like N^3/2 , where N is the list length. We consider success probabilities p<1 and prove bounds on the run-time with the same scaling as in the conditions for the p → 1 limit.</p><p>Adiabatic evolution: We generalize the adiabatic approximation to the case of open quantum systems in the joint limit of slow change and weak open system disturbances. </p>

The effects of open system evolution on single particle interferometry, quantum computation, and the adiabatic approximation are investigated. Single particle interferometry: Three concepts concerning completely positive maps (CPMs) and trace preserving CPMs (channels), named subspace preserving (SP) CPMs, subspace local channels, and gluing of CPMs, are introduced. SP channels preserve probability weights on given orthogonal sum decompositions of the Hilbert space of a quantum system. Subspace locality determines what channels act locally with respect to such decompositions. Gluings are the possible total channels obtainable if two evolution devices, characterized by channels, act jointly on a superposition of a particle in their inputs. It is shown that gluings are not uniquely determined by the two channels. We determine all possible interference patterns in single particle interferometry for given channels acting in the interferometer paths. It is shown that the standard interferometric setup cannot distinguish all gluings, but a generalized setup can. Quantum computing: The robustness of local and global adiabatic quantum search subject to decoherence in the instantaneous eigenbasis of the search Hamiltonian, is examined. In both the global and local search case the asymptotic time-complexity of the ideal closed case is preserved, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, it is shown that the local adiabatic quantum search performs as the classical search with scaling N, and that the global search scales like N3/2 , where N is the list length. We consider success probabilities p&lt;1 and prove bounds on the run-time with the same scaling as in the conditions for the p → 1 limit. Adiabatic evolution: We generalize the adiabatic approximation to the case of open quantum systems in the joint limit of slow change and weak open system disturbances.

Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and they have recently been extended to the setting of free noncommutative function theory. In this paper, we develop the subject further in a number of directions. We give a characterization of completely positive noncommutative kernels in the setting of Hilbert C*-modules and Hilbert W*-modules. We prove an Arveson-type extension theorem for completely positive noncommutative kernels, and we show that a uniformly bounded noncommutative kernel can be decomposed into a linear combination of completely positive noncommutative kernels. / Ph. D.