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].

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

- ABSTRACT - Noncommutative Choquet theory Let S be a linear subspace of a commutative C∗ -algebra C(X) that se- parates points of C(X) and contains identity. Then the closure of the Choquet boundary of the function system S is the Šilov boundary relati- ve to S. In the case of a noncommutative unital C∗ -algebra A, consider S a self-adjoint linear subspace of A that contains identity and generates A. Let us call S operator system. Then the noncommutative formulation of the stated assertion is that the intersection of all boundary representa- tions for S is the Šilov ideal for S. To that end it is sufficient to show that S has sufficiently many boundary representations. In the present work we make for the proof of that this holds for separable operator system.

<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.