In 1992, Xiao-Song Lin constructed an invariant h of knots in the 3-sphere via a signed count of the conjugacy classes of irreducible SU(2)-representations of the fundamental group of the knot exterior with trace-free meridians. Lin showed that h equals one-half times the knot signature. Using methods similar to Lin's, we construct an invariant of two-component links in the 3-sphere. Our invariant is a signed count of conjugacy classes of projective SU(2)-representations of the fundamental group of the link exterior with a fixed 2-cocycle and corresponding non-trivial second Stiefel--Whitney class. We show that our invariant is, up to a sign, the linking number. We further construct, for a two-component link in an integral homology sphere, an instanton Floer homology whose Euler characteristic is, up to sign, the linking number between the components of the link. We relate this Floer homology to the Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for two-component links in the 3-sphere, the Floer homology does not vanish unless the link is split.

In this thesis, we study the Fisher's zeros in lattice gauge theory. The analysis of singularities in the complex coupling plane is an important tool to understand the critical phenomena of statistical models. The Fisher's zero structure characterizes the scaling properties of the underlying models and has a strong influence on the complex renormalization group transformation flows in the region away from both the strong and weak coupling regimes. By reconstructing the density of states, we try to develop a systematical method to investigate these singularities and we apply the method to SU(2) and U(1) lattice gauge models with a Wilson action in the fundamental representation. We first take the perturbative approach. By using the saddle point approximation, we construct the series expansions of the density of states in both of the strong and weak regimes from the strong and weak coupling expansions of the free energy density. We analyze the SU(2) and U(1) models. The expansions in the strong and weak regimes for the two models indicate both possess finite radii of convergence, suggesting the existence of complex singularities. We then perform the numerical calculations. We use Monte Carlo simulations to construct the numerical density of states of the SU(2) and U(1) models. We also discuss the convergence of the Ferrenberg-Swendsen's method which we use for the SU(2) model and propose a practical method to find the initial values that improve the convergence of the iterations. The strong and weak series expansions are in good agreement with the numerical results in their respective limits. The numerical calculations also enable the discussion of the finite volume effects which are important to the weak expansion. We calculate the Fisher's zeros of the SU(2) and U(1) models at various volumes using the numerical entropy density functions. We compare different methods of locating the zeros. By the assumption of validity of the saddle point approximation, we find that the roots of the second derivative of the entropy density function have an interesting relation with the actual zeros and may possibly reveal the scaling property of the zeros. Using the analytic approximation of the numerical density of states, we are able to locate the Fisher's zeros of the SU(2) and U(1) models. The zeros of the SU(2) stabilize at a distance from the real axis, which is compatible with the scenario that a crossover instead of a phase transition is expected in the infinite volume limit. In contrast, with the precise determination of the locations of Fisher's zeros for the U(1) model at smaller lattice sizes L=4, 6 and 8, we show that the imaginary parts of the zeros decrease with a power law of L-3.07 and pinch the real axis at β= 1.01134, which agrees with results using other methods. Preliminary results at larger volumes indicate a first-order transition in the infinite volume limit.

Universiẗat, Diss., 1997--Gießen.

Doctor of Philosophy / Electrical and Computer Engineering / Sanjoy Das / Quantum circuit synthesis is one of the major areas of current research in the field of quantum computing. Analogous to its Boolean counterpart, the task involves constructing arbitrary quantum gates using only those available within a small set of universal gates that can be realized physically. However, unlike the latter, there are an infinite number of single qubit quantum gates, all of which constitute the special unitary group SU(2). Realizing any given single qubit gate using a given universal gate family is a complex task. Although gates can be synthesized to arbitrary degree of precision as long as the set of finite strings of the gate family is a dense subset of SU(2), it is desirable to accomplish the highest level of precision using only the minimum number of universal gates within the string approximation. Almost all algorithms that have been proposed for this purpose are based on the Solovay-Kitaev algorithm. The crux of the Solovay-Kitaev algorithm is the use of a procedure to decompose a given quantum gate into a pair of group commutators with the pair being synthesized separately. The Solovay-Kitaev algorithm involves group commutator decomposition in a recursive manner, with a direct approximation of a gate into a string of universal gates being performed only at the last level, i.e. in the leaf nodes of the search tree representing the execution of the Solovay-Kitaev algorithm. The main contribution of this research is in integrating conventional optimization procedures within the Solovay-Kitaev algorithm. Two specific directions of research have been studied. Firstly, optimization is incorporated within the group commutator decomposition, so that a more optimal pair of group commutators are obtained. As the degree of precision of the synthesized gate is explicitly minimized by means of this optimization procedure, the enhanced algorithm allows for more accurate quantum gates to be synthesized than what the original Solovay-Kitaev algorithm achieves. Simulation results with random gates indicate that the obtained accuracy is an order of magnitude better than before. Two versions of the new algorithm are examined, with the optimization in the first version being invoked only at the bottom level of Solovay-Kitaev algorithm and when carried out across all levels of the search tree in the next. Extensive simulations show that the second version yields better results despite equivalent computation times. Theoretical analysis of the proposed algorithm is able to provide a more formal, quantitative explanation underlying the experimentally observed phenomena. The other direction of investigation of this research involves formulating the group commutator decomposition in the form of bi-criteria optimization. This phase of research relaxed the equality constraint in the previous approach and with relaxation, a bi-criteria optimization is proposed. This optimization algorithm is new and has been devised primarily when the objective needs to be relaxed in different stages. This bi-criteria approach is able to provide comparably accurate synthesis as the previous approach.

No description available.

Os fundamentos para os métodos algébricos unidimensionais baseados na álgebra de momentum angular su(2)(clássica e deformada) são apresentados numa forma adequada à descrição de vibrações moleculares de sistemas diatômicos e de sistemas poliatômicos planares com simetria C_{2v}. Aplicações detalhadas envolvendo os estados vibracionais e seus respectivos potenciais foram exaustivamente desenvolvidas para os dois primeiros estados eletrônicos (A-X Sigma) dos sistemas diatômicos LiH (e seus isotopômeros), NaH, KH, RbH, CsH, e Li2. Alguns outros estados eletrônicos mais excitados também foram analisados algebricamente. Os potenciais vibracionais semi-clássicos foram calculados via estados coerentes do grupo SU(2). Calculamos também fatores de Franck-Condon usando as autofun{ções de Morse diretamente, sem qualquer necessidade de linearizações. O presente trabalho é a análise algébrica mais detalhada até o momento sobre moléculas diatômicas, pois envolve um grande número de moléculas distintas e seus estados eletrônicos excitados, os quais ainda não haviam sido analisados algebricamente. Além disto, outras propriedades físicas, como os fatores de Franck-Condon e potenciais vibracionais foram discutidos aqui em detalhes pela primeira vez. O propósito deste trabalho foi de servir como um laboratório para futuras aplicações de modelos algébricos unidimensionais a sistemas moleculares maiores, onde suas potencialidades e, principalmente, suas limitações pudessem ser testadas de forma clara. Além disto, devemos destacar também o aumento expressivo na quantidade de trabalhos sobre moléculas diatômicas e triatômicas devido à importância destas em questões de interesse mundial, como o efeito estufa, e questões de relevância acadêmica, como a espectroscopia em tempo real e de moléculas frias.

Pour un n\oe ud $K$ dans $S^3$, on construit dans l'esprit de Casson -- et plus précisément en s'inspirant des travaux ultérieurs de Lin (cf. J. Differential Geom. 35 (1992) 337-357) et Heusener (cf. Topology Appl. 127 (2003) 175-197) -- une forme volume sur l'espace des représentations du groupe $G_K$ du n\oe ud $K$ dans $SU(2)$. Plus exactement, si $\mathrm(Reg)(K)$ désigne l'ensemble des classes de conjugaison des représentations \emph(régulières) de $G_K$ dans $SU(2)$, alors $\mathrm(Reg)(K)$ est une variété unidimensionnelle et on établit qu'elle possède aussi une $1$-forme volume naturelle. On montre ensuite comment on peut interpréter cette forme volume en termes de torsion de Reidemeister non abélienne. On termine par des exemples : le calcul explicite de la forme volume que l'on vient de construire pour les n\oe uds toriques et les n\oe uds fibrés ainsi que celui de la torsion de Reidemeister des sphères d'homologie de Brieskorn à coefficients dans la représentation adjointe. On étudie également le comportement (à signe près) de la forme volume que l'on a construite sous l'effet d'une mutation.

Os fundamentos para os métodos algébricos unidimensionais baseados na álgebra de momentum angular su(2)(clássica e deformada) são apresentados numa forma adequada à descrição de vibrações moleculares de sistemas diatômicos e de sistemas poliatômicos planares com simetria C_{2v}. Aplicações detalhadas envolvendo os estados vibracionais e seus respectivos potenciais foram exaustivamente desenvolvidas para os dois primeiros estados eletrônicos (A-X Sigma) dos sistemas diatômicos LiH (e seus isotopômeros), NaH, KH, RbH, CsH, e Li2. Alguns outros estados eletrônicos mais excitados também foram analisados algebricamente. Os potenciais vibracionais semi-clássicos foram calculados via estados coerentes do grupo SU(2). Calculamos também fatores de Franck-Condon usando as autofun{ções de Morse diretamente, sem qualquer necessidade de linearizações. O presente trabalho é a análise algébrica mais detalhada até o momento sobre moléculas diatômicas, pois envolve um grande número de moléculas distintas e seus estados eletrônicos excitados, os quais ainda não haviam sido analisados algebricamente. Além disto, outras propriedades físicas, como os fatores de Franck-Condon e potenciais vibracionais foram discutidos aqui em detalhes pela primeira vez. O propósito deste trabalho foi de servir como um laboratório para futuras aplicações de modelos algébricos unidimensionais a sistemas moleculares maiores, onde suas potencialidades e, principalmente, suas limitações pudessem ser testadas de forma clara. Além disto, devemos destacar também o aumento expressivo na quantidade de trabalhos sobre moléculas diatômicas e triatômicas devido à importância destas em questões de interesse mundial, como o efeito estufa, e questões de relevância acadêmica, como a espectroscopia em tempo real e de moléculas frias.

In this thesis, we introduce EPOSIC channels, a class of SU(2) -covariant quantum channels. For each of them, we give a Stinespring representation, a Kraus representation, its Choi matrix, a complementary channel, and its dual map. We show that these channels are the extreme points of all SU(2) -irreducibly covariant channels. As an application of these channels to the theory of quantum information, we study the minimal output entropy of EPOSIC channels, and show that a large class of these channels is a potential example of violating the well-known problem, the additivity problem. We determine the cases where their minimal output entropy is not zero, and obtain some partial results on the fulfillment of their entanglement breaking property. We find a bound of the minimal output entropy of the tensor product of two SU(2) -irreducibly covariant channels. We also get an example of a positive map that is not completely positive.

In meiner Doktorarbeit habe ich das Potential zwischen zwei statischen Quarks in der Confinement ``Phase'' des SU(2) Higgs Modells untersucht. Statische Quarks sind externe Quellen in der fundamentalen Darstellung der Eichgruppe. In reinen nicht-Abelschen Eichtheorien w\"achst das Potential zwischen einem statischen Quark und einem statischen Anti-quark (statisches Potential genannt) linear mit dem Abstand zwischen den Quarks. Dieses Verhalten des Potentials wird lineares Confinement genannt und wurde mit Gittersimula tionen bis zu grossen Abst\"anden und nahe am Kontinuumslim es beobachtet. Wenn dynamische Materiefelder vorhanden sind, wird erwartet, dass das statische Potential bei grossen Abst\"anden abflacht: Der Grund ist die Abschirmung der statischen Quarks durch Paarerzeugung von leichten Quark Anti-quark Paaren. Die Abflachung des statischen Potentials nennt man String Breaking. Der Stand der Dinge am Anfang meiner Doktorarbeit war, dass String Breaking in nicht-Abelschen Eichtheorien mit Materiefeldern noch nicht beobachtet wurde. Im Gegenteil, die Gittersimulationen von QCD mit dynamischen Fermionen zeigten (und zeigen noch) einen linearen Zuwachs des Potentials bei Abst\"anden, wo das String Breaking eigentlich erwartet wird (aus einer Sch\"atzung in der quenched Approximation der QCD). Die Confinement ``Phase'' im SU(2) Higgs Modell hat Eigenschaften, die der QCD \"ahnlich sind, insbesondere wird das String Breaking erwartet. Deswegen ist die Bestimmung des statischen Potentials im SU(2) Higgs Modell eine wichtige Untersuchung der relevanten Eigenschaften des String Breaking Ph\"anomens. Ich habe das SU(2) Higgs Modell in der Confinement ``Phase'' auf dem Gitter simuliert: Die Resultate zeigen deutlich das String Breaking. Desweiteren kann auch das erste angeregte statische Potential bestimmt werden. Der entscheidende Punkt sind die Korrelationen, die man benutzt, um das statische Potential zu bestimmen. In der reinen Eichtheorie wird das statische Potential aus den Wilson Loops bestimmt, die den ``String Zustand'' des Eichfeldes gut beschreiben. String steht hier f\"ur die Eichfeldkonfiguration, die das lineare Confinement der Quarks verursacht. In Anwesenheit von Materiefeldern erwartet man, dass bei grossen Abst\"anden das statische Pot ential durch das Potential zwischen zwei statisch- leichten Mesonen (Bindungszust\"anden eines statischen Quarks mit dem dynamischen leichten Quarkfeld) beschrieben wird. Die Methode, die ich verwendet habe, um das statische Potential zu bestimmen, basiert auf eine Mischung von ``String-'' und ``Zwei-Meson Zust\"anden''. Mit einem Variationsprinzip wird die beste lineare Kombination solcher ``Zust\"anden'' bestimmt, welche die Eigenzust\"ande des Hamiltonoperators approximiert. Dank der Bestimmung des ersten angeregten Potentials, konnte auch die Interpretation des String Breakings als Level Crossing Ph\"anomen zwischen ``String'' und ``Meson Zust\"anden'' best\"atigt werden. In dem zweiten Teil meiner Doktorarbeit habe ich die Frage des ``Kontinuumlimes'' untersucht. Das String Breaking wurde f\"ur einen speziellen Satz von Parametern beobachtet: die Frage war, wie stark diese Resultate vom gew\"ahlten Gitterabstand abh\"angig sind (Cutoff-Effekte). Diese Frage f\"uhrt unmittelbar zur Untersuchung von Linien konstanter Physik im Parameterraum des SU(2) Higgs Modells. Obwohl in m einer Arbeit noch keine definitive Methode gefunden worden ist, um diese Linien zu konstruieren, konnte ich das Skalierungsverhalten der statischen Potentiale bei Variation des Gitterabstandes um einen Faktor zwei untersuchen. Die Resultate zeigen \"uberraschend kleine Cutoff-Effekte! Die M ethode, welche ich in meiner Arbeit verwendet habe, ist auch in der QCD zu verwenden, um das String Breaking zu beobachten. / The static potential in the confinement ``phase'' of the SU(2) Higgs model is studied. In particular, the observation of the screening (called {\em string breaking}) of the static quarks by the dynamical light quarks leading to the formation of two static-light mesons was not observed before my work in non-Abelian gauge theories. The tool that I employ is lattice gauge simulation. The observable from whic h the spectrum of the Hamiltonian in presence of two static quarks can be extracted, is a matrix correlation whose elements are constructed not only from string-type states represented by Wilson loops (like in pure gauge theories). Additional matrix elements representing transitions from string-type to meson-type states and the propagation of meson-type states are taken into account. From this basis of states it is possible to extract the ground state and first excited state static potentials employing a variational method. The crossing of these two energy levels in the string breaking region is clearly visible and the inadequacy of the Wilson loops alone can be demonstrated. I also address the question of the lattice artifacts. For this purpose lines of constant physics in the confinement ``phase'' of the model have to be constructed. This problem has only partially been solved. Nevertheless it is possible to show that the static potentials have remarkable scaling properties under a variation of the lattice spacing by a factor two and are almost independent of the quartic Higgs coupling.