No / Coding using quantum states in an angular-momentum (2j+1)-dimensional Hilbert space H is considered in this paper. A concatenated code is studied in two steps. In the first step the space HN is considered and the code is its subspace HA spanned by the direct products of N angular-momentum states with the same m. In the second step the space HAM is considered and the code is its subspace HB spanned by the direct products of M angle states with the same m. It is shown that the code introduces redundancy with respect to any transformation.

The evolution of classically chaotic quantum systems is analyzed within the formalism of Quantum Pseudo-Probability Distributions. Due to the deep connections that a quantum system shows with its classical correspondent in this representation, the Pseudo-Probability formalism appears to be a useful method of investigation in the field of "Quantum Chaos." In the first part of the thesis we generalize this formalism to quantum systems containing spin operators. It is shown that a classical-like equation of motion for the pseudo-probability distribution ρw can be constructed, dρw/dt = (L_CL + L_QGD)ρw, which is rigorously equivalent to the quantum von Neumann-Liouville equation. The operator L_CL is undistinguishable from the classical operator that generates the semiclassical equations of motion. In the case of the spin-boson system this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian the joint action of L_CL and L_QGD is illustrated. It is shown that the latter operator, L_QGD makes the spin system 'remember' its quantum nature, and competes with the irreversibility induced by the former operator. In the second part we test the idea of the enhancement of the quantum uncertainty triggered by the classical chaos by investigating the analogous effect of diffusive excitation in periodically kicked quantum systems. The classical correspondents of these quantum systems exhibit, in the chaotic region, diffusive behavior of the unperturbed energy. For the Quantum Kicked Harmonic Oscillator, in the case of quantum resonances, we provide an exact solution of the quantum evolution. This proves the existence of a deterministic drift in the energy increase over time of the system considered. More generally, this "superdiffusive" excitation of the energy is due to coherent quantum mechanical tunnelling between degenerate tori of the classical phase space. In conclusion we find that some of the quantum effects resulting from this fast increase do not have any classical counterpart, they are mainly tunnelling processes. This seems to be the first observation of an effect of this kind.

Orientador: Álvaro de Souza Dutra / Banca: Antonio Soares de Castro / Banca: Denis Dalmazi / Banca: Fabrício Augusto Barones Rangel / Banca: Edson Denis Leonel / Resumo: A busca por soluções exatas para sistemas quânticos vem despertando o interesse de muitos autores ao longo das décadas. Em particular para soluções que apresentam limite clássico. Nesta tese buscamos fazer um estudo sistemático da geração de soluções analíticas para uma classe de sistemas quânticos exatamente solúveis com massa dependente da posição. Analisamos o efeito da presença de campos magnéticos sobre alguns sistemas, discutimos o problema da ambiguidade de ordenamento quântico e apresentamos possíveis limite clássico para os sistemas em estudo. / Abstract: The search for exact solutions of quantum systems has been raising the interest of many authors along the decades. Particularly for nding solutions that present classical limit. In this thesis we make a systematic study of the generation of analytic solutions for a class of quantum exactly solvable systems with position-dependent masses. We analyze the e ect of the presence of magnetic elds on some of those systems. We discuss the problem of the ordering quantum ambiguity and present possible classical limit for the systems considered. / Doutor

Finite quantum systems with d-dimension Hilbert space, where position x and momentum p take values in Zd(the integers modulo d) are studied. An analytic representation of finite quantum systems, using Theta function is considered. The analytic function has exactly d zeros. The d paths of these zeros on the torus describe the time evolution of the systems. The calculation of these paths of zeros, is studied. The concepts of path multiplicity, and path winding number, are introduced. Special cases where two paths join together, are also considered. A periodic system which has the displacement operator to real power t, as time evolution is also studied. The Bargmann analytic representation for infinite dimension systems, with variables in R, is also studied. Mittag-Leffler function are used as examples of Bargmann function with arbitrary order of growth. The zeros of polynomial approximations of the Mittag-Leffler function are studied.

Finite quantum systems with d-dimension Hilbert space, where position x and momentum p take values in Zd(the integers modulo d) are studied. An analytic representation of finite quantum systems, using Theta function is considered. The analytic function has exactly d zeros. The d paths of these zeros on the torus describe the time evolution of the systems. The calculation of these paths of zeros, is studied. The concepts of path multiplicity, and path winding number, are introduced. Special cases where two paths join together, are also considered. A periodic system which has the displacement operator to real power t, as time evolution is also studied. The Bargmann analytic representation for infinite dimension systems, with variables in R, is also studied. Mittag-Leffler function are used as examples of Bargmann function with arbitrary order of growth. The zeros of polynomial approximations of the Mittag-Leffler function are studied. / Libyan Cultural Affairs

Yes / A quantum system with variables in Z(d) is considered. Coherent density matrices and coherent projectors of rank n are introduced, and their properties (e.g., the resolution of the identity) are discussed. Cooperative game theory and in particular the Shapley methodology, is used to renormalize coherent states, into a particular type of coherent density matrices (dressed coherent states). The Q-function of a Hermitian operator, is then renormalized into a physical analogue of the Shapley values. Both the Q-function and the Shapley values, are used to study the relocation of a Hamiltonian in phase space as the coupling constant varies, and its effect on the ground state of the system. The formalism is also generalized for any total set of states, for which we have no resolution of the identity. The dressing formalism leads to density matrices that resolve the identity, and makes them practically useful.

As quantum systems can never be isolated from their environment entirely, it is expected that the spacetime fluctuations will influence their evolution. In particular, the environmental interaction may cause the loss of quantum superpositions, or decoherence. In this thesis, we examine the effects of the quantised environmental background on a range of bosonic fields in the formalism of open quantum systems. We first quantise linearised gravity in a gauge invariant way, using Dirac's constraint quantisation. We then use the influence functional technique to obtain an exact master equation for general bosonic matter interacting with weak gravity. As an application of this, we investigate the decoherence of free scalar, electromagnetic and gravitational fields. For long-time decoherence, under the Markov approximation, the dissipative terms in the master equation vanish, leading to no decay of quantum interferences. As a short-time effect, we study the master equation for a many particle state of a free scalar field, massive or massless and relativistic or non-relativistic. We find that in this case, the particles exhibit a counterintuitive behaviour of bundling towards the same quantum state that is not shared by the single particle master equation. Such collective effects, as well as possible long-time decoherence for fields in an external potential may have important implications in setting limits for precision measurements and astronomical observations.

This thesis is divided into three chapters. In the first chapter we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems The second chapter outlines the density matrix embedding methodology in the context of electronic structure applications. We formulate analytical gradients for energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding and FCI-in-RHF embedding. The former involves solving the small embedded system at Restricted Hartree-Fock (RHF) level of theory. This serves to check the validity of the formulas by reproducing the RHF results on the full system for energies and gradients. The latter scenario employs full configuration interaction (FCI) as a high level solver for the small embedded system. Our results show that only Hellmann-Feynman terms, which involve derivatives of one- and two-electron terms in the atomic orbital basis, are required to calculate energy gradients in both cases. We applied our methodology to the problem of H 10 ring dissociation where the analytical gradients matched those obtained numerically. The gradient formulation is applicable to geometry optimization of strongly correlated molecules and solids. It can also be used in ab initio molecular dynamics where forces on nuclei are obtained from DMET energy gradients. In the final chapter we focus on the study of finite-temperature equilibrium properties of quantum systems in continuous space. We formulate an expansion of the partition function in continuous-time and use Monte Carlo to sample terms in the resulting infinite series. Such a strategy has been highly successful in quantum lattice models but has found scant application in off-lattice systems. The Monte Carlo estimate of the average energy of quantum particles in continuous space subject to simple model potentials is found to converge with low statistical error to the exact solutions even when very high perturbation order is required. We outline two ways in which the algorithm can be applied to more complex problems. First, by drawing an analogy between the formulation in continuous-time with the discretized, Trotter factorized version of standard path integral Monte Carlo (PIMC). This allows one to use the suite of standard PIMC moves to carry out the position sampling required to obtain the weight of each time configuration. Finally, we propose an alternate route by fitting the many-particle potential with multidimensional Gaussians which provides an analytical form for the position integrals.

This thesis covers open quantum systems and information transfer in the face of dissipation and disorder through numerical simulation. In Chapter 3 we present work on an open quantum system comprising a two-level system, single bosonic mode and dissipative environment; we have included the bosonic mode in the exact system treatment. This model allows us to gain an understanding of an environment's role in small energy transfer systems. We observe how the two-level system-mode coupling strength and the spectral density form characterising the environment interplay, affecting the system's coherent behaviour. We find strong coupling and a spectral density resonantly peaked on the two-level system oscillation frequency enhances the system's coherent oscillatory dynamics. Chapter 4 focusses on a physically motivated study of chain and ladder spin geometries and their use for entanglement transfer between qubits. We consider a nitrogen vacancy centre qubit implementation with nitrogen impurity spin-channels and demonstrate how matrix product operator techniques can be used in simulations of this physical system. We investigate coupling parameters and environmental decay rates with respect to transfer efficiency effects. Then, in turn, we simulate the effects of missing channel spins and disorder in the spin-spin coupling. We conclude by highlighting where our considered channel geometries outperform each other. The work in Chapter 5 is an investigation into the feasibility of routing entanglement between distant qubits in 2D spin networks. We no longer consider a physical implementation, but keep in mind the effects of dissipative environments on entanglement transfer systems. Starting with a single sending qubit-ancilla and multiple addressable receivers, we show it is possible to target a specific receiver and establish transferred entanglement between it and the sender's ancilla through eigenstate tunnelling techniques. We proceed to show that eigenstate tunnelling-mediated entanglement transfer can be achieved simultaneously from two senders across one spin network.

First-principles electronic structure theory explains properties of atoms, molecules and solids from underlying physical principles without input from empirical parameters. Time-dependent density functional theory (TDDFT) has emerged as arguably the most widely used first-principles method for describing the time-dependent quantum mechanics of many-electron systems. In this thesis, we will show how the fundamental principles of TDDFT can be extended and applied in two novel directions: The theory of open quantum systems (OQS) and quantum computation (QC). In the first part of this thesis, we prove theorems that establish the foundations of TDDFT for open quantum systems (OQS-TDDFT). OQS-TDDFT allows for a first principles description of non-equilibrium systems, in which the electronic degrees of freedom undergo relaxation and decoherence due to coupling with a thermal environment, such as a vibrational or photon bath. We then discuss properties of functionals in OQS-TDDFT and investigate how these differ from functionals in conventional TDDFT using an exactly solvable model system. Next, we formulate OQS-TDDFT in the linear-response regime, which gives access to environmentally broadened excitation spectra. Lastly, we present a hybrid approach in which TDDFT can be used to construct master equations from first-principles for describing energy transfer in condensed phase systems. In the second part of this thesis, we prove that the theorems of TDDFT can be extended to a class of qubit Hamiltonians that are universal for quantum computation. TDDFT applied to universal Hamiltonians implies that single-qubit expectation values can be used as the basic variables in quantum computation and information theory, rather than wavefunctions. This offers the possibility of simplifying computations by using the principles of TDDFT similar to how it is applied in electronic structure theory. Lastly, we discuss a related result; the computational complexity of TDDFT. / Physics