Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian.

We study the phenomenon of decoherence during the operation of one qubit transformation, controlled-not (CNOT) and controlled-controlled-not (C 2 NOT) quantum gates in a quantum computer model formed by a linear chain of three nuclear spins system. We make this study with different type of environments, and we determine the associated decoherence time as a function of the dissipative parameter. We found that the dissipation parameter to get a well defined quantum gates (without significant decoherence) must be within the range of . We also study the behavior of the purity parameter for these gates and different environments and found linear or quadratic decays of this parameter depending on the type of environments.

We use the phase space position-velocity (x, v) to deal with the statistical properties of velocity dependent dynamical systems, like dissipative ones. Within this approach, we study the statistical properties of an ensemble of harmonic oscillators in a linear weak dissipative media. Using the Debye model of a crystal, we calculate at first order in the dissipative parameter the entropy, free energy, internal energy, equation of state and specific heat using the classical and quantum approaches. For the classical approach we found that the entropy, the equation of state, and the free energy depend on the dissipative parameter, but the internal energy and specific heat do not depend of it. For the quantum case, we found that all the thermodynamical quantities depend on this parameter.

We make a numerical study of decoherence on the teleportation algorithm implemented in a linear chain of three nuclear spins system. We study different types of environments, and we determine the associated decoherence time as a function of the dissipative parameter. We found that the dissipation parameter to get a well defined quantum gates (without significant decoherence) must be within the range of γ≤4×10^-4 for not thermalized case, which was determined by using the purity parameter calculated at the end of the algorithm. For the thermalized case the decoherence is stablished for very small dissipation parameter, making almost not possible to implement this algorithm for not zero temperature.

We propose a method, based on the search and identification of complete subgraphs of a regular graph, to obtain sets of Pauli operators whose eigenstates form unextendible complete sets of mutually unbiased bases of n-qubit systems. With this method we can obtain results for complete and inextensible sets of mubs for 2, 3, 4 and 5 qubits.

We use the characteristic function of the Wigner function (its double Fourier transform) to give solution to any generic open quantum linear systems (systems whose Hamiltonian is at most quadratic). The solution is carried out in terms of the application of the transition matrix of the dynamical evolution in the Fourier space. We address two cases: the time-independent coefficients for which we give the solutions for several dissipative models of the quantum harmonic oscillator and the one-dimensional free particle. In the latter, we also derive a heuristic model for a pure damped motion with suppression of diffusion. For the time-dependent coefficient problem, we give some particular cases that are integrable and derive a second order approximation to the generic case in which all parameters are time-dependent. We additionally explore the solutions of the system when the diffusion processes of the dissipative model lie beyond the weak coupling limit.