On Coherent States and q-Deformed Algebras

We review some aspects of the relation between ordinary coherent states and q-deformed generalized coherent states with some of the simplest cases of quantum Lie algebras. In particular, new properties of (q-)coherent states are utilized to provide a path integral formalism allowing to study a modified form of q-classical mechanics, to probe some geometrical consequences of the q-deformation and finally to construct Bargmann complex analytic realizations for some quantum algebras.Comment: Presented at the 'International Symposium on Coherent States' June 1993, USA 14 pages, plain LATEX, FTUV/93-37, IFIC/93-2

Quantum Optical Random Walk: Quantization Rules and Quantum Simulation of Asymptotics

Rules for quantizing the walker+coin parts of a classical random walk are provided by treating them as interacting quantum systems. A quantum optical random walk (QORW), is introduced by means of a new rule that treats quantum or classical noise affecting the coin's state, as sources of quantization. The long term asymptotic statistics of QORW walker's position that shows enhanced diffusion rates as compared to classical case, is exactly solved. A quantum optical cavity implementation of the walk provides the framework for quantum simulation of its asymptotic statistics. The simulation utilizes interacting two-level atoms and/or laser randomly pulsating fields with fluctuating parameters.Comment: 18 pages, 3 figure

Pseudo Memory Effects, Majorization and Entropy in Quantum Random Walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.Comment: 3 figures include

Quantum Walks of SU(2)_k Anyons on a Ladder

We study the effects of braiding interactions on single anyon dynamics using a quantum walk model on a quasi-1-dimensional ladder filled with stationary anyons. The model includes loss of information of the coin and nonlocal fusion degrees of freedom on every second time step, such that the entanglement between the position states and the exponentially growing auxiliary degrees of freedom is lost. The computational complexity of numerical calculations reduces drastically from the fully coherent anyonic quantum walk model, allowing for relatively long simulations for anyons which are spin-1/2 irreps of SU(2)_k Chern-Simons theory. We find that for Abelian anyons, the walk retains the ballistic spreading velocity just like particles with trivial braiding statistics. For non-Abelian anyons, the numerical results indicate that the spreading velocity is linearly dependent on the number of time steps. By approximating the Kraus generators of the time evolution map by circulant matrices, it is shown that the spatial probability distribution for the k=2 walk, corresponding to Ising model anyons, is equal to the classical unbiased random walk distribution.Comment: 12 pages, 4 figure