Quantum random walk on the line as a markovian process

We analyze in detail the discrete--time quantum walk on the line by separating the quantum evolution equation into Markovian and interference terms. As a result of this separation, it is possible to show analytically that the quadratic increase in the variance of the quantum walker's position with time is a direct consequence of the coherence of the quantum evolution. If the evolution is decoherent, as in the classical case, the variance is shown to increase linearly with time, as expected. Furthermore we show that this system has an evolution operator analogous to that of a resonant quantum kicked rotor. As this rotator may be described through a quantum computational algorithm, one may employ this algorithm to describe the time evolution of the quantum walker.Comment: few typos corrected, 13 pages, 2 figures, to appear in Physica

Markovian Behaviour and Constrained Maximization of the Entropy in Chaotic Quantum Systems

The separation of the Schr\"{o}dinger equation into a Markovian and an interference term provides a new insight in the quantum dynamics of classically chaotic systems. The competition between these two terms determines the localized or diffusive character of the dynamics. In the case of the Kicked Rotor, we show how the constrained maximization of the entropy implies exponential localization.Comment: 8 pages, 2 figure