We investigate the quantum walk on the line when decoherences are introduced either through simultaneous measurements of the chirality and particle position, or as a result of broken links. Both mechanisms drive the system to a classical diffusive behavior. In the case of measurements, we show that the diffusion coefficient is proportional to the variance of the initially localized quantum random walker just before the first measurement. When links between neighboring sites are randomly broken with probability p per unit time, the evolution becomes decoherent after a characteristic time that scales as 1/p. The fact that the quadratic increase of the variance is eventually lost even for very small frequencies of disrupting events, suggests that the implementation of a quantum walk on a real physical system may be severely limited by thermal noise and lattice imperfections.

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.

The conditional shift in the evolution operator of a quantum walk generates entanglement between the coin and position degrees of freedom. This entanglement can be quantified by the von Neumann entropy of the reduced density operator (entropy of entanglement). In the long time limit, it converges to a well defined value which depends on the initial state. Exact expressions for the asymptotic (long-time) entanglement are obtained for (i) localized initial conditions and (ii) initial conditions in the position subspace spanned by | ± 1 .

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We consider a new model of quantum walk on a one dimensional momentum space that includes both discrete jumps and continuous drift. Its time evolution has two stages; a Markov diffusion followed by localized dynamics. As in the well known quantum kicked rotor, this model can be mapped into a localized one-dimensional Anderson model. For exceptional (rational) values of its scale parameter, the system exhibits resonant behavior and reduces to the usual discrete time quantum walk on the line.

We investigate the quantum walk and the quantum kicked rotor in resonance subjected to noise with a Lévy waiting time distribution. We find that both systems have a sub-ballistic wave function spreading as shown by a power-law tail of the standard deviation.PACS numbers: PACS: 03.67. 05.40.Fb; 05.45.Mt In the last decades the study of simple quantum systems, such as the quantum kicked rotor (QKR) [1] and the quantum walk (QW) [2], have exposed unexpected behaviors that suggest new challenges both theoretical and experimental in the field of quantum information processing [3]. The behavior of the QKR has two characteristic modalities: dynamical localization (DL) and ballistic spreading of the variance in resonance. These different behaviors depend on whether the period of the kick is a rational or irrational multiple of 4π. For rational multiples the behavior of the system is resonant and the average energy grows ballistically and for irrational multiples the average energy of the system grows, for a short time, in a diffusive manner and afterwards DL appears. Quantum resonance is a constructive interference phenomena and DL is a destructive one. The DL and the ballistic behavior have already been observed experimentally [4,5]. On the other hand the concept of QW introduced in [6,7] is a counterpart of the classical random walk. Its most striking property is its ability to spread over the line linearly in time, this means that the standard deviation grows as σ(t) ∼ t, while in the classical walk it grows as σ(t) ∼ t 1/2 . We have developed [8,9] a parallelism between the behavior of the QKR and a generalized form of the QW showing that these models have similar dynamics. In [10] we have investigated the resonances of the QKR subjected to an excitation that follows an aperiodic Fibonacci prescription; there we proved that the primary resonances retain their ballistic behavior while the secondary resonances show a sub-ballistic wave function spreading (σ(t) ∼ t c with 0.5 < c < 1) like the QW with the same prescription for the coin [11]. Casati et al. [12] have studied the dynamics of the QKR kicked according to a Fibonacci sequence outside the resonant regime, they found sub-diffusive behavior for small kicking strengths and a threshold above which the usual diffusion is recovered. More recently Schomerus and Lutz [13] investigated the QKR subjected to a Lévy noise [14] and they show that this decoherence never fully destroys the DL of the QKR but leads to a sub-diffusion regime * Corresponding author. E-mail address: alejo@fing.edu.uy for a short time before DL appears.In this article we investigate the QKR in resonant regime and the usual QW when both are subjected to decoherence with a Lévy noise. In the case of the QKR the model has two strength parameters whose action alternate in a such way that the time interval between them follows a power law distribution. In the case of QW the model uses two evolution operators whose alternation follows the same power law distribution. We show that this noise in the...

We study the resonances of the quantum kicked rotor subjected to an excitation that follows an aperiodic Fibonacci prescription. In such a case the secondary resonances show a sub-ballistic behavior like the quantum walk with the same aperiodic prescription for the coin. The principal resonances maintain the well-known ballistic behavior.

The separation of the Schrödinger 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. Key words: kicked rotor; markovian process; dynamical localizationIn the last few decades the field of Quantum Chaos has drawn the attention of researchers in several areas of science. Many interesting phenomena were studied in systems such as atom traps and microwave cavities. Furthermore, the recent advances in technology that allow to construct and almost perfectly preserve quantum states, has opened the field of quantum computation, where chaotic effects and their control play an essential role (1).In this work we develop a different and general approach to the subject of dynamical localization (DL) and the related issue of quantum diffusion. This approach leads to an improved understanding of why DL takes place in some systems while in others quantum diffusion continues for ever. The path we follow consists in rewriting the Schrödinger equation in a form in which the part responsible for DL is separated from the part responsible for quantum 1 also at: Facultad de Ciencias, Universidad de la República.