Asymptotic velocity of a position-dependent quantum walk

Alexandre C. Orthey, Edgard P. M. Amorim

We investigate the relation between transport properties and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following $\ensuremath{\delta}$-like (local) and Gaussian distributions. We find out that the knowledge of the limit velocity of the walker together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while this velocity with the quantum coin phases give it for highly delocalized states.

Toshikazu Sunada, Tatsuya Tate

L. C. Kwek, Setiawan

Quantum walks are interesting models with potential applications to quantum algorithms and physical processes such as photosynthesis. In this paper, we study two models of one-dimensional quantum walks, namely, quantum walks with a moving absorbing wall and quantum walks with one stationary and one moving absorbing wall. For the former, we calculate numerically the survival probability, the rate of change of average position, and the rate of change of standard deviation of the particle's position in the long time limit for different wall velocities. Moreover, we also study the asymptotic behavior and the dependence of the survival probability on the initial particle's state. While for the latter, we compute the absorption probability of the right stationary wall for different velocities and initial positions of the left wall boundary. The results for these two models are compared with those obtained for the classical model. The difference between the results obtained for the quantum and classical models can be attributed to the difference in the probability distributions.

Gustavo Segundo, Andrés Vallejo, R. Donangelo +1 more

Diego de Falco, Dario Tamascelli, Centro Interdipartimentale +1 more

Houssam Abdul‐Rahman, Thomas A. Jackson, Yousef Salah

We prove general sufficient conditions for zero velocity in position dependent one-dimensional quantum walks, and hence for the absence of ballistic transport. Our starting point is a general a priori upper bound on the velocity, formulated in terms of sparse bi-infinite sequences of sites, their gap structure, and the corresponding local coin parameters. This estimate yields several deterministic criteria for zero velocity that depend only on the behavior of suitable coin entries along selected subsequences and are independent of the values of the coins elsewhere. We also discuss the random case as an application of the general approach. All of our results remain valid in the CMV setting.