Properties of distribution and entanglement in discrete-time quantum walk with percolation

Abstract

We investigate one-dimensional discrete-time quantum walk on the line where the links between neighboring sites are randomly broken. Two link-broken ways, static percolation and dynamical percolation, are considered. The former means that the broken links are fixed in position space at each time step, while the latter is that broken links are varied with time step. Our attention focuses on the effects of these disorders on two physical quantities, the probability distribution and the entanglement between the coin degree of freedom and position degree of freedom. Choosing Hadamard coin operator and assuming the walker to start from the position eigenstate|0〉and attach itself to a coherent coin state 1/√2 (|↑〉+ i|↓〉), we give the statistical average results after making numerical calculations many times. The choices of coin operator and initial state, resulting in a symmetric probability distribution about origin in the ideal case, is helpful in comparing with different cases in different disorder strengths. It is shown that the probability distribution of static percolation quantum walk can change from a coherent behavior at short time to Anderson localization at longer time, while the dynamical percolation quantum walk can change to a classical diffusive behavior. With the decrease of the percolation probability, these transitions become faster. The entanglement for ideal case without disorder reaches a constant value after a short time evolution. The static percolation makes the entanglement less than that of ideal case and fluctuate irregularly around a certain value. The situation is very different for the dynamical percolation:the entanglement increases smoothly with the time step and can exceed the constant value in the ideal case at some time. Both of entanglements for two types of percolations decrease with reducing percolation probability. As a striking characteristic, the entanglement in dynamical case can tend to maximum regardless of percolation probability in long time limit, while the static case cannot. In the model for our study, the randomized unitary operations, induced by the static and dynamical percolations, can lead to some noticeable effects on the transport and entanglement of discrete time quantum walk. The results about the interplay between disorder and entanglement not only assist quantum information processing, but also give more options to further explore and understand disorder physical processes in nature.