One-dimensional quantum random walks with two entangled coins

Abstract

We offer theoretical explanations for some recent observations in numerical simulations of quantum random walks (QRW). Specifically, in the case of a QRW on the line with one particle (walker) and two entangled coins, we explain the phenomenon, called "localization", whereby the probability distribution of the walker's position is seen to exhibit a persistent major "spike" (or "peak") at the initial position and two other minor spikes which drift to infinity in either direction. Another interesting finding in connection with QRW's of this sort pertains to the limiting behavior of the position probability distribution. It is seen that the probability of finding the walker at any given location becomes eventually stationary and non-vanishing. We explain these observations in terms of the degeneration of some eigenvalue of the time evolution operator $U(k)$. An explicit general formula is derived for the limiting probability, from which we deduce the limiting value of the height of the observed spike at the origin. We show that the limiting probability decreases {\em quadratically} for large values of the position $x$. We locate the two minor spikes and demonstrate that their positions are determined by the phases of non-degenerated eigenvalues of $U(k)$. Finally, for fixed time $t$ sufficiently large, we examine the dependence on $t$ of the probability of finding a particle at a given location $x$.Comment: 11 page

Distribution of chirality in the quantum walk: Markov process and entanglement

“…(29) it is possible to obtain the initial conditions Eqs. (16,17) of the system that produce this entanglement asymptotically.…”

“…Now we evaluate Q 0 , Π L and Π R using the initial conditions Eqs. (16,17) in Eqs. (13,14,15) and noting that…”

“…Carneiro et al [11] investigate entanglement between the coin and the position calculating the entropy of the reduced density matrix of the coin. The relation between asymptotic entanglement and nonlocal initial conditions (in the one and two dimensional QW) is treated in [12][13][14][15][16][17]. Ref.…”

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

“…Almost all classes of the DTQW except for a one-dimensional one with a two-dimensional coin [24,38,39] have the aspect of the localization defined that the limit distribution of the DTQW divided by some power of the time variable has the probability density given by the Dirac delta function. In the one dimensional DTQW, the localization was shown in the models with a three-dimensional coin [29], a four-dimensional coin [28], a two-dimensional coin with memory [56], a two-dimensional coin in a random environment [31], two-dimensional coins [48], an spatially incommensurate coin [70] and a time-dependent two-dimensional coin on the Fibonacci quantum walk [63] and a random coin [30,58]. On the other hand, the nature of the localization in the two-dimensional DTQW has been studied numerically [51,78] and analytically [27,81].…”

The generator and quantum Markov semigroup for quantum walks

“…After it was initiated by Meyer [17], it attracted many interests and there are many works developing it in mathematically rigorous way on the one hand and explaining possible practical applications, e.g., in quantum computation (see [2,7,9,11,12,14,17], and references therein for more details).…”