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arXiv:1601.06483 · Physical Review A

Extraordinary behaviors in a two-dimensional decoherent alternative quantum walk

Tian Chen, Xiangdong Zhang

Published Jul 11, 2016• 19 citations
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Abstract

We reveal the quantum and classical behaviors of the two-dimensional (2D) alternative quantum walk (AQW) in the presence of decoherence. For different kinds of decoherence, the analytic expressions for the moments of position distribution of the AQW are obtained. Taking the broken line noise and coin decoherence as examples of decoherence, we find that when decoherence emerges in only one direction, the anisotropic position distribution pattern appears, and not all the motions of the walker exhibit the transition from quantum to classical behaviors. Considering the effect of decoherence, we reveal the anisotropic correlations between the $x\phantom{\rule{0.28em}{0ex}}(y)$ position of the 2D walker and the state of the coin in 2D AQWs.

DOI
10.1103/physreva.94.012316

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We reveal the quantum and classical behaviors of the two-dimensional (2D) alternative quantum walk (AQW) in the presence of decoherence. For different kinds of decoherence, the analytic expressions for the moments of position distribution of the AQW are obtained. Taking the broken line noise and coin decoherence as examples of decoherence, we find that when decoherence emerges in only one direction, the anisotropic position distribution pattern appears, and not all the motions of the walker exhibit the transition from quantum to classical behaviors. Considering the effect of decoherence, we reveal the anisotropic correlations between the x(y)x\phantom{\rule{0.28em}{0ex}}(y)x(y) position of the 2D walker and the state of the coin in 2D AQWs.
Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.
Abstract In this article we investigate the effects of shifting position decoherence, arising from the incoherent tunneling effect in the experimental realization of the quantum walk, on the one-dimensional discrete time quantum walk. We show that in the regime of this type of noise the quantum behavior of the walker does not vanish, in contrast to the coin decoherence for which the walker undergoes a quantum-to-classical transition even for weak noise. In particular, we show that the quadratic dependence of the variance on the time and also the coin–position entanglement, i.e. two important quantum aspects of the coherent quantum walk, are preserved in the presence of tunneling decoherence. Furthermore, we present an explicit expression for the probability distribution of the decoherent one-dimensional quantum walk in terms of the corresponding coherent probabilities, and show that this type of decoherence smooths the probability distribution.
We study a two-dimensional quantum walk with only one walker alternatively walking along the horizontal and vertical directions driven by a single two-side coin. We find the analytical expressions of the first two moments of the walker’s position distribution in the long-time limit, which indicates that the variance of the position distribution grows quadratically with walking steps, showing a ballistic spreading typically for quantum walks. Besides, we analyze the correlation by calculating the quantum mutual information and the measurement-induced disturbance respectively as the outcome of the walk in one dimension is correlated to the other with the coin as a bridge. It is shown that the quantum correlation between walker spaces increases gradually with the walking steps.
In this article we study decoherence in the discrete-time quantum walk on the line. We generalize the method of decoherent coin quantum walk, introduced by Brun et al. [Phys. Rev. A 67, 32304 (2003)]. Our analytical expressions are applicable for all kinds of decoherence. As an example of the coin-position decoherence, we study the broken line quantum walk and compare our results with the numerical one. We also show that our analytical results reduce to the Brun formalism when only the coin is subjected to decoherence.
Two-dimensional quantum walks using a two-state coin have simpler experimental implementation than two-dimensional quantum walks using a four-state coin. However, decoherence occurs inevitably during the evolution of quantum walks due to the coupling between the quantum systems and their environment. Thus, it is interesting to investigate the robustness against decoherence for two- and four-state two-dimensional quantum walks. Here, we investigate the effects of the decoherence on two- and four-state two-dimensional quantum walks produced by the broken-link-type noise and compare their robustness against the broken-link-type noise. Specifically, we analyze the quantum correlation between the two spatial dimensions x and y by using measurement-induced disturbance for the two-state quantum walks, i.e. the alternate walk and the Pauli walk, and the four-state quantum walks, i.e. the Grover, Hadamard and Fourier walks, respectively. Our analysis shows that the two-state walks are more robust against the broken-link-type noise than the four-state walks.
The behavior of one-dimensional quantum random walks is strikingly different from that of classical ones. However, when decoherence is involved, the limiting distributions take on many classical features over time. In this paper, we study the decoherence on both position and ``coin'' spaces of the particle. We propose an analytical approach to investigate these phenomena and obtain the generating functions which encode all the features of these walks. Specifically, from these generating functions, we find exact analytic expressions of several moments for the time and noise dependence of position. Moreover, the limiting position distributions of decoherent quantum random walks are shown to be Gaussian in an analytical manner. These results explicitly describe the relationship between the system and the level of decoherence.