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arXiv:0804.4311 · Physical Review A

Limiting distribution of decoherent quantum random walks

Kai Zhang

Published Jun 2, 2008• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.77.062302

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In this paper, we study the discrete-time quantum random walks on a line subject to decoherence. The convergence of the rescaled position probability distribution p(x,t)p(x,t)p(x,t) depends mainly on the spectrum of the superoperator Lkk{\mathcal{L}}_{kk}Lkk​. We show that if 1 is an eigenvalue of the superoperator with multiplicity one and there is no other eigenvalue whose modulus equals 1, then $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{P}(\frac{\ensuremath{\nu}}{\sqrt{t}},t)$ converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function ${p}_{t}(x,t)\ensuremath{\equiv}p(\sqrt{t}x,t)$, $x\ensuremath{\in}Z/\sqrt{t}$, converges in distribution to a continuous convex combination of normal distributions. We give a necessary and sufficient condition for a U(2) decoherent quantum walk that satisfies the eigenvalue conditions. We also give a complete description of the behavior of quantum walks whose eigenvalues do not satisfy these assumptions. Specific examples such as the Hadamard walk and walks under real and complex rotations are illustrated. For the O(2) quantum random walks, an explicit formula is provided for the scaling limit of p(x,t)p(x,t)p(x,t) and their moments. We also obtain exact critical exponents for their moments at the critical point and show universality classes with respect to these critical exponents.
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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.