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arXiv:1112.1487 · Physical Review A

Two quantum walkers sharing coins

Peng Xue, Barry C. Sanders

Published Feb 6, 2012• 48 citations
arXivHTMLPDF

Abstract

We consider two independent quantum walks on separate lines augmented by partial or full swapping of coins after each step. For classical random walks, swapping or not swapping coins makes little difference to the random-walk characteristics, but we show that quantum walks with partial swapping of coins have complicated yet elegant interwalker correlations. Specifically we study the joint position distribution of the reduced two-walker state after tracing out the coins and analyze total, classical, and quantum correlations in terms of mutual information, quantum mutual information, and measurement-induced disturbance. Our analysis shows intriguing quantum features without classical analogs.

DOI
10.1103/physreva.85.022307

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We consider two independent quantum walks on separate lines augmented by partial or full swapping of coins after each step. For classical random walks, swapping or not swapping coins makes little difference to the random-walk characteristics, but we show that quantum walks with partial swapping of coins have complicated yet elegant interwalker correlations. Specifically we study the joint position distribution of the reduced two-walker state after tracing out the coins and analyze total, classical, and quantum correlations in terms of mutual information, quantum mutual information, and measurement-induced disturbance. Our analysis shows intriguing quantum features without classical analogs.
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.
We present a mathematical formalism for the description of un- restricted quantum walks with entangled coins and one walker. The numerical behaviour of such walks is examined when using a Bell state as the initial coin state, with two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins as well as quantum walks with coins sharing different degrees of entanglement. We illustrate that the behaviour of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also demonstrate that simply by changing the shift operator, we can generate widely different distributions. We also compare the behaviour of quantum walks with maximally entangled coins with that of quantum walks with non-entangled coins. Finally, we show that the use of different shift operators on two and three qubit coins leads to different position probability distributions in one- and two-dimensional graphs.
We present a mathematical formalism for the description of unrestricted quantum walks with entangled coins and one walker. Furthermore, we examine the numerical behaviour of such walks using Bell states as initial coin states, two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins. We find the behavior of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also find that simply by changing the shift operator, we can generate widely different distributions.
Quantum Key Distribution (QKD) is an emerging cryptographic method designed for secure key sharing. Its security is theoretically guaranteed by fundamental principles of quantum mechanics, making it a leading candidate for future communication protocols. Quantum Random Walks (QRWs), on the other hand, are quantum processes that exhibit intriguing phenomena such as interference and superposition, enabling the generation of decentralized and asymmetric probability distributions. Inspired by both fields of study, we propose a novel QKD protocol based on two entangled quantum walkers. Our protocol exploits the unique correlations between the walkers at extremal positions of the walk to establish secret keys shared exclusively by the two parties. The security of the protocol is augmented by analyzing the joint probability distributions of the walkers' measured positions and their associated coin states.
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.