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arXiv:quant-ph/0507184 · Journal of Physics A Mathematical and General

Entanglement measurement with discrete multiple-coin quantum walks

Jochen Endrejat, H. Büttner

Published Oct 5, 2005• 16 citations
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Abstract

Within a special multi-coin quantum walk scheme, we analyse the effect of the entanglement of the initial coin state. For states with a special entanglement structure, it is shown that this entanglement can be measured with the mean value of the walk, which depends on the i-concurrence of the initial coin state. Further, the entanglement evolution is investigated and it is shown that the symmetry of the probability distribution is reflected by the symmetry of the entanglement distribution.

DOI
10.1088/0305-4470/38/42/008

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Within a special multi-coin quantum walk scheme, we analyse the effect of the entanglement of the initial coin state. For states with a special entanglement structure, it is shown that this entanglement can be measured with the mean value of the walk, which depends on the i-concurrence of the initial coin state. Further, the entanglement evolution is investigated and it is shown that the symmetry of the probability distribution is reflected by the symmetry of the entanglement distribution.
Quantum walks can be used either as tools for quantum algorithm development or as entanglement generators, potentially useful to test quantum hardware. We present a novel algorithm based on a discrete Hadamard quantum walk on a line with one coin and two walkers whose purpose is to generate entanglement between walkers. We provide several classical computer simulations of our quantum algorithm in which we show that, although the asymptotical amount of entanglement generated between walkers does not reach the highest degree of entanglement possible at each step for either coin measurement outcome, the entanglement ratio (entanglement generated/highest value of entanglement possible, for each step) tends to converge, and the actual convergence value depends on the coin initial state and on the coin measurement outcome. Furthermore, our numerical simulations show that, for the quantum walks used in our algorithm, the value towards which entanglement ratio converges also depends on the position probability distribution symmetry of a quantum walk computed with one single walker and the same coin initial state employed in the corresponding quantum walk with two walkers.
We introduce a generalization of quantum walks with entangled coins consisting of a model of discrete quantum walks with coin pairs under various degrees of entanglement and walkers in quantum superposition as initial states. We introduce novel position probability distributions that may be used for algorithm development based on quantum-mechanical phenomena. Also, we numerically show that the skewness of position probability distribution produced by using coin initial state with various degrees of entanglement cannot be easily inferred.
We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.
We report a systematic study of asymptotic behavior of entanglement between the position and coin degrees of freedom for a one-dimensional discrete quantum walk. Effects of coin bias, initial coin asymmetry and phase difference and non-locality of initial position state on the fluctuations and asymptotic value of entanglement are investigated by using Fourier approach. Fluctuations in entropy are found to die out as power law in time as t-α with α = 1/2,3/2 and 5/2 depending on the initial state parameters of the system.
Abstract We investigate the evolution of a discrete-time one-dimensional quantum walk driven by a position-dependent coin. The rotation angle, which depends upon the position of a quantum particle, parameterizes the coin operator. For different values of the rotation angle, we observe that such a coin leads to a variety of probability distributions, e.g. localized, periodic, classical-like, semi-classical-like, and quantum-like. Further, we study the Shannon entropy associated with position and the coin space of a quantum particle, and compare them with the case of the position-independent coin. Our results show that the entropy is smaller for most values of the rotation angle as compared to the case of the position-independent coin. We also study the effect of entanglement on the behavior of probability distribution and Shannon entropy by considering a quantum walk with two identical position-dependent entangled coins. We observe that in general, a wave function becomes more localized as compared to the case of the position-independent coin and hence the corresponding Shannon entropy is lower. Our results show that a position-dependent coin can be used as a controlling tool of quantum walks.