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Entropy 2016, 18(6), 228;

Discrete Time Dirac Quantum Walk in 3+1 Dimensions

QUIT group, Dipartimento di Fisica, via Bassi 6, Pavia 27100, Italy
INFN Gruppo IV, Sezione di Pavia, via Bassi 6, Pavia 27100, Italy
Author to whom correspondence should be addressed.
Academic Editors: Gregg Jaeger and Andrei Khrennikov
Received: 4 March 2016 / Revised: 28 May 2016 / Accepted: 30 May 2016 / Published: 14 June 2016
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In this paper we consider quantum walks whose evolution converges to the Dirac equation in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two, and three space dimensions. The behaviour of particle states—defined as states smoothly peaked in some wave-vector eigenstate of the walk—is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between the exactly evolved state and the approximated one. The jittering of the position operator expectation value for states having both a particle and an antiparticle component is analytically derived and observed in the numerical implementations. View Full-Text
Keywords: quantum walk; Dirac equation quantum walk; Dirac equation

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D’Ariano, G.M.; Mosco, N.; Perinotti, P.; Tosini, A. Discrete Time Dirac Quantum Walk in 3+1 Dimensions. Entropy 2016, 18, 228.

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