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Relativistic effects and rigorous limits for discrete- and continuous-time quantum walks

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10.1063/1.2759837

### Abstract

The mathematical relationship between the discrete-time and continuous-time quantum walks and the one-dimensional Dirac equation is explored by studying a class of solutions for each, expressed in terms of the generalized, regular, and modified Bessel functions, respectively. Rigorous limits connecting these solutions are established. In addition, new analytical and numerical results are presented for quantum walks and the Dirac equation, including entanglement, relativistic localization and wave packet spreading, and normal and anomalous *Zitterbewegung*.

© 2007 American Institute of Physics

Received 24 April 2007
Accepted 18 June 2007
Published online 02 August 2007

Article outline:

I. INTRODUCTION

II. EXACT SOLUTIONS OF MULTICOMPONENT WAVE EQUATIONS

A. Dirac equation

B. Discrete-time quantum walk

C. Continuous-time quantum walk

III. DIRAC–DISCRETE-TIME QUANTUM WALK–CONTINUOUS-TIME QUANTUM WALK LIMITS

A. Limit I: Discrete-time quantum walk to Dirac

B. Limit II: Discrete-time quantum walk to continuous-time quantum walk

IV. PROBABILITY DISTRIBUTIONS AND EXPECTATION VALUES

A. Dirac equation

B. Discrete-time quantum walk

C. Continuous-time quantum walk

V. ENTANGLEMENT

A. Dirac equation

B. Discrete-time quantum walk

VI. *ZITTERBEWEGUNG*

A. Dirac Equation

B. Discrete-time quantum walk

C. Continuous-time quantum walk

VII. CONCLUSION

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2007-08-02

2014-04-17

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