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Quantum Information Processing
Published in: Quantum Information Processing 9/2017

01-09-2017

Asymptotic entanglement in quantum walks from delocalized initial states

Authors: Alexandre C. Orthey Jr., Edgard P. M. Amorim

Published in: Quantum Information Processing | Issue 9/2017

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Abstract

We study the entanglement between the internal (spin) and external (position) degrees of freedom of the one-dimensional discrete time quantum walk starting from local and delocalized initial states whose time evolution is driven by Hadamard and Fourier coins. We obtain the dependence of the asymptotic entanglement with the initial dispersion of the state and establish a way to connect the asymptotic entanglement between local and delocalized states. We find out that the delocalization of the state increases the number of initial spin states which achieves maximal entanglement from two states (local) to a continuous set of spin states (delocalized) given by a simple relation between the angles of the initial spin state. We also carry out numerical simulations of the average entanglement along the time to confront with our analytical results.
previous article Equivalence of Szegedy’s and coined quantum walks
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Appendix
Available only for authorised users
Footnotes
1
In “Appendix” we extend the following calculations for a Fourier coin.
 
2
The Gaussian states were defined between \([-1000,1000]\), i.e., for a \(\sigma _0=30\) the discrete sum of the normalization condition gives an error below of 0.001%.
 
3
Since the numerical difference between the discrete sum and integration of the amplitudes with \(\sigma _0=1\) is around \(10^{-4}\), and this difference is even smaller for larger \(\sigma _0\).
 
4
The minimum entanglement condition for Gaussian and rectangular states obeys a power law \(\overline{S}_E\sim 0.3463\sigma _0^{-1.59}\) and \(\overline{S}_E\sim 0.4874\sigma _0^{-0.853}\), respectively, both obtained by curve fitting.
 
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Metadata
Title
Asymptotic entanglement in quantum walks from delocalized initial states
Authors
Alexandre C. Orthey Jr.
Edgard P. M. Amorim
Publication date
01-09-2017
Publisher
Springer US
Published in
Quantum Information Processing / Issue 9/2017
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-017-1672-1

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