Abstract
We discuss here the hydrodynamic limit of independent quantum random walks evolving on \(\mathbb {Z}\). As main result, we obtain that the time evolution of the local equilibrium is governed by the convolution of the chosen initial profile with a rescaled version of the limiting probability density obtained in the law of large numbers for a single quantum random walk.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Unitary matrix: its columns (or lines) compound an orthonormal basis for the space.
- 2.
In comparison with the classical random walk.
- 3.
The so-called infinite propagation speed in PDE’s, see [8, p. 49].
- 4.
Physically, to speak about finite propagation speed for QRW’s it is necessary to go further into the Lieb-Robinson bound, see [9]. We did not investigate such subject in this paper.
- 5.
See Ref. [10].
References
Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften. Fundamental Principles of Mathematical Sciences, vol. 320. Springer, Berlin (1999)
DeMasi, A., Presutti, E.: Mathematical methods for hydrodynamic limits. Lecture Notes in Mathematics, vol. 1501. Springer (1991)
Dobrushin, R., Siegmund-Schultze, R.: The hydrodynamic limit for systems of particles with independent evolution. Mathematische Nachrichten 105(1), 199–224 (1982)
Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48(2), 1687–1690 (1993)
Kempe, J.: Quantum random walks—an introductory overview. Contemp. Phys. 44(4), 307–327 (2003)
Grimmett, G., Janson, S., Scudo, P.: Weak limits of quantum random walks. Phys. Rev. E 69 (2004)
Durrett, R.: Probability: Theory and Examples, 4th edn. Cambridge University Press (2010)
Evans, L.: Partial Differential Equations. Graduate Studies in Mathematics V.19, 2nd edn. American Mathematical Society (1998)
Lieb, E., Robinson, D.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)
Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York (1999)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Baraviera, A., Franco, T., Neumann, A. (2016). Hydrodynamic Limit of Quantum Random Walks. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations III. Springer Proceedings in Mathematics & Statistics, vol 162. Springer, Cham. https://doi.org/10.1007/978-3-319-32144-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-32144-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-32142-4
Online ISBN: 978-3-319-32144-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)