Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0605195 · Physical Review A

Free-Dirac-particle evolution as a quantum random walk

Anthony J. Bracken, Demosthenes Ellinas, Ioannis Smyrnakis

Published Feb 21, 2007• 60 citations
arXivHTMLPDF

Abstract

It is known that any positive-energy state of a free Dirac particle that is initially highly localized evolves in time by spreading at speeds close to the speed of light. As recently indicated by Strauch, this general phenomenon, and the resulting ``two-horned'' distributions of position probability along any axis through the point of initial localization, can be interpreted in terms of a quantum random walk, in which the roles of ``coin'' and ``walker'' are naturally associated with the spin and translational degrees of freedom in a discretized version of Dirac's equation. We investigate the relationship between these two evolutions analytically and show how the evolved probability density on the $x$ axis for the Dirac particle at any time $t$ can be obtained from the asymptotic form of the probability distribution for the position of a ``quantum walker.'' The case of a highly localized initial state is discussed as an example.

DOI
10.1103/physreva.75.022322

Related papers

Time-evolution of highly localized positive-energy states of the free Dirac electron

Anthony J. Bracken, J.A Flohr, G.F. Melloy

Highly localized positive-energy states of the free Dirac electron are constructed and shown to evolve in a simple way under the action of Dirac's equation. When the initial uncertainty in position is small on the scale of the Compton wavelength, there is an associated uncertainty in the mean energy that is large compared with the rest mass of the electron. However, this does not lead to any breakdown of the one-particle description, associated with the possibility of pair-production, but rather leads to a rapid expansion of the probability density outwards from the point of localization, at speeds close to the speed of light.

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Sep 15, 2005•11 citations

A quantum walk on a line and a Weyl equation in a space

Takuya Machida

Discrete-time quantum walks are considered a discretization of Dirac equations and the study for them has been getting attention since around 2000. In this paper, we focus on a quantum walk on a line related to a Weyl equation in a space. The quantum walker with two inner states, also interpreted as a quantum particle with the down-spin and the up-spin state in physics, spreads on the line and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

•Apr 14, 2017•0 citations

Connecting Spreading and Entanglement in Quantum Walks

Alexandre C. Orthey, Edgard P. M. Amorim

We investigate the relation between the ballistic spreading and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following a delta-like (local) and Gaussian distributions. We find out that the knowledge of long-time expected position together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while the expected position with the quantum coin phases give it for highly delocalized states.

•Jul 3, 2018•1 citation

Asymptotic properties of the Dirac quantum cellular automaton

Armando Pérez

We show that the Dirac quantum cellular automaton [A. Bisio, G. M. D'Ariano, and A. Tosini, Ann. Phys. (N.Y.) 354, 244 (2015)] shares many properties in common with the discrete-time quantum walk. These similarities can be exploited to study the automaton as a unitary process that takes place at regular time steps on a one-dimensional lattice, in the spirit of general quantum cellular automata. In this way, it becomes an alternative to the quantum walk, with a dispersion relation that can be controlled by a parameter that plays a similar role to the coin angle in the quantum walk. The Dirac Hamiltonian is recovered under a suitable limit. We provide two independent analytical approximations to the long-term probability distribution. It is shown that, starting from localized conditions, the asymptotic value of the entropy of entanglement between internal and motional degrees of freedom overcomes the known limit that is approached by the quantum walk for the same initial conditions and is similar to the ones achieved by highly localized states of the Dirac equation.

arXiv:1504.07418•Physical Review A•Jan 15, 2016•18 citations

Relativistic quantum walks

Frederick W. Strauch

By pursuing the deep relation between the one-dimensional Dirac equation and quantum walks, the physical role of quantum interference in the latter is explained. It is shown that the time evolution of the probability density of a quantum walker, initially localized on a lattice, is directly analogous to relativistic wave-packet spreading. Analytic wave-packet solutions reveal a striking connection between the discrete and continuous-time quantum walks.

arXiv:quant-ph/0508096•Physical Review A•May 23, 2006•115 citations

Relationship between quantum walks and relativistic quantum mechanics

C. M. Chandrashekar, Subhashish Banerjee, R. Srikanth

Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This article revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the similarities of the mathematical structure of the decoupled and coupled forms of the discrete-time quantum walk to that of the Klein-Gordon and Dirac equations, respectively. In the latter case, the coin emerges as an analog of the spinor degree of freedom. Discrete-time quantum walk as a coupled form of the continuous-time quantum walk is also shown by transforming the decoupled form of the discrete-time quantum walk to the Schr\"odinger form. By showing the coin to be a means to make the walk reversible and that the Dirac-like structure is a consequence of the coin use, our work suggests that the relativistic causal structure is a consequence of conservation of information. However, decoherence (modeled by projective measurements on position space) generates entropy that increases with time, making the walk irreversible and thereby producing an arrow of time. The Lieb-Robinson bound is used to highlight the causal structure of the quantum walk to put in perspective the relativistic structure of the quantum walk, the maximum speed of walk propagation, and earlier findings related to the finite spread of the walk probability distribution. We also present a two-dimensional quantum walk model on a two-state system to which the study can be extended.

arXiv:1003.4656•Physical Review A•Jun 25, 2010•108 citations
It is known that any positive-energy state of a free Dirac particle that is initially highly localized evolves in time by spreading at speeds close to the speed of light. As recently indicated by Strauch, this general phenomenon, and the resulting two-horned'' distributions of position probability along any axis through the point of initial localization, can be interpreted in terms of a quantum random walk, in which the roles of coin'' and walker'' are naturally associated with the spin and translational degrees of freedom in a discretized version of Dirac's equation. We investigate the relationship between these two evolutions analytically and show how the evolved probability density on the $x$ axis for the Dirac particle at any time $t$ can be obtained from the asymptotic form of the probability distribution for the position of a quantum walker.'' The case of a highly localized initial state is discussed as an example.
Highly localized positive-energy states of the free Dirac electron are constructed and shown to evolve in a simple way under the action of Dirac's equation. When the initial uncertainty in position is small on the scale of the Compton wavelength, there is an associated uncertainty in the mean energy that is large compared with the rest mass of the electron. However, this does not lead to any breakdown of the one-particle description, associated with the possibility of pair-production, but rather leads to a rapid expansion of the probability density outwards from the point of localization, at speeds close to the speed of light.
Discrete-time quantum walks are considered a discretization of Dirac equations and the study for them has been getting attention since around 2000. In this paper, we focus on a quantum walk on a line related to a Weyl equation in a space. The quantum walker with two inner states, also interpreted as a quantum particle with the down-spin and the up-spin state in physics, spreads on the line and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.
We investigate the relation between the ballistic spreading and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following a delta-like (local) and Gaussian distributions. We find out that the knowledge of long-time expected position together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while the expected position with the quantum coin phases give it for highly delocalized states.
We show that the Dirac quantum cellular automaton [A. Bisio, G. M. D'Ariano, and A. Tosini, Ann. Phys. (N.Y.) 354, 244 (2015)] shares many properties in common with the discrete-time quantum walk. These similarities can be exploited to study the automaton as a unitary process that takes place at regular time steps on a one-dimensional lattice, in the spirit of general quantum cellular automata. In this way, it becomes an alternative to the quantum walk, with a dispersion relation that can be controlled by a parameter that plays a similar role to the coin angle in the quantum walk. The Dirac Hamiltonian is recovered under a suitable limit. We provide two independent analytical approximations to the long-term probability distribution. It is shown that, starting from localized conditions, the asymptotic value of the entropy of entanglement between internal and motional degrees of freedom overcomes the known limit that is approached by the quantum walk for the same initial conditions and is similar to the ones achieved by highly localized states of the Dirac equation.
Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This article revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the similarities of the mathematical structure of the decoupled and coupled forms of the discrete-time quantum walk to that of the Klein-Gordon and Dirac equations, respectively. In the latter case, the coin emerges as an analog of the spinor degree of freedom. Discrete-time quantum walk as a coupled form of the continuous-time quantum walk is also shown by transforming the decoupled form of the discrete-time quantum walk to the Schr\"odinger form. By showing the coin to be a means to make the walk reversible and that the Dirac-like structure is a consequence of the coin use, our work suggests that the relativistic causal structure is a consequence of conservation of information. However, decoherence (modeled by projective measurements on position space) generates entropy that increases with time, making the walk irreversible and thereby producing an arrow of time. The Lieb-Robinson bound is used to highlight the causal structure of the quantum walk to put in perspective the relativistic structure of the quantum walk, the maximum speed of walk propagation, and earlier findings related to the finite spread of the walk probability distribution. We also present a two-dimensional quantum walk model on a two-state system to which the study can be extended.