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arXiv:1205.4357

Discrete Quantum Walk on a Line with Two Entangled Particles

J. Nsofini

Published May 19, 2012• 1 citation
arXivHTMLPDF

Abstract

A review of discrete quantum walk with two particle is given. The use of different states encountered in identical particle, and the idea of entanglement and superposition is explored to explored the interesting dynamics of two particle quantum walk. Boundary conditions can specify certain dynamics and so a survey of periodic boundary condition (circle) is presented. A simulation for a Hadamard walk for different periods of a circle is considered and results are compared for various periods and for a case with absorbing boundaries.

DOI
10.48550/arxiv.1205.4357

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A review of discrete quantum walk with two particle is given. The use of different states encountered in identical particle, and the idea of entanglement and superposition is explored to explored the interesting dynamics of two particle quantum walk. Boundary conditions can specify certain dynamics and so a survey of periodic boundary condition (circle) is presented. A simulation for a Hadamard walk for different periods of a circle is considered and results are compared for various periods and for a case with absorbing boundaries.
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We introduce the concept of a quantum walk with two particles and study it for the case of a discrete time walk on a line. A quantum walk with more than one particle may contain entanglement, thus offering a resource unavailable in the classical scenario and which can present interesting modifications on quantum walks with single particles. In this work, we show both numerically and analytically how the entanglement and the relative phase between the states describing the coin degree of freedom of each particle will influence the evolution of the quantum walk. In particular, the probability to find at least one particle in a certain position after NNN steps of the walk, as well as the average distance and the squared distance between the two particles, can be larger or smaller than the case of two unentangled particles, depending on the initial conditions we choose. This resource can then be tuned according to our needs to modify the features of a quantum walk. Experimental implementations are briefly discussed.
In most widely discussed discrete time quantum walk model, after every unitary shift operator, the particle evolves into the superposition of position space and settles down in one of its basis states, loosing entanglement in the coin space in the new position. The Hadamard operation is applied to let the particle to evolve into the superposition in the coin space and the walk is iterated. We present a model with a additional degree of freedom for the unitary shift operator U′U^{\prime}U′. The unitary operator with additional degree of freedom will evolve the quantum particle into superposition of position space retaining the entanglement in coin space. This eliminates the need for quantum coin toss (Hadamard operation) after every unitary displacement operation as used in most widely studied version of the discrete time quantum walk model. This construction is easily extended to a multiple particle quantum walk and in this article we extend it for a pair of particles in pure state entangled in coin degree of freedom by simultaneously subjecting it to a pair of unitary displacement operators which were constructed for single particle. We point out that unlike for single particle quantum walk, upon measurement of its position after NNN steps, the entangled particles are found together with 1/2 probability and at different positions with 1/2 probability. This can act as an advantage in applications of the quantum walk. A special case is also treated using a complex physical system such as, inter species two-particle entangled Bose-Einstein condensate, as an example.
In this paper discrete quantum walks with different coins used for odd and even time steps are studied. These coins are called hybrid. The calculation results are compared with the most frequently used coin, the Hadamard transform. Furthermore, quantum walks on the line which involve two or more quantum walkers with hybrid coins are studied. Quantum walks with entangled walkers and hybrid coins are also studied. The results of these calculations show that the proposed types of quantum walks can be used for quantum search, because the walker can be directed towards preferred directions and can also be confined in certain segments of the line.
Abstract Asymptotic dynamics of a Hadamard walk of two non-interacting quantum particles on a dynamically percolated finite line or a circle is investigated. We construct a basis of the attractor space of the corresponding random-unitary dynamics and prove the completeness of our solution. In comparison to the one-particle case, the structure of the attractor space is much more complex, resulting in intriguing asymptotic dynamics. General results are illustrated on two examples. First, for circles of length not divisible by 4 the boundary conditions reduces the number of attractors considerably, allowing for fully analytic solution. Second, we investigate line of length 4 and determine the asymptotic cycle of reduced coin states and position distributions, focusing on the correlations between the two particles. Our results show that a random unitary evolution, which is a combination of quantum dynamics and a classical stochasticity, leads to correlations between initially uncorrelated particles. This is not possible for purely unitary evolution of non-interacting quantum particles. The shared dynamically percolated graph can thus be considered as a weak form of interaction.