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arXiv:quant-ph/0411065 · Physical Review A

Quantum walk on a line with two entangled particles

Yasser Omar, Nikola Paunković, Lana Sheridan, Sougato Bose

Published Oct 3, 2006• 149 citations
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Abstract

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 $N$ 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.

DOI
10.1103/physreva.74.042304

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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.
We introduce the idea of a quantum walk with two particles and study it for the case of a discrete time walk on a line. We consider both separable and maximally entangled initial conditions, and show 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, these factors will have consequences for the distance between the particles and the probability of finding them at a given point, yielding results that cannot be obtained from a separable initial state, be it pure or mixed. Finally, we review briefly proposals for implementations.
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.
Quantum walks can be used either as tools for quantum algorithm development or as entanglement generators, potentially useful to test quantum hardware. We present a novel algorithm based on a discrete Hadamard quantum walk on a line with one coin and two walkers whose purpose is to generate entanglement between walkers. We provide several classical computer simulations of our quantum algorithm in which we show that, although the asymptotical amount of entanglement generated between walkers does not reach the highest degree of entanglement possible at each step for either coin measurement outcome, the entanglement ratio (entanglement generated/highest value of entanglement possible, for each step) tends to converge, and the actual convergence value depends on the coin initial state and on the coin measurement outcome. Furthermore, our numerical simulations show that, for the quantum walks used in our algorithm, the value towards which entanglement ratio converges also depends on the position probability distribution symmetry of a quantum walk computed with one single walker and the same coin initial state employed in the corresponding quantum walk with two walkers.
We study the evolution of quantum correlations in two-particle discrete-time non-unitary quantum walks on a line with gain and loss. The two particles are initially prepared in a maximally entangled state and evolve independently. Using numerically exact calculations of position probability densities, average interparticle distance, entanglement entropy, and concurrence, we examine how dissipation and particle-exchange symmetry of the state (symmetric or antisymmetric) influence the resulting correlations. Notably, we find that the entanglement entropy of the antisymmetric state decays slower than that of the symmetric state at the exceptional point of the model. We also show that, in the antisymmetric state, the concurrence measured on the one-particle reduced density operator is unaffected by dissipation. Furthermore, we discuss how these correlations indicate quantum-to-classical transitions through the appearance of Gaussian position probability densities, sub-linear growth of average interparticle distance after a few steps, and decay of the entanglement entropy. On the other hand, the robustness of the qubit entanglement entropy can differentiate the model's topological phases better than in the one-particle quantum walk scenario.
In this paper, we present a simulator for two-particle quantum walks on the line and one-particle on a two-dimensional squared lattice. It can be used to investigate the equivalence between the two cases (one- and two-particle walks) for various boundary conditions (open, circular, reflecting, absorbing and their combinations). For the case of a single walker on a two-dimensional lattice, the simulator can also implement the Möbius strip. Furthermore, other topologies for the walker are also simulated by the proposed tool, like certain types of planar graphs with degree up to 4, by considering missing links over the lattice. The main purpose of the simulator is to study the genuinely quantum effects on the global properties of the two-particle joint probability distribution on the entanglement between the walkers/axis. For that purpose, the simulator is designed to compute various quantities such as: the entanglement and classical correlations, (classical and quantum) mutual information, the average distance between the two walkers, different hitting times and quantum discord. These quantities are of vital importance in designing possible algorithmic applications of quantum walks, namely in search, 3-SAT problems, etc. The simulator can also implement the static partial measurements of particle(s) positions and dynamic breaking of the links between certain nodes, both of which can be used to investigate the effects of decoherence on the walker(s). Finally, the simulator can be used to investigate the dynamic Anderson-like particle localization by varying the coin operators of certain nodes on the line/lattice. We also present some illustrative and relevant examples of one- and two-particle quantum walks in various scenarios. The tool was implemented in C and is available on-line at http://qwsim.weebly.com/ .
We study discrete-time quantum walks on the line and on general undirected graphs with two interacting or noninteracting particles. We introduce two simple interaction schemes and show that they both lead to a diverse range of probability distributions that depend on the correlations and relative phases between the initial coin states of the two particles. We investigate the characteristics of these quantum walks and the time evolution of the entanglement between the two particles from both separable and entangled initial states. We also test the capability of two-particle discrete-time quantum walks to distinguish nonisomorphic graphs. For strongly regular graphs, we show that noninteracting discrete-time quantum walks can distinguish some but not all nonisomorphic graphs with the same family parameters. By incorporating an interaction between the two particles, all nonisomorphic strongly regular graphs tested are successfully distinguished.