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Journal of Quantum Information Science

Quantum Walk of Two Quantum Particles on One Dimensional System

Yongmei Zhang, Junjie Xiao, Dongsheng Hu

Published Jan 1, 2013• 0 citations
Publisher

Abstract

We study two particle quantum walks on one dimensional chain.Probability distribution of two particle quantum walks is dependent on the initial state, and symmetric quantum walk or asymmetric quantum walk is analogous to that of one particle quantum walk.The quantum correlation probability is much different from classical coincidence probability.The difference reflects quantum interference between two particles.

DOI
10.4236/jqis.2013.34018

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We study two particle quantum walks on one dimensional chain.Probability distribution of two particle quantum walks is dependent on the initial state, and symmetric quantum walk or asymmetric quantum walk is analogous to that of one particle quantum walk.The quantum correlation probability is much different from classical coincidence probability.The difference reflects quantum interference between two particles.
We investigate the two-component quantum walk in one-dimensional lattice. We show that the inter-component interaction strength together with the hopping imbalance between the components exhibit distinct features in the quantum walk for different initial states. When the walkers are initially on the same site, both the slow and fast particles perform independent particle quantum walks when the interaction between them is weak. However, stronger inter-particle interactions result in quantum walks by the repulsively bound pair formed between the two particles. For different initial states when the walkers are on different sites initially, the quantum walk performed by the slow particle is almost independent of that of the fast particle, which exhibits reflected and transmitted components across the particle with large hopping strength for weak interactions. Beyond a critical value of the interaction strength, the wave function of the fast particle ceases to penetrate through the slow particle signalling a spatial phase separation. However, when the two particles are initially at the two opposite edges of the lattice, then the interaction facilitates the complete reflection of both of them from each other. We analyze the above mentioned features by examining various physical quantities such as the on-site density evolution, two-particle correlation functions and transmission coefficients.
We investigate the quantum walk of two interacting particles of different mass in a one dimensional lattice. We show that the inter-particle interaction together with the mass imbalance exhibit distinct features in the quantum walk for different initial states. When the walkers are initially on the same site, both the heavy and light particles perform independent particle quantum walks for small interaction. However, stronger interactions lead to the quantum walk of the repulsively bound pair. For a different initial states when the walkers are on different sites initially, the heavy particle performs quantum walk almost independently and the light particle shows reflected and transmitted components across the heavy particle for small interactions. After a critical interaction strength, the light particle wave function ceases to penetrate through the heavy particle. However, when the quantum walkers are initially kept at the two opposite edges of the lattice, then the interaction facilitates the complete reflection of both heavy and light particles from each other. We analyze these features by examining various physical quantities such as the on-site density evolution, two-particle correlation functions and transmission coefficients.
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.
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 present a mathematical formalism for the description of un- restricted quantum walks with entangled coins and one walker. The numerical behaviour of such walks is examined when using a Bell state as the initial coin state, with two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins as well as quantum walks with coins sharing different degrees of entanglement. We illustrate that the behaviour of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also demonstrate that simply by changing the shift operator, we can generate widely different distributions. We also compare the behaviour of quantum walks with maximally entangled coins with that of quantum walks with non-entangled coins. Finally, we show that the use of different shift operators on two and three qubit coins leads to different position probability distributions in one- and two-dimensional graphs.
In this work, we theoretically investigate a one-dimensional bosonic and fermionic discrete in time quantum Walk, affected by a temporal disorder. Specifically, we study how the wave-function symmetry and the phase disorder distribution affect the final coincidence probability and transport properties of a 100-step quantum Walk. The degree of randomness is suitably adjusted by the disorder distribution width and different diffusion regimes are achieved, ranging from the classical diffusion to the ballistic, associated with quantum states propagation.