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arXiv:2203.17046 · Physical Review A

Enhancing entanglement with the generalized elephant quantum walk from localized and delocalized states

Caio B. Naves, Marcelo A. Pires, Diogo O. Soares-Pinto, Sı́lvio M. Duarte Queirós

Published Oct 6, 2022• 12 citations
arXivHTMLPDF

Abstract

Recently, a generalization of a nonstandard step operator named the elephant quantum walk (EQW) was introduced. With proper statistical distribution for the steps, that generalized EQW (gEQW) can be tuned to exhibit a myriad of dynamical scaling behavior ranging from standard diffusion to hyperballistic spreading. In this work, we study the influence of the statistics of the step size and the delocalization of the initial states on the entanglement entropy of the coin. Our results show that the gEQW generates maximally entangled states for almost all initial coin states and coin operators considering initially localized walkers, and for the delocalized ones, taking the proper limit, the same condition is guaranteed. Differently from all the previous protocols that produce highly entangled states via QWs, this model is not upper bounded by ballistic spreading and hence opens prospects for applications of dynamically disordered QWs as a robust maximal entanglement generator in programmable setups that ranges from slower than ballistic to faster than ballistic.

DOI
10.1103/physreva.106.042408

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Recently, a generalization of a nonstandard step operator named the elephant quantum walk (EQW) was introduced. With proper statistical distribution for the steps, that generalized EQW (gEQW) can be tuned to exhibit a myriad of dynamical scaling behavior ranging from standard diffusion to hyperballistic spreading. In this work, we study the influence of the statistics of the step size and the delocalization of the initial states on the entanglement entropy of the coin. Our results show that the gEQW generates maximally entangled states for almost all initial coin states and coin operators considering initially localized walkers, and for the delocalized ones, taking the proper limit, the same condition is guaranteed. Differently from all the previous protocols that produce highly entangled states via QWs, this model is not upper bounded by ballistic spreading and hence opens prospects for applications of dynamically disordered QWs as a robust maximal entanglement generator in programmable setups that ranges from slower than ballistic to faster than ballistic.
Quantum walks have been a platform for the study and modeling of many processes and physical systems, going from its application on the development of quantum algorithms to the study of energy transport in biological molecules.In the same way that in general aspects of quantum theory, it also aroused the necessity of studying the introduction of noise in the evolution of quantum walks, given that it is a major factor in experimental applications.In this context, in 2018 G. D. Molfetta et al. presented a quantum model of the classical elephant random walk, the elephant quantum walk, that analyzes a noisy unitary evolution.Posteriorly, M. A. Pires et al. generalized this walk into the generalized elephant quantum walk, showing to have an interesting property of controlling the diffusive behavior of the walker while maintaining the maximum generation of entanglement between its degrees of freedom, something unknown until then.In this work we proposed ourselves to study, through numerical experiments, the generation of entanglement in the generalized elephant quantum walk in general initial settings and evolutions.For such, first we introduce classical random walks, reviewing the necessary probability concepts.After, we approach quantum walks and restrict the description to coined discrete time quantum walks, the type of quantum walk to be studied.Our results indicate that this type of quantum walk potentially generates maximally entangled states for almost all initial states and evolutions, even with a low degree of noise.These results indicate that a dynamically random evolution, being it either in the coin operator, as previously studied for other authors, or in the shift operator as in this walk, generates maximally correlated states.
Abstract We study a 2D disordered discrete-time quantum walk (QW) based on 1D generalized elephant QW where an entangling coin operator is assumed. We show that considering a given disorder in one direction, it is possible to control the degree of spreading and entanglement in the other direction. This observation helps assert that the random QWs of this ilk serve as a controllable decoherence channel with the degree of randomness being the tunable parameter and highlight the role of dimensionality in quantum systems regarding information and transport.
A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.
Disorder in coined quantum walks generally leads to localization. We investigate the influence of the localization on the entanglement properties of coined quantum walks. Specifically, we consider quantum walks on the line and explore the effects of quenched disorder in the coin operations. After confirming that our choice of disorder localizes the walker, we study how the localization affects the properties of the coined quantum walk. We find that the mixing properties of the walk are altered nontrivially with mixing being improved at short time scales. Special focus is given to the influence of coin disorder on the properties of the quantum state and the coin-walker entanglement. We find that disorder alters the quantum state significantly even when the walker probability distribution is still close to the nondisordered case. We observe that, generically, coin disorder decreases the coin-walker entanglement and that the localization leaves distinct traces in the entanglement entropy and the entanglement negativity of the coined quantum walk.
Abstract We extend to the gamut of functional forms of the probability distribution of the time-dependent step-length a previous model dubbed Elephant Quantum Walk, which considers a uniform distribution and yields hyperballistic dynamics where the variance grows cubicly with time, σ2 ∝ t3, and a Gaussian for the position of the walker. We investigate this proposal both locally and globally with the results showing that the time-dependent interplay between interference, memory and long-range hopping leads to multiple transitions between dynamical regimes, namely ballistic → diffusive → superdiffusive → ballistic → hyperballistic for non-hermitian coin whereas the first diffusive regime is quelled for implementations using the Hadamard coin. In addition, we observe a robust asymptotic approach to maximal coin-space entanglement.
We study maximal coin-position entanglement generation via a discrete-time quantum walk, in which the coin operation is randomly selected from one of two coin operators set at each step. We solve maximal entanglement generation as an optimization problem with quantum process fidelity as the cost function. Then we determine the maximal entanglement that can be rigorously generated for any step beyond the second regardless of initial conditions with appropriate coin sequences. The simplest coin sequence comprising Hadamard and identity operations is equivalent to the generalized elephant quantum walk, which exhibits an increasingly faster spreading in terms of probability distribution. Experimentally, we demonstrate a ten-step quantum walk driven by such coin sequences with linear optics and thereby show the desired high-dimensional bipartite entanglement as well as the transport behavior of faster spreading.