Two-site quantum random walk

Reviews in Mathematical Physics

In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution [Formula: see text] of OQRW. This sheds new light on some properties of the measure [Formula: see text]. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of [Formula: see text]-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.

Journal of Statistical Physics, 2019

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties such as reducibility/irreducibility, recurrence/transience, accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on the discussion of the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We will see also that the classical Markov chains can be reconstructed as quantum Markov chains.

arXiv: Mathematical Physics, 2018

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. We then discuss the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. The quantum Markov chain was introduced by Accardi by using transition expectation, and the construction of the quantum Markov chains for the open quantum random walks was introduced by Dhahri and Mukhamedov. Here we construct nonhomogeneous quantum Markov chains for the open quantum random walks so that we can naturally express the evolution of any open quantum random walk by the corresponding quantum Markov chain. We provide with some examples. In particular, we show that the classical Markov chains are reconstructed as quantum Markov chains as well and the reducibility and irreducibility properties of classical Markov chains can be investigated in the language of quantum Markov chains.

Journal of Statistical Physics, 2012

We consider the limit distributions of open quantum random walks on one-dimensional lattice space. We introduce a dual process to the original quantum walk process, which is quite similar to the relation of Schrödinger-Heisenberg representation in quantum mechanics. By this, we can compute the distribution of the open quantum random walks concretely for many examples and thereby we can also obtain the limit distributions of them. In particular, it is possible to get rid of the initial state when we consider the evolution of the walk, it appears only in the last step of the computation. Recently Attal et al. [3, 4] introduced and investigated the open quantum random walk (OQRW) on graphs, which shows various different dynamical behaviors comparing to usual quantum walks and it includes the classical random walk as a special case. Then they considered the limit theorems for OQRW's: they have shown the central limit theorem [2]. The purpose of this paper is to further investigate the distribution and the limit theorem of OQRW's. A quantum analog of the classical random walk, called quantum walk (QW), has been intensively studied for the last decade (see [6, 7, 11, 13]). The most remarkable difference between classical random walk and quantum walk appears in the central limit theorem. In the classical random walk, the limit distribution is Gaussian with scaling speed √ n. On the other hand, in quantum walk the speed is linear in time, i.e., n, and moreover, the limit Abbr. title: Dynamics for open quantum random walks

2015

An explicit description of the decoherence-free subalgebra is obtained in a type of Quantum Markov Semigroups, called continuous-time open quantum random walks, representing a natural quantum analogue of continuous-time Markov chains. This leads to an analysis of the asymptotic behavior and of the decoherence of these semigroups in many interesting cases.

Quantum walks are powerful tools not only to construct the quantum speedup algorithms but also to describe specific models in physical processes. Furthermore, the discrete time quantum walk has been experimentally realized in various setups. We show the limit distribution of the discrete time quantum walk with the periodic position measurement under the time scale transformation.

Annales Henri Poincaré, 2014

Open Quantum Random Walks, as developed in Attal et al. (J. Stat. Phys. 147(4):832-852, 2012), are a quantum generalization of Markov chains on finite graphs or on lattices. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in quantum information theory (such as the wellknown Hadamard random walk). Typically, in the case of open quantum random walks on lattices, their distribution seems to always converge to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous open quantum random walks on Z d , we prove such a central limit theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a central limit theorem with explicit drift and explicit covariance matrix. The interesting point which appears with our construction and result is that it applies actually to a wider setup: it provides a central limit theorem for the sequence of recordings of the quantum trajectories associated wih any completely positive map. This is what we show and develop as an application of our result. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block diagonal in a common basis.

NATO Science Series II: Mathematics, Physics and Chemistry, 2006

We give a short overview over recent developments on quantum graphs and outline the connection between general quantum graphs and so-called quantum random walks.

This report summarizes our research process on the behaviour of quantum random walks within the specific model of the Creutz ladder. We begin by introducing fundamental concepts related to classical random walks, quantum random walks and some key differences between them. We then focus on our specific case study, examining the property of localization. By utilizing combinatorial methods and providing some visualization, we obtained interesting results regarding the confinement of the particle on the quantum lattice

Applied Mathematics, 2014

We review (not exhaustively) the quantum random walk on the line in various settings, and propose some questions that we believe have not been tackled in the literature. In a sense, this article invites the readers (beginner, intermediate, or advanced), to explore the beautiful area of quantum random walks.