Quantum Markov Chains Associated with Open Quantum Random Walks

arXiv (Cornell University), 2022

In the present paper, we construct QMC (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 of OQRW. Furthermore, we first propose a new construction of QMC on trees, which is an extension of QMC considered in Ref. [9]. Using such a construction, we are able to construct QMCs on tress associated with OQRW. Our investigation leads to the detection of the phase transition phenomena within the proposed scheme. This kind of phenomena appears first time in this direction. Moreover, mean entropies of QMCs are calculated.

Cornell University - arXiv, 2020

We propose an intermediate walk continuously connecting an open quantum random walk and a quantum walk with parameters M ∈ N controlling a decoherence effect; if M = 1, the walk coincides with an open quantum random walk, while M = ∞, the walk coincides with a quantum walk. We define a measure which recovers usual probability measures on Z for M = ∞ and M = 1 and we observe intermediate behavior through numerical simulations for varied positive values M. In the case for M = 2, we analytically show that a typical behavior of quantum walks appears even in a small gap of the parameter from the open quantum random walk. More precisely, we observe both the ballistically moving towards left and right sides and localization of this walker simultaneously. The analysis is based on Kato's perturbation theory for linear operator. We futher analyze this limit theorem in more detail and show that the above three modes are described by Gaussian distributions.

Physical Review D, 2005

One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we "quantize" the classical random walk by finding, subject to a certain condition of "strong positivity", the most general Markovian, translationally invariant "decoherence functional" with nearest neighbor transitions.

Open Systems & Information Dynamics, 2022

In the present paper, we construct QMC (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 P ρ of OQRW. However, we are going to look at the probability distribution as a Markov field over the Cayley tree. Such kind of consideration allows us to investigated phase transition phenomena associated for OQRW within QMC scheme. Furthermore, we first propose a new construction of QMC on trees, which is an extension of QMC considered in Ref. [10]. Using such a construction, we are able to construct QMCs on tress associated with OQRW. Our investigation leads to the detection of the phase transition phenomena within the proposed scheme. This kind of phenomena appears first time in this direction. Moreover, mean entropies of QMCs are calculated.

Axioms, 2023

Quantum Markov chains (QMCs) and open quantum random walks (OQRWs) represent different quantum extensions of the classical Markov chain framework. QMCs stand as a more profound layer within the realm of Markovian dynamics. The exploration of both QMCs and OQRWs has been a predominant focus over the past decade. Recently, a significant connection between QMCs and OQRWs has been forged, yielding valuable applications. This bridge is particularly impactful when studying QMCs on tree structures, where it intersects with the realm of phase transitions in models naturally arising from quantum statistical mechanics. Furthermore, it aids in elucidating statistical properties, such as recurrence and clustering. The objective of this paper centers around delving into the intricate structure of QMCs on Cayley trees in relation to OQRWs. The foundational elements of this class of QMCs are built upon using classical probability measures that encompass the hierarchical nature of Cayley trees. This exploration unveils the pivotal role that the dynamics of OQRWs play in shaping the behavior of the Markov chains under consideration. Moreover, the analysis extends to their classical counterparts. The findings are further underscored by the examination of notable examples, contributing to a comprehensive understanding of the outcomes.

International Journal of Modern Physics A, 1993

Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated with a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realized in Lin(H) in such a way that Δh=W(h⊗1)W−1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantum-mechanically to the system at time t+δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a novel kind of CTP theorem.

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.

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

General Relativity and Gravitation, 2011

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure µ n on the space of n-paths, and the µ n in turn induce a quantum measure µ on the cylinder sets within the space Ω of untruncated paths. Although µ cannot be extended to a continuous quantum measure on the full σ-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of Ω in a systematic way. We begin an investigation of this problem by showing that µ can be extended to a quantum measure on a "quadratic algebra" of subsets of Ω that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the n-path space.