scholarly journals A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

2018 ◽  
Vol 326 ◽  
pp. 352-464 ◽  
Author(s):  
F.A. Grünbaum ◽  
L. Velázquez
Keyword(s):  
Random Walks ◽  
Orthogonal Polynomials ◽  
Schur Functions ◽  
Quantum Walks ◽  
Open Quantum ◽  
Nevanlinna Functions

scholarly journals Homogeneous open quantum walks on the line: criteria for site recurrence and absorption

2021 ◽  
Vol 21 (1&2) ◽  
pp. 0037-0058
Author(s):  
Thomas S. Jacq ◽  
Carlos F. Lardizabal
Keyword(s):  
Random Walks ◽  
Nearest Neighbor ◽  
Trace Class ◽  
Quantum Walks ◽  
Internal Degree ◽  
Quantum Random Walks ◽  
Open Quantum ◽  
Infinite Line ◽  
Trace Class Operators ◽  
Internal Degree Of Freedom

In this work, we study open quantum random walks, as described by S. Attal et al.. These objects are given in terms of completely positive maps acting on trace-class operators, leading to one of the simplest open quantum versions of the recurrence problem for classical, discrete-time random walks. This work focuses on obtaining criteria for site recurrence of nearest-neighbor, homogeneous walks on the integer line, with the description presented here making use of recent results of the theory of open walks, most particularly regarding reducibility properties of the operators involved. This allows us to obtain a complete criterion for site recurrence in the case for which the internal degree of freedom of each site (coin space) is of dimension 2. We also present the analogous result for irreducible walks with an internal degree of arbitrary finite dimension and the absorption problem for walks on the semi-infinite line.

Higher-dimensional open quantum walk in environment of quantum Bernoulli noises

2021 ◽  
pp. 2250001
Author(s):  
Ce Wang
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Initial State ◽  
Quantum Random Walks ◽  
Open Quantum ◽  
Higher Dimensional ◽  
Gauss Type ◽  
Open Quantum Random Walks ◽  
Limit Probability ◽  
Creation Operators

Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.

scholarly journals Analysis of random walks using orthogonal polynomials

1998 ◽  
Vol 99 (1-2) ◽  
pp. 387-399 ◽  
Author(s):  
Pauline Coolen-Schrijner ◽  
Erik A. van Doorn
Keyword(s):  
Random Walks ◽  
Orthogonal Polynomials

scholarly journals One dimensional quantum walks with memory

2010 ◽  
Vol 10 (5&6) ◽  
pp. 509-524
Author(s):  
M. Mc Gettrick
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Walks ◽  
One Dimensional ◽  
With Memory ◽  
Previous Step

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of one previous step. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

scholarly journals Microscopic derivation of open quantum walks

2014 ◽  
Author(s):  
Ilya Sinayskiy ◽  
Francesco Petruccione
Keyword(s):  
Quantum Walks ◽  
Open Quantum ◽  
Microscopic Derivation

scholarly journals Open quantum random walks, quantum Markov chains and recurrence

2019 ◽  
Vol 31 (07) ◽  
pp. 1950020 ◽  
Author(s):  
Ameur Dhahri ◽  
Farrukh Mukhamedov
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Markov Processes ◽  
Transition Operator ◽  
Quantum Random Walks ◽  
Open Quantum ◽  
Commutative Subalgebra ◽  
Open Quantum Random Walks ◽  
Quantum Markov Chains

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.

scholarly journals A limit theorem for a splitting distribution of a quantum walk

2018 ◽  
Vol 16 (03) ◽  
pp. 1850023
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Random Walks ◽  
Discrete Time ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Unitary Evolution ◽  
Time Quantum ◽  
Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

scholarly journals Generalized Schur–Nevanlinna functions and their realizations

2020 ◽  
Vol 92 (5) ◽  
Author(s):  
Lassi Lilleberg
Keyword(s):  
Transfer Functions ◽  
Schur Functions ◽  
Inner Function ◽  
Pontryagin Space ◽  
State Spaces ◽  
Limit Values ◽  
Adjoint Systems ◽  
Generalized Schur Functions ◽  
Nevanlinna Functions ◽  
Weak Similarity

Abstract Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which allows one to generalize the notion of an inner function to Pontryagin space operator valued setting. Transfer functions of self-adjoint systems such that their state spaces are Pontryagin spaces, are generalized Nevanlinna functions, and symmetric generalized Schur functions can be realized as transfer functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion when a symmetric generalized Schur function is also a generalized Nevanlinna function is given. The criterion involves the negative index of the weak similarity mapping between an optimal minimal realization and its dual. In the special case corresponding to the generalization of an inner function, a concrete model for the weak similarity mapping can be obtained by using the canonical realizations.

scholarly journals Relation between random walks and quantum walks

2015 ◽  
Vol 91 (5) ◽  
Author(s):  
Stefan Boettcher ◽  
Stefan Falkner ◽  
Renato Portugal
Keyword(s):  
Random Walks ◽  
Quantum Walks
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