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.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.

Stochastics and DynamicsJul 3, 20214 citations

Higher-Dimensional Open Quantum Walk Constructed from Quantum Bernoulli Noises

Ce Wang

Quantum Bernoulli noises are annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relations (CAR) in equal-time. In this paper, we use quantum Bernoulli noises to introduce a model of open quantum walk on the $d$ -dimensional integer lattice $\mathbb{Z}^d$ for a general positive integer $d\geq 2$ , which we call the $d$ -dimensional open QBN walk. We obtain a quantum channel representation of the $d$ -dimensional open QBN walk, and find that it admits the ``separability-preserving'' property. We prove that, for a wide range of choices of its initial state, the $d$ -dimensional open QBN walk has a limit probability distribution of $d$ -dimensional Gauss type. Finally we unveil links between the $d$ -dimensional open QBN walk and the unitary quantum walk recently introduced in [Ce Wang and Caishi Wang, Higher-dimensional quantum walk in terms of quantum Bernoulli noises, Entropy 2020, 22, 504].Quantum Bernoulli noises are annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relations (CAR) in equal-time. In this paper, we use quantum Bernoulli noises to introduce a model of open quantum walk on the $d$-dimensional integer lattice $\mathbb{Z}^d$ for a general positive integer $d\geq 2$, which we call the $d$-dimensional open QBN walk. We obtain a quantum channel representation of the $d$-dimensional open QBN walk, and find that it admits the ``separability-preserving'' property. We prove that, for a wide range of choices of its initial state, the $d$-dimensional open QBN walk has a limit probability distribution of $d$-dimensional Gauss type. Finally we unveil links between the $d$-dimensional open QBN walk and the unitary quantum walk recently introduced in [Ce Wang and Caishi Wang, Higher-dimensional quantum walk in terms of quantum Bernoulli noises, Entropy 2020, 22, 504].

arXiv:2006.08090Jun 15, 20200 citations

Open Quantum Random Walks

Stéphane Attal, Francesco Petruccione, Christophe Sabot +1 more

arXiv:1402.3253Journal of Statistical PhysicsMay 30, 2012165 citations

d

-dimensional integer lattice

\mathbb{Z}^d

for a general positive integer

d\geq 2

, which we call the

d

-dimensional open QBN walk. We obtain a quantum channel representation of the

d

-dimensional open QBN walk, and find that it admits the ``separability-preserving'' property. We prove that, for a wide range of choices of its initial state, the

d

-dimensional open QBN walk has a limit probability distribution of

d

-dimensional Gauss type. Finally we unveil links between the

d

-dimensional open QBN walk and the unitary quantum walk recently introduced in [Ce Wang and Caishi Wang, Higher-dimensional quantum walk in terms of quantum Bernoulli noises, Entropy 2020, 22, 504].