Quantum Bernoulli Noises Research Articles

Quantum Exclusion Process driven by Brownian Motions in terms of quantum Bernoulli noises

Quantum Bernoulli noises are the family of annihilation and creation operators on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. The paper is devoted to the study of Quantum Exclusion Process driven by Brownian Motions in terms of Quantum Bernoulli Noises. We first study the existence and uniqueness of regular solutions to the classical Quantum Exclusion Process driven by Brownian Motions. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the Quantum Exclusion Process of classical Markov chains. Finally, we prove that the existence of regular invariant measures for Quantum Exclusion Process driven by Brownian Motions in terms of Quantum Bernoulli Noises.

Quantum data compression under localized features

In this paper, we first introduce the local quantum information processing task by compressing the density operator based on local quantum Bernoulli noises. Then, the local quantum compression theorem is given, that is, the local quantum entropy is the minimum achievable rate of local compression. Finally, we prove the theorem with the proof of the direct coding theorem and the inverse theorem. The direct coding theorem shows that a scheme with such a local compression rate exists, and that this local compression rate converges to the local quantum entropy. The inverse theorem shows that the compression scheme with the rate below the local entropy is unachievable. With the continuous development of quantum technology, local quantum data compression technology will have broad application prospects and development space.

Quantum Bernoulli noises approach to quantum master equations and applications to Ehrenfest-type theorems

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables.

Entropy exchange and coherent information based on local quantum Bernoulli noises

Entropy exchange and coherent information based on local quantum Bernoulli noises

Quantum conditional entropy and classical-quantum conditional entropy with localization characteristics

In the era of rapid development of quantum mechanics, some issues in the field of quantum information have become increasingly important. Local quantum Bernoulli noises (LQBNs) is a localized Bernoulli functional, and it is the localization of quantum Bernoulli noises (QBNs). The distinctive feature of LQBNs lies in its localization properties, which make it highly valuable in the field of quantum information. In this paper, we study three new types of quantum conditional entropy with localization characteristics from three dimensions: local quantum conditional entropy, local quantum conditional entropy at the same position, and classical-quantum conditional entropy. Based on the locality of LQBNs, we prove that the local quantum conditional entropy is concave and unitary invariant. Surprisingly, we find that the local quantum mutual information (entropy) corresponding to the same position is 0. The local quantum conditional entropy defined based on the local quantum state at a specific position, proposed in this paper, provides new insights into understanding the impact of local quantum noise on the information transmission in quantum channels. This discovery not only deepens our understanding of the internal structure of local quantum entropy, but also reveals its influence on the performance of quantum communication.

Environment Decoherence of Quantum Exclusion Semigroups in Terms of Quantum Bernoulli Noises

In this paper, we study the environment induced decoherence of quantum exclusion semigroups constructed by quantum Bernoulli noises. We first study the structure of the decoherence-free algebra and discuss some of its properties, and give the conditions for its possible mutual inclusions with the set of fixed points. Then we describe the structure of the set of fixed points and use the new results obtained here to discuss the environment induced decoherence properties of the semigroup.

Quantum conditional entropy based on local quantum Bernoulli noises

Quantum noise has always been an important issue in the field of quantum information transmission. As the localization of quantum noise, local quantum Bernoulli noises (LQBNs) have attracted extensive attention in recent years. In this paper, the quantum conditional entropy based on LQBNs is deeply discussed, and a new definition of quantum conditional entropy is given. Then we find that this quantum conditional entropy has some basic properties such as concavity, conditional reduced entropy and unitary invariance. This research is of great significance to the field of quantum information, which can help us to understand the influence of local quantum noises on quantum information transmission.

Recurrence and transience of quantum Markov semigroups constructed form quantum Bernoulli noises

Quantum exclusion semigroup constructed form quantum Bernoulli noises can be written on diagonal and off-diagonal operators space, respectively. The diagonal parts describe a classical Markov process. In this paper, we first produce a decomposition into the sum of irreducible components determined by the class states of the associated classical Markov process and prove that each irreducible component is recurrent or transient if and only if its diagonal restriction is recurrent or transient. We then give conditions for existence of invariant states and study properties of invariant states such as detailed balance, uniqueness and convergence towards as time approaches infinity.

Quantum Markov semigroup for open quantum system interacting with quantum Bernoulli noises

Quantum Bernoulli noises (QBNs) refer to the annihilation and creation operators acting on the space [Formula: see text] of square integrable Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. In this paper, we consider the Markov evolution of an open quantum system interacting with QBNs. Let [Formula: see text] be the system space of an open quantum system interacting with QBNs. Then [Formula: see text] just describes the coupled quantum system. In the framework of [Formula: see text], we first construct a quantum Markov semigroup that respects the interactions between the system and QBNs, and then we prove that under some mild conditions the semigroup has faithful invariant states. To support our main results mentioned above, we prove several technical propositions and theorems about operators defined in [Formula: see text]. Some other results are also obtained.

Interacting Stochastic Schrödinger Equation

Being the annihilation and creation operators on the space h of square integrable Bernoulli functionals, quantum Bernoulli noises (QBN) satisfy the canonical anti-commutation relation (CAR) in equal time. Let K be the Hilbert space of an open quantum system interacting with QBN (the environment). Then K⊗h just describes the coupled quantum system. In this paper, we introduce and investigate an interacting stochastic Schrödinger equation (SSE) in the framework K⊗h, which might play a role in describing the evolution of the open quantum system interacting with QBN (the environment). We first prove some technical propositions about operators in K⊗h. In particular, we obtain the spectral decomposition of the tensor operator IK⊗N, where IK means the identity operator on K and N is the number operator in h, and give a representation of IK⊗N in terms of operators IK⊗∂k*∂k, k≥0, where ∂k and ∂k* are the annihilation and creation operators on h, respectively. Based on these technical propositions as well as Mora and Rebolledo’s results on a general SSE, we show that under some mild conditions, our interacting SSE has a unique solution admitting some regularity properties. Some other results are also proven.

Ergodicity of exclusion semigroups constructed from quantum Bernoulli noises

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an Abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.

The uniform measure for quantum walk on hypercube: A quantum Bernoulli noises approach

In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube, and then, based on the alternative description, we find that the operators ∂k*+∂k behave actually as the shift operators, where ∂k and ∂k* are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the above-mentioned operators as the shift operators on the position space, we introduce a discrete-time quantum walk model on a general hypercube and obtain an explicit formula for calculating its probability distribution at any time. We also establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution. Finally, we show that the walk produces the uniform measure as its stationary measure on the hypercube provided its initial state satisfies some mild conditions. Some other results are also proven.

Quantum channel measurement with local quantum Bernoulli noises

As an important stochastic process, quantum Bernoulli noises has a very important physical background and is an important research object in the field of quantum information. In this paper, we review local quantum Bernoulli noises and local quantum mutual entropy, then introduce quantum channel measurement with local quantum Bernoulli noises. On this basis, we give the channel structure between two systems, and prove the completely positivity of this quantum channel. We also give a channel application on the local quantum mutual entropy.

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

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.

Quantum mutual entropy in terms of local quantum Bernoulli noises

Localization of quantum Bernoulli noises(LQBNs) are the family of local annihilation and local creation operator acting on Bernoulli functionals. In this paper, we give the local quantum mutual entropy based on the local quantum entropy Furthermore, we demonstrate that this quantum mutual entropy also has the basic properties.

Continuous-time quantum walk on graph associated with quantum Bernoulli noises

Continuous-time quantum walk on graph associated with quantum Bernoulli noises

Quantum entropy in terms of local quantum Bernoulli noises and related properties

Localization of quantum Bernoulli noises (LQBNs) are the family of local annihilation and local creation operators acting on Bernoulli functionals. In this paper, we construct a density operator ρk represented by LQBNs, and define a new quantum entropy based on LQBNs. Furthermore, we demonstrate that this quantum entropy also has the basic properties of von Neumann entropy, such as concavity, subadditivity, and nonnegativity. In particular, we obtain the necessary and sufficient condition for

Quantum Feller semigroup in terms of quantum Bernoulli noises

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.

Quantum Bernoulli noises approach to Stochastic Schrödinger equation of exclusion type

Stochastic Schrödinger equations are a special type of stochastic evolution equations in complex Hilbert spaces, which arise in the study of open quantum systems. Quantum Bernoulli noises refer to annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we investigate a linear stochastic Schrödinger equation of exclusion type in terms of quantum Bernoulli noises. Among others, we prove the well-posedness of the equation, illustrate the results with examples, and discuss the consequences. Our main work extends that of Chen and Wang [J. Math. Phys. 58(5), 053510 (2017)].

Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises.

As a discrete-time quantum walk model on the one-dimensional integer lattice , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.