A Kind of Four-Mode Continuous-Variable Entangled State Generated by Beamsplitter and Parametric Down-Conversions

We introduce a new four-mode continuous-variable entangled state in Fock space which can compose a new quantum mechanical representation. This state is important and can be common used because it can be generated by using one symmetrical beamsplitter and a pair of parameter down-convertors. Applying this new state in quantum information is described.     

New Tripartite Nonlinear Entangled State Representation in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators, we construct a new kind of tripartite nonlinear entangled state ｜α,γ）λ in 3-mode Fock space, which can make up a complete set. We also simply discuss its properties and application.     

New Tripartite Nonlinear Entangled State Representation in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators, we construct a new kind of tripartite nonlinear entangled state |α,γ〉_λ in 3-mode Fock space, which can make up a complete set. We also simply discuss its properties and application.     

Nonlinear Tripartite Entangled State Representation and Related Generalized Wigner Function

We construct the nonlinear tripartite entangled state representation and the related generalized Wigner operator. Then we discussed the Wigner functions of the nonlinear tripartite entangled state and the three-mode nonlinear squeezed vacuum state, and obtained the classical Weyl corresponding function of the three-mode nonlinear squeezed state.     

Nonlinear Tripartite Entangled State Representation and Related Generalized Wigner Function

We construct the nonlinear tripartite entangled state representation and the related generalized Wigner operator. Then we discussed the Wigner functions of the nonlinear tripartite entangled state and the three-mode nonlinear squeezed vacuum state, and obtained the classical Weyl corresponding function of the three-mode nonlinear squeezed state.     

New Three-Mode Squeezing Operators Gained via Tripartite Entangled State Representation

We show that the Agarwal-Simon representation of single-mode squeezed states can be generalized to find new form of three-mode squeezed states. We use the tripartite entangled state representations ｜p, y, z） and ｜x, u, v） to realize this goal.     

Relationship Between Wave Function and Corresponding Wigner Function Studied in Entangled State Representation

By using the explicit form of the entangled Wigner operator and the entangled state representation we derive the relationship between wave function and corresponding Wigner function for bipartite entangled systems. The technique of integration within an ordered product （IWOP） of operators is employed in our discussions.     

Coordinate Eigenstate Representation in Noncommutative Space as a New Entangled State in Quantum Mechanics

For the first time we construct the eigenstate ｜τ〉 of noncommutatlve coordinate. It turns out that｜τ〉 is an entangled state in the ordinary space. Remarkably, its Schmidt decomposition has definite expression in the coordinate representation and the momentum representation. The 〈τ｜ representation can simplify some calculations for obtaining energy level of two-dimensional oscillator in noncommutative space.     

A New Kind of Bipartite Entangled State Representation and its Application

We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new intermediate coordinate-momentum representation for a two-particle system is proposed and applied to some two-body dynamic problems.     

Tripartite Nonlinear Entangled State Representations in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators we construct a kind of tripartite nonlinear entangled states, which can make up a complete set. As its application, we also derive nonlinear 3-mode charge-related entangled state. The essential point for constructing these states lies in choosing the appropriate charge operator.     

Tripartite Nonlinear Entangled State Representations in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators we construct a kind of tripartite nonlinear entangled states, which can make up a complete set. As its application, we also derive nonlinear 3-mode charge-related entangled state. The essential point for constructing these states lies in choosing the appropriate charge operator.     

基于纠缠态表象和自旋相干态的新复变函数空间

本文利用有序算符内的积分(IWOP)技术,构造了一个基于单变量厄米多项式H2j(ξ*+τξ/2√τ)的新复变函数空间,该空间与纠缠态表象及施温格玻色环境下的自旋相干态有关。推导出了包含二元厄米多项式的二项式定理,有助于构造新的复变函数空间。同时还提出了一种新的基于H2j(ξ*+τξ/2√τ)的积分变换及其逆变换,这对于推导某些算符恒等式是很有用的。     

On the Entangled Fractional Fourier Transform in Tripartite Entangled State Representation

Based on the newly constructed two mutually conjugate 3-mode entangled states of continuum variablesin three-mode Fock space we introduce entangled fractional Fourier transform (EFFT) for the tripartite entangled staterepresentations, which are not a direct product of three 1-dimensional FFTs. The eigenmodes of EFFT are obtained,which is different from the usual Hermite polynomials. The EFFT of the three-mode squeezed state is derived.     

Application of Entangled State Representation to Deriving Normally Ordered Expansion of 1-Dimensional Coulomb Potential

Guided by Dirac’s advice that “When one has a particular problem to work out in quantum mechanics, one can minimize the labour by using a representation in which the representatives of the more important abstract quantities occurring in that problem are as simple as possible,’’ we construct the entangled state representation to derive the normally ordered expansion formula of the 1-dimensional two-body Coloumb potential. The method of integration within an ordered product of operators is also used. Further application of the new formula in some perturbation calculation is discussed.     

Entangled Fractional Fourier Transform for the Multipartite Entangled State Representation

We deduce entangled fractional Fourier transformation (EFFT) for the multipartite entangled state representation, which was newly constructed with two mutually conjugate n-mode entangled states of continuum variables in n-mode Fock space. We establish a formalism of EFFT for quantum mechanical wave functions, which provides us a convenient way to derive some wave functions. We find that the eigenmode of EFFT is different from the usual Hermite Polynomials. We also derive the EFFT of the n-mode squeezed state.     

从混合态构造相应的系统-热库纠缠纯态的新方法

在量子统计学和低温物理学中,常用密度算符(或混合态)来表示物理系综.本文旨在提出一个新方法能将混合态转化为系统与热库相纠缠的量子纯态,这样做不但可以彰显环境对系统的影响,探寻新的物理态,尤其是纠缠态,而且能将求系综平均的复杂计算转化为较为简洁的求纯态平均.我们的新方法是将相干态和有序算符内的积分理论结合起来,就可直接将混合态扩充为自由度加倍的纯态.     

Effective Approach to Constructing n-Mode Wigner Operator in the Entangled State Representation

By extending Fan-Klauder entangled state representation to multipartite case. We construct n-mode Wigner operator in the common eigenvector of the multipartite centre-of-mass coordinate and two mass-weighted relative momenta, and its canonical conjugate state, they are both more complicated entangled state of continuum variables. the technique of integration within an ordered product (IWOP) of operators is essential in our derivation.     

On the Entangled Fractional Fourier Transform in Tripartite Entangled State Representation

Based on the newly constructed two mutually conjugate 3-mode entangled states of continuum variables in three-mode Fock space we introduce entangled fractional Fourier transform (EFFT) for the tripartite entangled state representations, which are not a direct product of three 1-dimensional FFTs. The eigenmodes of EFFT are obtained,which is different from the usual Hermite polynomials. The EFFT of the three-mode squeezed state is derived.     

On Fermionic Entangled State Representation and Fermionic Entangled Wigner Operator

By analogy with the bosonic bipartite entangled state we construct fermionic entangled state with the Grassmann numbers. The Wigner operator in the fermionic entangled state representation is introduced, whose marginal distributions are understood in an entangled way. The technique of integration within an ordered product (IWOP) of Fermi operators is used in our discussion.