Entangled State Representation Research Articles

Wigner function of two-mode squeezed number state and its properties

By means of the entangled state representation as well as the Wigner operator in such a representation, the Wigner function of the two-mode squeezed number states is derived. The properties of corresponding Wigner function in the phase space are discussed by numerical calculation. The results indicate that Wigner function distribution in the phase space should be influenced by the correlation and the entanglement between two modes of two-mode squeezed number state.

A new finite-dimensional thermal coherent state and its Wigner distribution function

This paper constructs a new type of finite-dimensional thermal coherent states (FDTCS), which differs from the proceeding thermal coherent state in construction, as realisations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, it investigates the orthonormality and completeness relation of the FDTCS. Based on the thermal Wigner operator in the thermal entangled state representation, the Wigner function of the FDTCS is obtained. The nonclassical properties of the FDTCS are discussed in terms of the negativity of its Wigner function.

Optical entangled fractional Fourier transform derived via non-unitary SU(2) bosonic operator realization and its convolution theorem

Based on the two mutually conjugate tripartite entangled state representations (TESR), we introduce the 3-mode optical entangled fractional Fourier transform (EFrFT) through a new non-unitary bosonic operator realization of the SU(2) generators and the tripartite squeezing operator. The EFrFT, which is characteristic of the eigenmodes being three-variable Hermite polynomials, is not the direct product of three single-variable FrFTs. We show that the EFrFT adapts to the transform between two mutually conjugate quantum mechanical TESR, i.e., we can recast the EFrFT into the matrix element of expiπ2−αJ++J− in the TESR. In addition, we define the two functions' convolution in the EFrFT scheme and obtain the convolution theorem using the TESR. The derivation is concise and rigorous because our calculation is based on Dirac's powerful representation theory.

Two mutually conjugated tripartite entangled states and their fractional Fourier transformation kernel

We newly construct two mutually-conjugate tripartite entangled state representations, based on which we propose the formulation of three-mode entangled fractional Fourier transformation (EFFT) and derive the transformation kernel. The EFFT's additivity property is proved and the eigenmode of EFFT is derived. As an application, we calculate the EFFT of the three-mode squeezed vacuum state.

Solving nonlinear master equation describing quantum damping by virtue of the entangled state representation

This paper solves the newly constructed nonlinear master equation dρ/dt = κ[2f (N) aρ (1/f (N − 1))a† − a†aρ−ρa†a], where f(N) is an operator-valued function of N = a†a, for describing amplitude damping channel, and derives the infinite operator sum representation of quasi-Kraus operators for the density operator. It also shows that in this nonlinear process the initial pure number state density operator will evolve into the binomial field (a mixed state) when

Entangled state representation for describing both squeezing and entanglement involved in the parametric down-conversion process

By constructing a new orthogonal-complete quantum mechanical entangled state representation, we show that it can provide us with a combinative view of both squeezing and quantum entanglement involved in the parametric down-conversion process. The two-mode Wigner operator in this representation is also found.

Nonlinear two-mode squeezing obtained by analysing two-mode exponential quadrature operators in entangled state representation

By analysing the properties of two-mode quadratures in an entangled state representation (ESR) we derive from ESR some complicated exponential quadrature operators for nonlinear two-mode squeezing, which directly leads to wave function of the nonlinear squeezed state in ESR.

A New Approach for Constructing New Coherent-Entangled State Representations

A convenient approach for deriving new quantum mechanical coherent-entangled state representations is proposed to construct the integration of unity in normally ordered Gaussian operator forms and then decomposing it as projection operators. As examples, the n-mode complicated entangled state representations are derived and their application is presented.

From the Thermo Wigner Operator to the Thermo Husimi Operator in Thermo Field Dynamics

Using the thermo entangled state representation and based on the thermo Wigner operator we first introduce the thermo Husimi operator, which turns out to be a projective operator of the thermo squeezed coherent state, therefore the positive definite property of Husimi distribution function can be insured. In addition, we find the use of the thermo Husimi operator in simplifying the form of master equations.

Some New Properties of Wigner Function Studied in Entangled State Representation

Based on the Wigner operator in the entangled state representation we study some new important properties of Wigner function for bipartite entangled systems, such as size of an entangled state, upper bound of Wigner functions, etc. These discussions demonstrate the beauty and elegance of the entangled state representation.

Adaption of optical Fresnel transform to optical Wigner transform

Enlightened by the algorithmic isomorphism between the rotation of the Wigner distribution function (WDF) and the αth fractional Fourier transform, we show that the optical Fresnel transform performed on the input through an ABCD system makes the output naturally adapting to the associated Wigner transform, i.e. there exists algorithmic isomorphism between ABCD transformation of the WDF and the optical Fresnel transform. We prove this adaption in the context of operator language. Both the single-mode and the two-mode Fresnel operators as the image of classical Fresnel transform are introduced in our discussions, while the two-mode Wigner operator in the entangled state representation is introduced for fitting the two-mode Fresnel operator.

Inversion formula and Parseval theorem for complex continuous wavelet transforms studied by entangled state representation

In a preceding letter (2007 Opt. Lett. 32 554) we propose complex continuous wavelet transforms and found Laguerre-Gaussian mother wavelets family. In this work we present the inversion formula and Parseval theorem for complex continuous wavelet transform by virtue of the entangled state representation, which makes the complex continuous wavelet transform theory complete. A new orthogonal property of mother wavelet in parameter space is revealed.

New transformation of Wigner operator in phase space quantum mechanics for the two-mode entangled case

As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ (μ, ν) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to (a†1 – a2) and (a†1 – a2) can be solved and the contents of phase space quantum mechanics can be enriched, where ai, a†i are bosonic creation and annihilation operators, respectively.

New Approach for Solving Master Equations of Density Operators for the Raman-Coupled Model with Cavity Damping

By virtue of the thermo entangled state representation, in which one mode is fictitious accompanying the system mode, we exhibit a novel approach to deriving density operator for a Raman-coupled model with damping of the cavity mode. The normal ordering forms of density matrix elements can be obtained, and the corresponding Wigner functions are also derived.

Complex-Variable Fokker–Planck Equation Solved in Entangled State Representation

We find that the Fokker–Planck equation in complex variables can he conveniently solved in the context of bipartite entangled state representation and its relationship with SU(2) Lie algebraic generators' new realization {(1/4)[(Q1 – Q2)2 + (P1 + P2)2], (1/4)[(Q1 + Q2)2 + (P1 – P2)2], and – (i/2)(Q1P2 + Q2P1)}, the quadratic combination of canonical operators.

A new finite-dimensional pair coherent state studied by virtue of the entangled state representation and its statistical behavior

In this paper we construct a new type of finite-dimensional pair coherent states | ξ, q〉 as realizations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, the nonorthogonality and completeness properties of the state | ξ, q〉 are investigated. Based on the Wigner operator in the entangled state | τ〉 representation, the Wigner function of | ξ, q〉 is obtained. The properties of | ξ, q〉 are discussed in terms of the negativity of its Wigner function. The tomogram of | ξ, q〉 is calculated with the aid of the Radon transform between the Wigner operator and the projection operator of the entangled state | η, κ 1, κ 2〉. In addition, using the entangled state | τ〉 representation of | ξ, q〉 to show that the states | ξ, q〉 are just a set of energy eigenstates of time-independent two coupled oscillators.

The s-parameterized Weyl–Wigner correspondence in the entangled form and its applications

Based on the theory of integration within s-ordering of operators and the bipartite entangled state representation we introduce s-parameterized Weyl–Wigner correspondence in the entangled form. Some of its applications in quantum optics theory are presented as well.

Multipartite entangled state representation and squeezing of the n-pair entangled state

We show that the best way to study the relationship between quantum entanglement and quantum squeezing for the multimode case is through constructing the new n-pair entangled state representation . The explicit form of the 2n-mode squeezing operator which squeezes the n-pair entangled state can be directly derived via the transition from to where M is an n × n complex matrix, and the technique of integration within an ordered product of operators. We also analyze the squeezing properties of the 2n-mode squeezed state for M being Hermitian, and obtain the variances of the 2n-mode quadrature operators in the 2n-mode squeezed vacuum state with a concise result; the condition for reaching the minimum of the uncertainty relationship is also investigated.

Optical Operator Method in Two-Mode Case and Entangled Fresnel Operator's Decomposition

Based on the entangled Fresnel operator (EFO) proposed in [Commun. Theor. Phys. 46 (2006) 559], the optical operator method studied by the IWOP technique (Ma et al., Commun. Theor. Phys. 49 (2008) 1295) is extended to the two-mode case, which gives the decomposition of the entangled Fresnel operator, corresponding to the decomposition of ray transfer matrix [A, B, C, D]. The EFO can unify those optical operators in two-mode case. Various decompositions of EFO into the exponential canonical operators are obtained. The entangled state representation is useful in the research.

A New Kind of Integration Transformation in Phase Space Related to Two Mutually Conjugate Entangled-State Representations and Its Uses in Weyl Ordering of Operators

Based on the two mutually conjugate entangled state representations |ξ⟩ and |η⟩, we propose an integration transformation in ξ — η phase space , and its inverse transformation, which possesses some well-behaved transformation properties, such as being invertible and the Parseval theorem. This integral transformation is a convolution, where one of the factors is fixed as a special normalized exponential function. We generalize this transformation to a quantum mechanical case and apply it to studying the Weyl ordering of bipartite operators, regarding to (Q1 – Q2) ↔ (P1 – P2) ordered and simultaneously (P1 + P2) ↔ (Q1 + Q2) ordered operators.