The Fourier slice transformation of the Wigner operator and the quantum tomogram of the density operator

Using the Weyl quantization scheme and based on the Fourier slice transformation (FST) of the Wigner operator, we construct a new expansion formula of the density operator ρ, with the expansion coefficient being the FST of ρ's classical Weyl correspondence, and the latter the Fourier transformation of ρ's quantum tomogram. The coordinate-momentum intermediate representation is used as the Radon transformation of the Wigner operator.     

Optical complex integration-transform for deriving complex fractional squeezing operator

We find a new complex integration-transform which can establish a new relationship between a two-mode operator's matrix element in the entangled state representation and its Wigner function. This integration keeps modulus invariant and therefore invertible. Based on this and the Weyl–Wigner correspondence theory, we find a two-mode operator which is responsible for complex fractional squeezing transformation. The entangled state representation and the Weyl ordering form of the two-mode Wigner operator are fully used in our derivation which brings convenience.     

量子扩散通道中Wigner算符的演化规律

众所周知,量子态的演化可用与其相应的Wigner函数演化来代替.因为量子态的Wigner函数和量子态的密度矩阵一样,都包含了概率分布和相位等信息,因此对量子态的Wigner函数进行研究,可以更加快速有效地获取量子态在演化过程的重要信息.本文从经典扩散方程出发,利用密度算符的P表示,导出了量子态密度算符的扩散方程.进一步通过引入量子算符的Weyl编序记号,给出了其对应的Weyl量子化方案.另外,借助于密度算符的另一相空间表示-Wigner函数,建立了Wigner算符在扩散通道中演化方程,并给出了其Wigner算符解的形式.本文推导出了Wigner算符在量子扩散通道中的演化规律,即演化过程中任意时刻Wigner算符的形式.在此结论的基础上,讨论了相干态经过量子扩散通道的演化情况.     

New optical field operator expansion in number state representation

By virtue of the Weyl ordering method,we find a new formalism of optical field operator expansion in number state representation.Miscellaneous optical fields’(coherent state,squeezed field,Wigner operator,etc.) new expansions are therefore exhibited.Some new generating functions of special polynomials are derived herewith.     

Radon transforms of the Wigner operator on hyperplanes

The generalization of tomographic maps to hyperplanes is considered. We find that the Radon transform of the Wigner operator in multi-dimensional phase space leads to a normally ordered operator in binomial distribution---a mixed-state density operator. Reconstruction of the Wigner operator is also feasible. The normally ordered form and the Weyl ordered form of the Wigner operator are used in our derivation. The operator quantum tomography theory is expressed in terms of some operator identities, with the merit of revealing the essence of the theory in a simple and concise way.     

复合函数算符的微商法则及其在量子物理中的应用

给出了在量子物理学、量子统计学、算符排序理论、矩阵论以及控制理论中有着重要用途的复合函数算符的一般微分法则,利用这一法则研究了Wigner算符和Weyl对应规则中的积分问题,证明了两类典型的算符恒等公式.给出了Wigner算符的有序算符内的微分形式,并得到了一些重要函数的新的微分式.最后,引入了一个参数型的Wigner算符来统一正规序、Weyl编序以及反正规序三种算符排序.     

Evolution of quantum states via Weyl expansion in dissipative channel

Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.     

On the Radon transformation of Wigner function altered with various optical processes

We discuss what happens to the Radon transformation of signal's Wigner functions (i.e., signal's Wigner transformation (WT)) if the signal function undergoes various optical processes, such as Fraunhofer diffraction, lens transformation and Fresnel diffraction, etc. Because the usual Wigner transforms can be studied via their corresponding transforms of the Wigner operator, we use the Weyl ordered form of the Wigner operator and the Weyl ordering invariance under similar transformations to derive the result, we find that the alteration of Radon transformation of signal's Wigner function (or named the variation of tomogram function), through these optical processes, can be ascribed to the variation of Radon transformation parameters once the parameter of WT is given.     

New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, \emph{ J. Phys.} A {\bf 25} (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator $\varDelta \left( q',p'\right) $ ($\mathrm{q}$-number transform) in phase space quantum mechanics, $\iint_{-\infty}^{\infty}\frac{{\rm d}p'{\rm d}q'}{\pi }\varDelta \left( q',p'\right) \e^{-2\i\left( p-p'\right) \left( q-q'\right) }=\delta \left( p-P\right) \delta \left( q-Q\right),$ and its inverse% $ \iint_{-\infty}^{\infty}{\rm d}q{\rm d}p\delta \left( p-P\right) \delta \left( q-Q\right) \e^{2\i\left( p-p'\right) \left( q-q'\right) }=\varDelta \left( q',p'\right),$ where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among $Q$--$P$ ordering, $P$--$Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.     

Applications of Weyl Ordered Two-Mode Wigner Operator for Quantum Mechanical Entangled System

Based on the technique of integral within a Weyl ordered product of operators, we present applications of the Weyl ordered two-mode Wigner operator for quantum mechanical entangled system, e.g., we derive the complex Wigner transform and its relation to the complex fractional Fourier transform, as well as the entangled Radon transform.     

Entanglement Rule of Wigner Operators at a Beamsplitter

It is known that beamsplitter can be used to produce quantum entanglement, in this paper we examine this topic from the point of view of Wigner operators. Using Weyl-ordering of the Wigner operator and the Weyl ordering invariance of Weyl ordered operators under similarity transformation we derive the entanglement rule of Wigner operators at a beamsplitter.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for derivingmiscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can alsobe easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transformsand the squeezing transforms in quantum optics is investigated.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for deriving miscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can also be easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transforms and the squeezing transforms in quantum optics is investigated.     

Weyl Ordering of Two-Mode Fresnel Operator and Its Applications

By using the Weyl ordering operator formula and the Weyl transformation rule, we derive Weyl ordering of the two-mode Fresnel operator, and then obtain its matrix element in the coordinate representation, which is the integral kernel of the generalized two-mode Fresnel transformation in classical optics. By means of the optical operator method, we obtain a decomposition of the two-mode Fresnel operator’s Weyl ordering and an operator identity.     

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,v ) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a₁⁺-a₂) and (a₁+a₂⁺) can be solved and the contents of phase space quantum mechanics can be enriched, where a_i,a_i⁺ are bosonic creation and annihilation operators, respectively.     

Unifying the theory of integration within normal-, Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators(which considers normally ordered,antinormally ordered and Weyl ordered product of operators as its special cases).The s-ordered operator expansion(denoted by...) formula of density operators is derived,which is ρ = 2 1 s ∫ d2βπβ|ρ |β exp { 2 s 1(s|β|2 β a + βa a a) }.The s-parameterized quantization scheme is thus completely established.     

Some Evolution Formulas on the Optical Fields Propagation in Realistic Environments

Evolution formulas of the density operator, the photon number distribution, and the Wigner function are derived for the problem on the optical fields propagation in realistic environments. Using the idea “reservoir modeled by beam splitter (BS)” and the Weyl expansion of the density operator, we obtain these formulas cleverly, which are very useful for quantum optics and quantum statistics. As an application, we study the time evolution of the photon number distribution and the Wigner function for single-photon-added coherent state in thermal environment.     

Mutual transformations between the P–Q,Q–P,and generalized Weyl ordering of operators

Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ok （p, q） with a real k parameter and can unify the P-Q, Q-P, and Weyl ordering of operators in k = 1, - 1,0, respectively, we find the mutual transformations between 6 （p - P） （q - Q）, （q - Q） 3 （p - P）, and （p, q）, which are, respectively, the integration kernels of the P-Q, Q-P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The - and - ordered forms of （p, q） are also derived, which helps us to put the operators into their - and - ordering, respectively.     

New equation for deriving pure state density operators by Weyl correspondence and Wigner operator

By virtue of the completeness of Wigner operator and Weyl correspondence we construct a general equation for deriving pure state density operators. Several important examples are considered as the applications of this equation, which shows that our approach is effective and convenient for deducing these entangled state representations.     

Re-explanation of Weyl Quantization Scheme via Weyl Ordering Approach

We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl ordered product of operators, i.e., the Weyl correspondence rule can be reconstructed by classical functions' Fourier transformation followed by an inverse Fourier transformation within Weyl ordering of operators. As an application of this reconstruction, we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.