Wigner Operator Research Articles

Generalized Entangled Wigner Operator for Unifying Three Quantization Schemes of Entangled Systems

For entangled systems since wave function for single particle is not physical, marginal distributions of the Wigner function for entangled states are only meaningful in the entangled state representation, we propose generalized Weyl quantization scheme which relies on the generalized entangled Wigner operator \({\Omega }_{k}\left (\sigma ,\gamma \right ) \) with a real k-parameter and which can unify \(\mathfrak {P}-\)ordering, \(\mathfrak {X}-\) ordering and Weyl ordering of operators in k = 1, −1, 0 respectively, we also find the mutual transformations among the integration kernel of \( \mathfrak {P}-\)ordering, \(\mathfrak {X}-\) ordering and generalized Weyl quantization schemes. The mutual transformations provides us with a new approach for deriving Wigner function of entangled quantum states. The \( \mathfrak {P}-\)ordered and \(\mathfrak {X}-\)ordered form of \({\Omega }_{k}\left (\sigma ,\gamma \right ) \) are also derived which helps to put operators into their \(\mathfrak {P}-\)ordering and \(\mathfrak {X}-\)ordering respectively.

From Weyl Expansion of Operator to Ordering of Operator and its New Applications

By using the IWOP technique, Weyl expansion of operator is derived. Based on this, three new ordering formulas of operators are presented, which have been applied to calculate Q-P and P-Q ordering of some operators. As other applications of Weyl expansion, the formula of photon counting is also easily obtained, which are related to Wigner and Q-function. In particular, noticing the Weyl ordering of displacement operator is itself, Weyl ordering invariance under similarity transformations is conveniently proved, instead of using of Wigner operator.

Time evolution of the coarse-graining-smoothed Wigner operator in an amplitude dissipative channel: from a pure state to a mixed state

Based on the solution to the master equation of the density operator describing the amplitude dissipative channel, we derive the time evolution law of the coarse-graining-smoothed Wigner operator in this channel, which demonstrates how an initial pure state evolves into a mixed state, exhibiting decoherence.

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 Ωk (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 δ(p — P)δ(q — Q), δ(q — Q)δ(p — P), and Ωk(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 \U0001d513— and \U0001d514— ordered forms of Ωk (p, q) are also derived, which helps us to put the operators into their \U0001d513— and \U0001d514— ordering, respectively.

Differential quotient rules of operator in composite function and its applications in quantum physics

Differential quotient rule of composite function operator and its applications in quantum physics, quantum statistics, operator ordering theory, matrix theory and control theory are given. The integration problem of Wigner operator and Weyl corresponding rules are studied. Two kinds of typical operator identity formulas are proved. The differential form of Wigner operator in ordered product of operators and new differential form of important functions are obtained. Finally, a Wigner operator with parameter for unifying regular order, Weyl sequencing and abnormal order is introduced.

Optical Tomography for Single- and Two-Mode Squeezed Chaotic Fields

Different from the previous methods of obtaining optical tomography of quantum states, in this paper we use the Radon transform between the single- (two-)mode Wigner operator and the pure-state density operator |q〉 f,g f,g 〈q| (|η,τ 1,τ 2〉〈η,τ 1,τ 2|) to analytically obtain optical tomograms for single- (two-)mode squeezed chaotic fields, where the states |q〉 f,g (|η,τ 1,τ 2〉) refer to the newly introduced intermediate coordinate-momentum (entangled) states. More interestingly, we find that the tomograms of squeezed chaotic fields and those of coherent states are related by a integration. Thus, we establish a new convenient approach of obtaining tomograms for quantum states.

New entangled state representation and its characteristics for the three compatible operators

The eigenvector set of |φ(x, r 1, r 2)〉 with three compatible operators (\(\hat x_1 + \hat x_2 + \hat x_3\), \(\hat p_2 + \hat p_3 - 2\hat p_1\) and \(\hat p_3 - \hat p_2\)) is constructed by virtue of Radon transformation of the Wigner operator and the technique of integration within an ordered product (IWOP) of operators. Its entanglement property is then revealed by deriving its standard Schmidt decomposition. |φ(x, r 1, r 2)〉 makes up a new quantum mechanical representation. A new three-mode squeezing operator is found by using this entangled state representation.

Tripartite entangled state, tripartite entangled Wigner operator and their generalization to an n-mode case

For entangled three particles one should treat their wave function as a whole. There is no physical meaning talking about the wave function (or Wigner function) for any one of the tripartite, and therefore considering the entangled Wigner function (Wigner operator) is of necessity. In this paper, we introduce a pair of mutually conjugate tripartite entangled state representations for defining the Wigner operator of entangled tripartite. Its marginal distributions and the Wigner function of the three-mode squeezed vacuum state are presented. Deriving wave function from its corresponding tripartite entangled Wigner function is also discussed. Moreover, through establishing the n-mode entangled state representation, we introduce the n-mode entangled Wigner operator, which would be more generally useful in quantum physics.

Wigner function, optical tomography of two-variable Hermite polynomial state, and its decoherence effects studied by the entangled-state representations

We analytically investigate the Wigner function (WF) and the optical tomography for the two-variable Hermite polynomial state (THPS) and the effect of decoherence on the THPS via the entangled-state representations. The nonclassicality of the THPS is investigated in terms of the partial negativity of the WF, which depends much on the polynomial orders m, n and the squeezing parameter r. We also extend recent theoretical studies of optical tomography and introduce the radon transformation between the Wigner operator and the projection operator of the entangled state |η,τ1,τ2〉 to derive the tomogram of the THPS. Furthermore, we investigate how the WF for the THPS evolves undergoing the thermal channel. The results show that quantum dissipation in the decoherence channel can thoroughly deteriorate the nonclassicality of the THPS, and thermal noise leads to much quicker decoherence than amplitude damping.

Entangled Projector and a New Approach for Obtaining the Entangled Wigner Operator

Like the coordinate projector |q〉〈q|=δ(q−Q), where Q is coordinate operator, we find that \(\pi\delta( \eta_{1}-\frac{Q_{1}-Q_{2}}{\sqrt{2}}) \delta( \eta_{2}-\frac{P_{1}+P_{2}}{\sqrt{2}}) \) is an entangled projector |η〉〈η|, where |η〉 is the bipartite entangled state and η=η1+iη2. We then derive the entangled Wigner operator in terms of the properties of the entangled projector. This seems a new approach for obtaining the entangled Wigner operator.

Squeezed displaced Wigner operator and its applications

In this paper, we introduce the squeezed displaced Wigner operator. We proved that the squeezed displaced Wigner operator can bring more convenience to calculate the Wigner functions of squeezed states. Finally, we give some new applications of such squeezed displaced Wigner operator.

Entanglement Swapping Transform Rule of Entangled Wigner Operators

For mixed input fields quantum information processing, it is very convenient to investigate a specified protocol by employ quasi-probability functions and characteristic functions in phase space. In this work, considering a nonlocal swapping operation labelled by \(\hat{E}_{s}\), we derive the entanglement swapping transform rule for entangled Wigner operators. The same rule can be obtained by implementing this nonlocal swapping operation via two entangled pairs channels. And then we apply this rule to examine how does the Wigner function for output states change to demonstrate the entanglement swapping. As a result, this transform rule can be utilized to investigate swapping operation for any two-body entangled system.

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.

New Properties of the Wigner Function of the Tripartite Entangled State

For entangled three particles one should treat their wave function as a whole, there is no physical meaning talking about the wave function (or Wigner function) for any one of the tripartite, therefore thinking of the entangled Wigner function (Wigner operator) is of necessity, we introduce the entangled Wigner operator related to a pair of mutually conjugate tripartite entangled state representations and discuss some of its new properties, such as the trace product rule, the size of an entangled quantum state and the upper bound of the three-mode Wigner function. Deriving wave function from its corresponding tripartite entangled Wigner function is also presented. Those new properties of the tripartite entangled Wigner function play significant role in quantum physics because they provide us deeper insight into the shape of quantum states.

Wilson lines and orbital angular momentum

We present an explicit realization of the Chen et al. approach to the proton spin decomposition in terms of Wilson lines, generalizing the light-front gauge-invariant extensions discussed recently by Hatta. Particular attention is drawn to the residual gauge freedom by further separating the pure-gauge term into contour and residual terms. We show that the kinetic orbital angular momentum operator can be expressed in terms of the Wigner operator only when the momentum variable is integrated over. Finally, we confirm from twist-2 arguments that the advanced, retarded and antisymmetric light-front canonical orbital angular momenta are the same.

逆坐标算符激发相干态的非经典性

In this paper,we introduce a new nonclassical quantum state,an inverse coordinate operator excited coherent state.By using the IWOP technique,the normally ordering form of inverse coordinate operator and the normalization factor are derived.It is shown that the normally ordering form is related to Hermit polynomial.On the basis of the coherent state representation of the Wigner operator,the Wigner function is also calculated,whose negative property is used to discuss the nonclassical property of the nonclassical quantum state.

A new bipartite entangled state describing the parametric down-conversion process and its applications in quantum optics

We construct a new bipartite entangled state (NBES), which describes both the squeezing and the entanglement involved in the parametric down-conversion process and can be produced using a symmetric beam splitter. Constructing asymmetric ket—bra integrations based on the NBES leads to some new squeezing operators, which clearly exhibit the relationships between squeezing and entangled state transformations. Moreover, an entangled Wigner operator with a definite physical meaning is also presented.

Localizability in de Sitter space

An analogue of the Newton–Wigner position operator is defined for a massive neutral scalar field in de Sitter space. The one-particle subspace of the theory, consisting of positive-energy solutions of the Klein–Gordon equation selected by the Hadamard condition, is identified with an irreducible representation of the de Sitter group. Postulates of localizability analogous to those written by Wightman for fields in Minkowski space are formulated on it, and a unique solution is shown to exist. Representations in both the principal and the complementary series are considered. A simple expression for the time evolution of the Newton–Wigner operator is presented.

S-Ordered Exponential of Quadratic Forms Gained via IWSOP Technique

Using the generalized \(\bar{s}\)-ordered Wigner operator, in which \(\bar{s}\) is a vector over the field of complex numbers, the technique of integration within an s-ordered product of operators (IWSOP) has been extended to multimode case. We derive the \(\bar{s}\)-ordered form of the widely applicable multimode exponential of quadratic form \(\exp\{\sum_{i,j = 1}^{n} a_{i}^{\dag}\varLambda_{ij}{a_{j}}\} \), each mode being in some particular order si, applying this method.

A generalized Weyl-Wigner quantization scheme unifying P—Q and Q—P ordering and Weyl ordering of operators

By extending the usual Wigner operator to the s-parameterized one aswith s being a real parameter, we propose a generalized Weyl quantization scheme which accompanies a new generalized s-parameterized ordering rule. This rule recovers P-Q ordering, Q-P ordering, and Weyl ordering of operators in s = 1, - 1, 0 respectively. Hence it differs from the Cahill-Glaubers' ordering rule which unifies normal ordering, antinormal ordering, and Weyl ordering. We also show that in this scheme the s-parameter plays the role of correlation between two quadratures Q and P. The formula that can rearrange a given operator into its new s-parameterized ordering is presented.