A polarization correlation experiment with two maximally entangled photons created by spontaneous parametric down-conversion is studied in the Weyl-Wigner formalism, that reproduces the quantum predictions. An interpretation is proposed in terms of stochastic processes assuming that the quantum vacuum fields are real. This proves that local realism is compatible with the violation of Bell inequalities, thus rebutting the claim that local realism has been refuted by entangled photon experiments. Entanglement appears as a correlation between fluctuations of a signal field and vacuum fields.
An extension of the Weyl-Wigner-Moyal formulation of quantum mechanics suitable for a Dirac quantized constrained system is proposed. In this formulation, quantum observables are described by equivalent classes of Weyl symbols. The Weyl product of these equivalent classes is defined. The new Moyal bracket is shown to be compatible with the Dirac bracket for constrained systems.
The general Weyl -- Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd -- dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number -- phase Wigner function in quantum optics. Analogous results for the `almost' symmetric ordering in an even -- dimensional Hilbert space are obtained. Relations between the discrete Wigner funct…
The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space $L^{2}(\mathbb{R}^{3})\otimes{\mathcal{H}}^{(s+1)}$ and functions on the phase space $\mathbb{R}^{3}\times\mathbb{R}^{3}\times \{0,...,s\} \times\{0,...,s\}$ is found. The expressions for the Stratonovich-Weyl quantizer, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin $\frac{1}{2}$ nonrel…
Classical model of light in helicity formalism is presented. Then quantum point of view at photons -- construction and interpretation of photon wave function is proposed. Quantum mechanics of photon is investigated. The Białynicki -- Birula scalar product $\langle {\bf Ψ}_{1}|{\bf Ψ}_{2}\rangle_{BB}$ and the generalized Hermitian conjugation $\widehatγ^{+\hspace{-0.45em}+}$ of linear operator $\widehatγ$ are discussed. Quantum description of light on a phase space is developed. A photon Wigner function is built.
Evaluating the Wigner function of quantum states in the entangled state representation is introduced, based on which we present a new approach for deriving time evolution formula of Wigner function in laser process. Application of this fomula to calculating time evolution of photon number is also presented, as an example, the case when the initial state is photon-added coherent state is discussed.
The metaplectic covariance for all forms of the Weyl-Wigner-Groenewold-Moyal quantization is established with different realizations of the inhomogeneous symplectic algebra. Beyond that, in its most general form $W_{\infty}$ -covariance of this quantization scheme is investigated, and explicit expressions for the quantum-deformed Hamiltonian vector fields are presented. In a general basis the structure constants of the $W_{\infty}$-algebra are obtained and its subalgebras are analyzed.
The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group $G$ is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a `semiquantised phase space', a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space $T^*G$ and the Hilbert space of square integrable functions on $G$. General expressions for the star product for Weyl symbols are presented and explicitly worked ou…
As a natural extension of Fan's paper (arXiv: 0903.1769vl [quant-ph]) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation we find new two-fold complex integration transformation about the Wigner operator (in its entangled form) in phase space quantum mechanics and its inverse transformation. In this way, some operator ordering problems can be solved and the contents of phase space quantum mechanics can be enriched.
Based on the correspondence between Collins diffraction formula (optical Fresnel transform) and the transformation matrix element of a three-parameters two-mode squeezing operator in the entangled state representation (Opt. Lett. 31 (2006) 2622) we further explore the relationship between output field intensity determined by the Collins formula and the input field's probability distribution along an infinitely thin phase space strip both in spacial domain and frequency domain. The entangled Wigner function is introduced for recapitulating the result.
The Weyl-Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations is still missing. In this paper we present a covariant generalization of the Weyl order prescription and of the Weyl-Wigner map and use them to derive Vey quantum mechanics directly from the standard op…
We construct an explicit Wigner function for N-mode squeezed state. Based on a previous observation that the Wigner function describes correlations in the joint measurement of the phase-space displaced parity operator, we investigate the non-locality of multipartite entangled state by the violation of Zukowski-Brukner N-qubit Bell inequality. We find that quantum predictions for such squeezed state violate these inequalities by an amount that grows with the number N.
By introducing a new entangled state representation, we show that the Laguerre-Gaussian (LG) mode is just the wave function of the common eigenvector of the orbital angular momentum and the total photon number operators of 2-d oscillator, which can be generated by 50:50 beam splitter with the phase difference phi=Pi/2ϕ between the reflected and transmitted fields. Based on this and using the Weyl ordering invariance under similar transforms, the Wigner representation of LG is directly obtained, which can be considered as the generalized Wigner transform of Hermite Gaussian modes.
We derive the steady state solution of the Fokker-Planck equation that describes the dynamics of the nondegenerate optical parametric oscillator in the truncated Wigner representation of the density operator. The adiabatic limit of strong pump damping is assumed. This phase space image provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the OPO. A nongaussian distribution is obtained for the above threshold solution.
We first introduce the Wigner-Weyl-Moyal formalism for a theory whose phase-space is an arbitrary Lie algebra. We also generalize to quantum Lie algebras and to supersymmetric theories. It turns out that the non-commutativity leads to a deformation of the classical phase-space: instead of being a vector space it becomes a manifold, the topology of which is given by the commutator relations. It is shown in fact that the classical phase-space, for a semi-simple Lie algebra, becomes a homogenous symplectic manifold. The symplectic product is also deformed. We finally make some comments on how to …
By introducing the thermo entangled state representation, we convert the calculation of Wigner function (WF) of density operator to an overlap between "two pure" states in a two-mode enlarged Fock space. Furthermore, we derive a new WF evolution formula of any initial state in self-Kerr Medium with photon loss and find that the photon number distribution for any initial state is independent of the coupling factor with Kerr Medium, where the number state is not affected by the Kerr nonlinearity and evolves into a density operator of binomial distribution.
A description of scalar charged particles, based on the Feshbach-Villars formalism, is proposed. Particles are described by an object that is a Wigner function in usual coordinates and momenta and a density matrix in the charge variable. It is possible to introduce the usual Wigner function for a large class of dynamical variables. Such an approach explicitly contains a measuring device frame. From our point of view it corresponds to the Copenhagen interpretation of quantum mechanics. It is shown how physical properties of such particles depend on the definition of the coordinate operator. The…
Via the Weyl correspondence approach, we construct multipartite entangled state which is the common eigenvector of their center-of-mass coordinate and mass-weighted relative momenta. This approach is concise and effective for setting up the Fock representation of continuous multipartite entangled states. The technique of integration within an ordered product (IWOP) of operators is also essential in our derivation.
A Bell-type inequality is proposed in terms of Wigner-Yanase skew information, which is quadratic and involves only one local spin observable at each site. This inequality presents a hierarchic classification of all states of multipartite quantum systems from separable to fully entangled states, which is more powerful than the one presented by quadratic Bell inequalities from two-entangled to fully entangled states. In particular, it is proved that the inequality provides an exact test to distinguish entangled from nonentangled pure states of two qubits. Our inequality sheds considerable light…
The Weyl-Wigner-Moyal formalism of fermionic classical systems with a finite number of degrees of freedom is considered. This correspondence is studied by computing the relevant Stratonovich-Weyl quantizer. The Moyal $\star$-product, Wigner functions and normal ordering are obtained for generic fermionic systems. Finally, this formalism is used to perform the deformation quantization of the Fermi oscillator and the supersymmetric quantum mechanics.
We derive the steady state solution of the Fokker-Planck equation that describes the dynamics of the nondegenerate optical parametric oscillator in the truncated Wigner representation of the density operator. We assume that the pump mode is strongly damped, which permits its adiabatic elimination. When the elimination is correctly executed, the resulting stochastic equations contain multiplicative noise terms, and do not admit a potential solution. However, we develop an heuristic scheme leading to a satisfactory steady-state solution. This provides a clear view of the intracavity two-mode ent…
The differential structure of operator bases used in various forms of the Weyl-Wigner-Groenewold-Moyal (WWGM) quantization is analyzed and a derivative-based approach, alternative to the conventional integral-based one is developed. Thus the fundamental quantum relations follow in a simpler and unified manner. An explicit formula for the ordered products of the Heisenberg-Weyl algebra is obtained. The W(infty) -covariance of the WWGM-quantization in its most general form is established. It is shown that the group action of W(infty) that is realized in the classical phase space induces on bases…
In this paper, we use the displacement operator together with parity operation to construct the superposition of two coherent states. By transmitting this superposition from 50-50 beam splitter the two-mode qubit like ECS is generated. Moreover, we introduce a controllable method for producing qutrit like ECS using atom-field interaction in cavity QED and beam splitter. We will show that the distances of peaks of Wigner functions for reduced density matrices of two-mode ECS's are entanglement sensitive and can be a witness for entanglement. To confirm the results we use concurrence measure to …
The generation of entangled states that display negative values of the Wigner function in the quantum phase space is a challenging task, particularly elusive for massive, and possibly macroscopic, systems such as mechanical resonators. In this work, we propose two schemes based on reservoir engineering for generating Wigner-negative entangled states unconditionally. We consider two non-interacting mechanical resonators that are radiation-pressure coupled to either one or two common cavity fields; the optomechanical coupling with the field(s) features both a linear and quadratic part in the mec…
We show that qubit stabilizer states can be represented by non-negative quasi-probability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive state-independent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal formalism to three generators instead of two---producing an exterior, or Grassmann, algebra---which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a non-negative probability distribution can be associated with each …
The experimental test of Bell's inequality is mainly focused on Clauser-Horne-Shimony-Holt (CHSH) form, which provides a quantitative bound, while little attention has been pained on the violation of Wigner inequality (WI). Based on the spin coherent state quantum probability statistics we in the present paper extend the WI and its violation to arbitrary two-spin entangled states with antiparallel and parallel spin-polarizations. The local part of density operator gives rise to the WI while the violation is a direct result of non-local interference between two components of the entangled state…
A generalized Weyl quantization formalism for a particle on the circle investigated in \cite{1} is developed. A Wigner function for the state $\hat{\varrho}$ and the kernel $\mathcal{K}$ for a particle on the circle is defined and its properties are analyzed. Then it is shown how this Wigner function can be easily modified to give the number-phase Wigner function in quantum optics. Some examples of such number-phase Wigner function are considered.
The Weyl-Wigner-Moyal formalism is developed for spin by means of a correspondence between spherical harmonics and spherical harmonic tensor operators. The analogue of the Moyal expansion is developed for the Weyl symbol of the product of two operators in terms of the symbols for the individual operators, and it is shown that in the classical limit, the Weyl symbol for a commutator equals $i$ times the Poisson bracket of the corresponding Weyl symbols. It is also found that, to the same order, there is no correction in the symbol for the anticommutator.
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
A fluid analog of the information flux in the phase-space associated to purity and von Neumann entropy are identified in the Weyl-Wigner formalism of quantum mechanics. Once constrained by symmetry and positiveness, the encountered continuity equations provide novel quantifiers for non-classicality (non-Liouvillian fluidity) given in terms of quantum decoherence, purity and von Neumann entropy fluxes. Through definitions in the Weyl-Wigner formalism, one can identify the quantum fluctuations that distort the classical-quantum coincidence regime, and the corresponding quantum information profil…
Here, we clarify the physical aspects between the discrete Weyl-Wigner (W-W) formalism, well developed in condensed matter physics, and the so-called 'precise Weyl-Wigner calculus for lattice models' recently appearing in the literature. We point out that the use of compact continuous momentum space for a discrete lattice model is unphysically founded. It has an incommensurate phase space, highly unphysical, lacks the finite fields aspects, as exemplified by the Born-von Karman boundary condition of compactified Bravais lattice of solid-state physics, and leads to several ambiguities. This new…
We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a specific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.
Higher (2nd)-order Wigner distribution function in quantum phase space for entangled bi-modal coherent states, a representative of higher (2nd)-order optical-polarization, is introduced by generalizing kernel (transiting) operator in Cahill-Glauber C(s)-correspondence rule. The nature is analyzed which reveals the occurrence of oscillating three peaks: 'two' for individual bi-modes and third for interference between modes. Also, the graphics of 2nd-order polarization-Wigner distribution function, incisively, demonstrates that it is of non-Gaussian nature attaining non-negative values in quantu…
Classical and quantum mechanics across representations: an operational reading of the Wigner Weyl correspondence
University of Vienna, Faculty of Physics（维也纳大学物理学院）；Vienna Center for Quantum Science and Technology（维也纳量子科学与技术中心）；Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Constructor University（Constructor大学）；University of Geneva, Group of Applied Physics（日内瓦大学应用物理组）
The Weyl-Wigner representations for canonical thermal equilibrium quantum states are obtained for the whole class of quadratic Hamiltonians through a Wick rotation of the Weyl-Wigner symbols of Heisenberg and metaplectic operators. The behavior of classical structures inherently associated to these unitaries is described under the Wick mapping, unveiling that a thermal equilibrium state is fully determined by a complex symplectic matrix, which sets all of its thermodynamical properties. The four categories of Hamiltonian dynamics (Parabolic, Elliptic, Hyperbolic, and Loxodromic) are analyzed. …
A comprehensive theory of the Weyl-Wigner formalism for the canonical pair angle-angular momentum is presented, with special emphasis in the implications of rotational periodicity and angular-momentum discreteness.
Wigner's argument inferring Bell-type inequality for the EPR-Bohm entangled state is generalized here for any N-partite state. This is based on assuming for the relevant dichotomic observables the existence of the overall joint probability distributions, satisfying the locality condition, that would yield the measurable marginal probabilities. For any N, such Generalized Wigner's Inequality (GWI) is violated by quantum mechanics for all pure entangled states. The efficacy of GWI is probed, comparing with the Seevinck-Svetlichny multipartite Bell-type inequality, by calculating threshold visibi…
A general canonical transformation of mechanical operators of position and momentum is considered. It is shown that it automatically generates a parameter s which leads to a generalized (or s-parameterized) Wigner function. This allows one to derive a generalized (s-parameterized) Moyal brackets for any dimensions. In the classical limit the s-parameterized Wigner averages of the momentum and its square yield the respective classical values. Interestingly enough,in the latter case the classical Hamilton-Jacobi equation emerges as a consequence of such a transition only if there is a non-zero p…
Using operators' Weyl ordering expansion formula (Hong-yi Fan,\emph{\}J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator $Δ(q',p')$ ($q$-number transform) in phase space quantum mechanics, \[ \iint_{-\infty}^\infty dp' dq'/πΔ(q',p') e^{-2i(p-p') (q-q')} =δ(p-P) δ(q-Q), \] and its inverse \[\iint_{-\infty}^\infty dq dp δ(p-P) δ(q-Q) e^{2i(p-p') (q-q')}=Δ(q',p'), \] where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl ordering of op…
We present a family of methods, which can describe behaviour of quantum ensembles and demonstrate the creation of nontrivial (meta) stable states (patterns), localized, chaotic, entangled or decoherent from basic localized modes in collective models arising from the quantum hierarchy of Wigner-von Neumann-Moyal equations.
Squeezed state of light is one of the important subjects in quantum optics which is generated by optical nonlinear interactions. In this paper, we especially focus on qubit like entangled squeezed states (ESS's) generated by beam splitters, phase-shifter and cross Kerr nonlinearity. Moreover the Wigner function of two-mode qubit and qutrit like ESS are investigated. We will show that the distances of peaks of Wigner functions for two-mode ESS are entanglement sensitive and can be a witness for entanglement. Like the qubit cases, monogamy inequality is fulfilled for qutrit like ESS. These trend…
Based on Takahashi-Umezawa thermo field dynamics and the order-invariance of Weyl ordered operators under similar transformations, we present a new approach to deriving the exact Wigner functions of thermo number state, photon subtracted and added thermo vacuum state. We find that these Wigner functions are related to the Gaussian-Laguerre type functions of temperature, whose statistical properties are then analysed.
Similar in spirit to the preceding work [Opt. Commun. 282 (2009) 3734] where the relation between optical Fresnel transformation and quantum tomography is revealed, we study this kind of relationship in the two-mode entangled case. We show that under the two-mode Fresnel transformation the bipartite entangled state density |eta><eta| becomes density operator F_2|eta><eta|F_2 ^{dag}=|eta>_{r,s}<eta|, which is just the Radon transform of the two-mode Wigner operator (sigma,gama) in entangled form, where F_2 is an two-mode Fresnel operator in quantum optics, and s,r are the complex-value expressi…
We studied the effect of delocalized single-photon addition (DPA) on two input modes containing four cases: two independent coherent states (CSs), two independent thermal states (TSs), two independent single-mode squeezed vacuums (SVs), and an entangled two-mode squeezed vacuum (TMSV). In essence, four types of new non-Gaussian entangled light states are generated. We studied three different resources (including entanglement, discorrelation and Wigner negativity) for each two-mode light state. The output states after DPA are entangled, with more parameters and complex structures, characterizin…
Many non-classicality indicators are used to measure quantum effects of different systems. Kenfack's and Sadeghi's non-classicality indicators are introduced in terms of the amount of Wigner function's negativities and interferences in phase space quantum mechanics, respectively. They are, effectively, applied for some real distribution functions. In this paper, we compared these non-classicality indicators for the entangled state of photonic number states in the Wigner, Husimi and Rivier representations. It is shown that for a two-level entangled state, Sadeghi's indicator has more benefits w…
We derive a quantum kinetic model describing the dynamics of graphene electrons in phase space based on the Wigner--Weyl formalism. To take into account the quantum nature of the carriers, we make use of the quantum Liouville equation for the density matrix. By relating the density matrix elements with the Wigner function, the equation of motion for the latter is established, with the Coulomb interaction being introduced self-consistently (i.e., in the Hartree approximation). The long-wavelength limit for the plasmon dispersion relation is obtained, for both ungated and gated situations. As an…
For the first time we introduce the Husimi operator Delta_h(gamma,varepsilon;kappa) for studying Husimi distribution in phase space(gamma,varepsilon) for electron's states in uniform magnetic field, where kappa is the Gaussian spatial width parameter. Using the Wigner operator in the entangled state |lambda> representation [Hong-Yi Fan, Phys. Lett. A 301 (2002)153; A 126 (1987) 145) we find that Delta_h(gamma,varepsilon;kappa) is just a pure squeezed coherent state density operator |gamma,varepsilon>_kappa kappa<gamma,varepsilon|, which brings convenience for studying and calculating the Husim…
A relativistic phase-space representation for a class of observables with matrix-valued Weyl symbols proportional to the identity matrix (charge-invariant observables)is proposed. We take into account the nontrivial charge structure of the position and momentum operators. The evolution equation coincides with its analog in relativistic quantum mechanics with nonlocal Hamiltonian under conditions where particle-pair creation does not take place (free particle and constant magnetic field). The differences in the equations are connected with peculiarities of the constraints on the initial conditi…
We present uncertainty relations based on Wigner--Yanase--Dyson skew information with quantum memory. Uncertainty inequalities both in product and summation forms are derived. \mbox{It is} shown that the lower bounds contain two terms: one characterizes the degree of compatibility of two measurements, and the other is the quantum correlation between the measured system and the quantum memory. Detailed examples are given for product, separable and entangled states.
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 normal ordered, antinormally ordered and Weyl ordered product of operators as its special cases). The s-ordered operator expansion formula of density operators is derived, and the s-parameterized quantization scheme is thus completely established.
Quantum-mechanically correlated (entangled) states of many particles are of interest in quantum information, quantum computing and quantum metrology. Metrologically useful entangled states of large atomic ensembles have been experimentally realized, but these states display Gaussian spin distribution functions with a non-negative Wigner function. Non-Gaussian entangled states have been produced in small ensembles of ions, and very recently in large atomic ensembles. Here, we generate entanglement in a large atomic ensemble via the interaction with a very weak laser pulse; remarkably, the detec…
By introducing the thermo entangled state representation, we derived four new photocount distribution formulas for a given density operator of light field. It is shown that these new formulas, which is convenient to calculate the photocount, can be expressed as such integrations over Laguree-Gaussian function with characteristic function, Wigner function, Q-function, and P-function, respectively.
In continuous-variable quantum information, non-Gaussian entangled states that are obtained from Gaussian entangled states via photon subtraction are known to contain more entanglement. This makes them better resources for quantum information processing protocols, such as, quantum teleportation. We discuss the teleportation of non-Gaussian, non-classical Schrödinger-cat states of light using two-mode squeezed vacuum light that is made non-Gaussian via subtraction of a photon from each of the two modes. We consider the experimentally realizable cat states produced by subtracting a photon from t…
Tomographic probability representation of multimode electromagnetic field states in the scheme of center-of-mass tomography is reviewed. Both connection of the field state Wigner function and observable Weyl symbols with the center-of-mass tomograms as well as connection of Grönewold kernel with the center-of-mass tomographic kernel determining the noncommutative product of the tomograms are obtained. The dual center-of-mass tomogram of the photon states are constructed and the dual tomographic kernel is obtained. The models of other generalised center-of-mass tomographies are discussed. Examp…
A mapping between operators on the Hilbert space of $N$-dimensional quantum system and the Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich-Weyl kernel. It is shown that the moduli space of the Stratonovich-Weyl kernel is given by an intersection of the coadjoint orbit space of the $SU(N)$ group and a unit $(N-2)$-dimensional sphere. The general consideration is exemplified by a detailed description of the moduli space of 2, 3 and …
The family of unitary non-equivalent Weyl-Stratonovich kernels determining the Wigner probability distribution function of an arbitrary N-level quantum system is constructed.
Using the Wigner-Weyl mapping of quantum mechanics to phase space we consider exactly the quantum mechanics of an harmonic oscillator driven by an external white noise force or whose frequency is time dependent, either adiabatically or parametrically. We find several transition probabilities exactly. We also consider the (quantum mechanical) randomizing effects of the external white noise force on the Weyl quantized phase angle and upon other Weyl quantized quantities.
