The Wigner function shares several properties with classical distribution functions on phase space, but is not positive-definite. The integral of the Wigner function over a given region of phase space can therefore lie outside the interval [0,1]. The problem of finding best-possible upper and lower bounds for a given region is the problem of finding the greatest and least eigenvalues of an associated Hermitian operator. Exactly solvable examples are described, and possible extensions are indicated.
To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modelling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner-Weyl transformation, the modelling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conserv…
We discuss the case of a Markovian master equation for an open system, as it is frequently found from environmental decoherence. We prove two theorems for the evolution of the quantum state. The first one states that for a generic initial state the corresponding Wigner function becomes strictly positive after a finite time has elapsed. The second one states that also the P-function becomes exactly positive after a decoherence time of the same order. Therefore the density matrix becomes exactly decomposable into a mixture of Gaussian pointer states.
We investigate the relationship between two properties of quantum transformations often studied in popular subtheories of quantum theory: covariance of the Wigner representation of the theory and the existence of a transformation noncontextual ontological model of the theory. We consider subtheories of quantum theory specified by a set of states, measurements and transformations, defined specifying a group of unitaries, that map between states (and measurements) within the subtheory. We show that if there exists a Wigner representation of the subtheory which is covariant under the group of uni…
We derive simple formulas connecting the generalized Wigner functions for $s$-ordering with the density matrix, and vice-versa. These formulas proved very useful for quantum mechanical applications, as, for example, for connecting master equations with Fokker-Planck equations, or for evaluating the quantum state from Monte Carlo simulations of Fokker-Planck equations, and finally for studying positivity of the generalized Wigner functions in the complex plane.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators, which is a distribution uniquely characterized by the property that the marginals for all linear combinations of the given operators agree with the quantum mechanical distributions. Its role as a joint quasi-probability distribution is underlined by the property that its support always lies in the set of expectation value tuples of the operators. We characterize the set of singularities and positivity, and provide some basic examples.
We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current~$\bm J$ as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. Our central result is an explicit, elementary proof which shows that only systems up to quadratic in their pote…
The Wigner function W(q,p) is formulated as a phase-space path integral, whereby its sign oscillations can be seen to follow from interference between the geometrical phases of the paths. The approach has similarities to the path-centroid method in the configuration-space path integral. Paths can be classified by the mid-point of their ends; short paths where the mid-point is close to (q,p) and which lie in regions of low energy (low P function of the Hamiltonian) will dominate, and the enclosed area will determine the sign of the Wigner function. As a demonstration, the method is applied to a…
Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly-solvable eigenvalue prob…
A coarse grained Wigner distribution p_{W}(x,u) obeying positivity derives out of information-theoretic considerations. Let p(x,u) be the unknown joint PDF (probability density function) on position- and momentum fluctuations x,u for a pure state particle. Suppose that the phase part Psi(x,z) of its Fourier transform F.T.[p(x,u)]=|Z(x,z)|exp[iPsi(x,z)] is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an ext…
The integral of the Wigner function over a subregion of the phase-space of a quantum system may be less than zero or greater than one. It is shown that for systems with one degree of freedom, the problem of determining the best possible upper and lower bounds on such an integral, over all possible states, reduces to the problem of finding the greatest and least eigenvalues of an hermitian operator corresponding to the subregion. The problem is solved exactly in the case of an arbitrary elliptical region. These bounds provide checks on experimentally measured quasiprobability distributions.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators. We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. …
The Wigner function's behavior of accelerated and non-accelerated Greenberger Horne Zeilinger (GHZ) state is discussed. For the non-accelerated GHZ state, the minimum/maximum peaks of the Wigner function depends on the distribution's angles, where they are displayed regularly at fixed values of the distribution's angles. We show that, for the accelerated GHZ state, the minimum bounds increases as the acceleration increases. The increasing rate depends on the number of accelerated qubits. Due to the positivity/ negativity behavior of the Wigner function, one can use it as an indicators of the p…
The integral of the Wigner function of a quantum mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0,1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric disks and circles of given radius a. Thei…
We show that quantum circuits where the initial state and all the following quantum operations can be represented by positive Wigner functions can be classically efficiently simulated. This is true both for continuous-variable as well as discrete variable systems in odd prime dimensions, two cases which will be treated on entirely the same footing. Noting the fact that Clifford and Gaussian operations preserve the positivity of the Wigner function, our result generalizes the Gottesman-Knill theorem. Our algorithm provides a way of sampling from the output distribution of a computation or a sim…
We discuss the entanglement properties of bipartite states with Gaussian Wigner functions. Separability and the positivity of the partial transpose are characterized in terms of the covariance matrix of the state, and it is shown that for systems composed of a single oscillator for Alice and an arbitrary number for Bob, positivity of the partial transpose implies separability. However, this implications fails with two oscillators on each side, as we show by a five parameter family of explicit counterexamples.
We present a filtered backprojection algorithm for reconstructing the Wigner function of a system of large angular momentum j from Stern-Gerlach-type measurements. Our method is advantageous over the full determination of the density matrix in that it is insensitive to experimental fluctuations in j, and allows for a natural elimination of high-frequency noise in the Wigner function by taking into account the experimental uncertainties in the determination of j, its projection m, and the quantization axis orientation. No data binning and no arbitrary smoothing parameters are necessary in this …
By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of Wigner function (W). A non-integral representation is presented in terms of a quadratic form V*FV, where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending in a simple recursive way on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of Wigner function and the spacial uncertainty of the coherent state basis we us…
For Dirac equation, operator-invariants containing explicit time-dependence in parallel to known time-dependent invariants of nonrelativistic Schrödinger equation are introduced and discussed. As an example, a free Dirac particle is considered and new invariants are constructed for it. The integral of motion, which is initial Newton-Wigner position operator, is obtained explicitly for a free Dirac particle. For such particle with kick modeled by delta-function of time, the time-depending integral, which has physical meaning of initial momentum, is found.
The Lindblad equation governs general markovian evolution of the density operator in an open quantum system. An expression for the rate of change of the Wigner function as a sum of integrals is one of the forms of the Weyl representation for this equation. The semiclassical description of the Wigner function in terms of chords, each with its classically defined amplitude and phase, is thus inserted in the integrals, which leads to an explicit differential equation for the Wigner function. All the Lindblad operators are assumed to be represented by smooth phase space functions corresponding to …
The relation of the Wigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner--Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of…
Quantum correlations of observables for two particle states have demonstrated the nonlocal character of the quantum mechanics. However nonlocality can be exhibited even for noncommuting observables of a single particle system. In this paper we show nonlocality of position-momentum correlations of a single particle in the double-slit experiment modeled by an initially correlated Gaussian wavepacket. The positivity or negativity of the Wigner function for the state of the particle at the detection screen is related with the $σ_{xp}$ covariances. A Bell's inequality is constructed from the Wigner…
Starting from Feynman's Lagrangian description of quantum mechanics, we propose a method to construct explicitly the propagator for the Wigner distribution function of a single system. For general quadratic Lagrangians, only the classical phase space trajectory is found to contribute to the propagator. Inspired by Feynman's and Vernon's influence functional theory we extend the method to calculate the propagator for the reduced Wigner function of a system of interest coupled to an external system. Explicit expressions are obtained when the external system consists of a set of independent harmo…
The Wigner Phase Operator (WPO) is identified as an operator valued measure (OVM) and its eigen states are obtained. An operator satisfying the canonical commutation relation with the Wigner phase operator is also constructed and this establishes a Wigner distribution based operator formalism for the Wigner Phase Distribution. The operator satisfying the canonical commutation relation with the Wigner Phase Operator valued measure (WP-OVM) is found to be not the usual number operator. We show a way to overcome the non-positivity problem of the WP-OVM by defining a positive OVM by means of a pro…
On the basis of the phase states, we present the correct integral expressions of the two number-phase Wigner functions discovered so far. These correct forms are derived from those defined in the extended Fock space with negative number states. The analogous conditions to Wigner's original ones cannot lead to the number-phase function uniquely. To show this fact explicitly, we propose another function satisfying all these conditions. It is also shown that the ununiqueness of the number-phase Wigner function result from the phase-periodicity problem.
Wigner distribution function has much importance in quantum statistical mechanics. It finds applications in various disciplines of physics including condense matter, quantum optics, to name but a few. Wigner distribution function is introduced by E. Wigner in 1932. However, there is no analytical derivation of Wigner distribution function in the literatures, to date. In this paper, a simple analytical derivation of Wigner distribution function is presented. Our derivation is based on two assumptions, these are A) by taking the integral of Wigner distribution function, with respect to configura…
We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton's identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schrödinger's uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy.
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson's Theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We re…
Bell conjectured that a positive Wigner function does not allow violation of the inequalities imposed by local hidden variable theories. A requirement for this conjecture is "when phase space measurements are performed". We introduce the theory-independent concept of "operationally local transformations" which refers to the change of the switch on a local measurement apparatus. We show that two separated parties, performing only phase space measurements on a composite quantum system with a positive Wigner function and performing only operationally local transformations that preserve this posit…
For an ergodic system, the time average of a classical observable coincides with that obtained via the Liouville probability density, a delta-function on the energy shell. Reinterpreting this distribution as a Wigner function, that is, the Weyl representation of a density operator, the quantum ergodic conjecture identifies this classical construction as an approximate representation of the quantum eigenstate of the same energy. It is found that this reasonable hypothesis, as far as expectations for observables are concerned, does not satisfy the requirement of positivity, so that the delta-fun…
Phase space reflection operators lie at the core of the Wigner-Weyl representation of density operators and observables. The role of the corresponding classical reflections is known in the construction of semiclassical approximations to Wigner functions of pure eigenstates and their coarsegrained microcanonical superpositions, which are not restricted to classically integrable systems. In their active role as unitary operators, they generate transitions between pairs of eigenstates specified by transition Wigner functions (or cross-Wigner functions): The square modulus of the transition Wigner…
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）
We consider a set of operators hat{x}=(hat{x}_1,..., hat{x}_N) with diagonal representatives P(n) in the space of generalized coherent states |n>; hat{x}=int dn P(n) |n><n|. We regularize the coherent-state path integral as a limit of a sequence of averages <.>_L over polygonal paths with L vertices {n_1...L}. The distribution of the path centroid bar{P}=(1/L) sum_{i=1}^{L}P(n_i) tends to the Wigner function W(x), the joint distribution for the operators: W(x)=lim_{L->infinity} <delta(x-bar{P})>_{L}. This result is proved in the case where the Hamiltonian commutes with hat{x}. The Wigner funct…
We study the high-temperature behavior of quantum-mechanical path integrals. Starting from the Feynman-Kac formula, we derive a new functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities. As shown by its applications to different potentials, the presented expansion turns out to be quite efficient in generating analytic form of the higher-order expansion coefficients. To put some flesh on the bare bones we apply the expansion to obtain basic thermodynamic functions of the one-dimensional anharmonic oscillator. Further salient issues, such as gene…
In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on $\mathbb R^2$ which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < η\leq 1$, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the $\mathbb L_\infty$-risk over a class of infinitely differentiable. We compute also the lower …
Propagation of the Wigner function is studied on two levels of semiclassical propagation, one based on the van-Vleck propagator, the other on phase-space path integration. Leading quantum corrections to the classical Liouville propagator take the form of a time-dependent quantum spot. Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic. It can be interpreted as the result of interference of a \emph{pair} of classical trajectories, indicating how quantum coherences are to be propagated semiclassically in phase space. The phase-space path-integral…
This article has to do with the derivation and solution of the Fokker--Planck equation associated to the momentum-independent Wigner function, of a particle subjected to a harmonic external field and immersed in a ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung and Grabert of interpreting the operators as c-numbers to derive the adjoint equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the in…
The phase-space path-integral approach to the damped harmonic oscillator is analyzed beyond the Markovian approximation. It is found that pairs of nonclassical trajectories contribute to the path-integral representation of the Wigner propagating function. Due to the linearity of the problem, the sum coordinate of a pair still satisfies the classical equation of motion. Furthermore, it is shown that the broadening of the Wigner propagating function of the damped oscillator arises due to the time-nonlocal interaction mediated by the heat bath.
We employ the Wigner function formalism to simulate partially coherent, dissipative electron transport in biased semiconductor superlattices. We introduce a model collision integral with terms that describe energy dissipation, momentum relaxation, and the decay of spatial coherences (localization). Based on a particle-based solution to the Wigner transport equation with the model collision integral, we simulate quantum electronic transport at 10 K in a GaAs/AlGaAs superlattice and accurately reproduce its current density vs field characteristics obtained in experiment.
The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. General features of time-independent Wigner functions are explored here, including the functional ("star") eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux ("supersymmetric") isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear pot…
We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems. The state is represented through the Wigner function, a generalized probability density on $\mathbb{R}^2$ which may take negative values and must respect intrinsic positivity constraints imposed by quantum physics. The effect of the losses due to detection inefficiencies, which are always present in a real experiment, is the addition to the tomographic data of independent Gaussian noise. We construct a kernel estimator for the Wigner function, prove tha…
A well known result of P. Flandrin states that a Gaussian uniquely maximizes the integral of the Wigner distribution over every centered disc in the phase plane. While there is no difficulty in generalizing this result to higher-dimensional poly-discs, the generalization to balls is less obvious. In this note we provide such a generalization.
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equatio…
Using the quadrature bases that incorporate the spatiotemporal degrees of freedom, we develop a Wigner functional theory for quantum optics, as an extension of the Moyal formalism. Since the spatiotemporal quadrature bases span the complete Hilbert space of all quantum optical states, it does not require factorization as a tensor product of discrete Hilbert spaces. The Wigner functions associated with such a space become functionals and operations are expressed by functional integrals -- the functional version of the star product. The resulting formalism enables tractable calculations for scen…
A gauge-invariant Wigner quantum mechanical theory is obtained by applying the Weyl-Stratonovich transform to the von Neumann equation for the density matrix. The transform reduces to the Weyl transform in the electrostatic limit, when the vector potential and thus the magnetic field are zero. Both cases involve a center-of-mass transform followed by a Fourier integral on the relative coordinate introducing the momentum variable. The latter is continuous if the limits of the integral are infinite or, equivalently, the coherence length is infinite. However, the quantum theory involves Fourier t…
