Positive Wigner Function Research Articles

Continuous majorization in quantum phase space

We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson's theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.

Decoherence and nonclassicality of photon-added and photon-subtracted multimode Gaussian states

Photon addition and subtraction render Gaussian states non-Gaussian. We provide a quantitative analysis of the change in nonclassicality produced by these processes by analyzing the Wigner negativity and quadrature coherence scale (QCS) of the resulting states. The QCS is a recently introduced measure of nonclassicality [Phys. Rev. Lett. 122, 080402 (2019); Phys. Rev. Lett. 124, 090402 (2020)], which we show to undergo a relative increase under photon addition and subtraction that can be as large as 200%. This implies that the degaussification and the concomitant increase of nonclassicality come at a cost. Indeed, the QCS is proportional to the decoherence rate of the state so that the resulting states are considerably more prone to environmental decoherence. Our results are quantitative and rely on explicit and general expressions for the characteristic and Wigner functions of photon-added and -subtracted single- and multimode Gaussian states for which we provide a simple and straightforward derivation. These expressions further allow us to certify the quantum non-Gaussianity of the photon-subtracted states with positive Wigner function.

On a Recent Conjecture by Z. Van Herstraeten and N. J. Cerf for the Quantum Wigner Entropy

We address a recent conjecture stated by Z. Van Herstraeten and N. J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positive Wigner functions. Moreover, we are able to prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant, which is not very distant from the constant suggested by Van Herstraeten and Cerf. We also prove an analogous result for another conjecture stated by the same authors for the Rényi entropy of positive Wigner functions. As a by-product we prove a new inequality for the radar-ambiguity function (and for the Wigner distribution) which is reminiscent of Lieb’s inequalities.

Quantum Wigner entropy

We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $ln\ensuremath{\pi}+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics where it allows us to establish a Wigner entropy-power inequality. It also opens a way towards stronger entropic uncertainty relations. Finally, we define the Wigner-R\'enyi entropy of Wigner-positive states and conjecture an extended lower bound that is reached for Gaussian pure states.

Witnessing Wigner Negativity

Negativity of the Wigner function is arguably one of the most striking non-classical features of quantum states. Beyond its fundamental relevance, it is also a necessary resource for quantum speedup with continuous variables. As quantum technologies emerge, the need to identify and characterize the resources which provide an advantage over existing classical technologies becomes more pressing. Here we derive witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, which can be reliably measured using standard detection setups. They possess a threshold expectation value indicating whether the measured state has a negative Wigner function. Moreover, the amount of violation provides an operational quantification of Wigner negativity. We phrase the problem of finding the threshold values for our witnesses as an infinite-dimensional linear optimisation. By relaxing and restricting the corresponding linear programs, we derive two hierarchies of semidefinite programs, which provide numerical sequences of increasingly tighter upper and lower bounds for the threshold values. We further show that both sequences converge to the threshold value. Moreover, our witnesses form a complete family – each Wigner negative state is detected by at least one witness – thus providing a reliable method for experimentally witnessing Wigner negativity of quantum states from few measurements. From a foundational perspective, our findings provide insights on the set of positive Wigner functions which still lacks a proper characterisation.

Wigner negativity in the steady-state output of a Kerr parametric oscillator

The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, depending on the properties of the selected mode.

Violation of Bell inequalities by stochastic simulations of Gaussian States based on their positive Wigner representation

At first sight, the use of an everywhere positive Wigner function as a probability density to perform stochastic simulations in quantum optics seems equivalent to the introduction of local hidden variables, thus preventing any violation of Bell inequalities. However, because of the difference between symmetrically and normally ordered operators, some trajectories in stochastic simulations can imply negative intensities, despite a positive mean. Hence, Bell inequalities do not apply. Here, we retrieve for a weakly squeezed Gaussian state the maximum violation on polarization states allowed by quantum mechanics, for the Clauser-Horn-Shimony-Holt (CHSH), as well as for the Clauser-Horn Bell inequalities. For the case of the Clauser-Horn Bell inequality, the influence of the quantum efficiency of the detectors is studied, and for both inequalities, the influence of the degree of squeezing is assessed, as well as the uncertainty range versus the number of trajectories used in the simulations.

Analysis of single-particle nonlocality through the prism of weak measurements

Although regarded today as an important resource in quantum information, nonlocality has yielded over the years many conceptual conundrums. Among the latter are nonlocal aspects of single particles which have been of major interest. In this paper, the nonlocality of single quanta is studied in a square nested Mach–Zehnder interferometer with spatially separated detectors using a delayed choice modification of quantum measurement outcomes that depend on the complex-valued weak values. We show that if spacelike separated Bob and Alice are allowed to freely control their quantum devices, the geometry of the setup constrains the local hidden variable models. In particular, hidden signaling and a list of contextual instructions are required to split a quantum state characterized by a positive Wigner function into two quantum states with nonpositive Wigner functions. This implies that local hidden variable models could rely neither on only two hidden variables for position and momentum, nor on simultaneous factorizability of both the hidden probability densities and weights of splitting to reproduce the correct quantum distributions. While our analysis does not fully exclude the existence of nonfactorizable local hidden variable models, it demonstrates that the recently proposed weak values of quantum histories necessitate contextual splitting of prior commitments to measurement outcomes, due to functional dependence on the total Feynman sum that yields the complex-valued quantum probability amplitude for the studied quantum transition. This analysis also highlights the quantum nature of weak measurements.

Numerical study of Wigner negativity in one-dimensional steady-state resonance fluorescence

In a numerical study, we investigate the steady-state generation of nonclassical states of light from a coherently driven two-level atom in a one-dimensional waveguide. Specifically, we look for states with a negative Wigner function, since such nonclassical states are a resource for quantum information processing applications, including quantum computing. We find that a waveguide terminated by a mirror at the position of the atom can provide Wigner-negative states, while an infinite waveguide yields strictly positive Wigner functions. Moreover, our investigation reveals a connection between the purity of a quantum state and its Wigner negativity. We also analyze the effects of decoherence on the negativity of a state.

Disentropy of the Wigner function

In this work, the usage of the disentropy of the Wigner function as order and quantumness measure is investigated. Furthermore, the relative disentropy of the Wigner function as a distance measure between two Wigner functions is also discussed. The numerical calculation of the dynamic of the disentropy and relative disentropy of the Wigner function are realized considering an initially pure quantum state that evolves to a classical mixture and when an initially coherent state evolves to a Schrodinger cat state. It is shown that in all cases the disentropy is minimal for the quantum states with maximal quantumness, and it is maximal for maximally ordered states (pure states with positive Wigner function).

Arbitrarily large violations of non-contextuality in single mode photon states with positive Wigner function

Banaszek, Wódkiewicz and others (Banaszek and Wdkiewicz 1998 Phys. Rev. A 58 4345; Chen et al 2002 Phys. Rev. Lett. 88 040406; Chen and Zhang 2002 Phys. Rev. A 65 044102) made the surprising discovery that Einstein–Bell locality inequalities can be violated by the two mode squeezed vacuum by a factor , in spite of the fact that the state has a positive Wigner function. I use here the more general Gleason–Kochen–Specker assumption of non-contextuality (Gleason 1957 J. Math. Mech. 6 885) to express classicality. I then derive non-contextuality Bell inequalities for correlations of N pseudo spins embedded in an infinite dimensional continuous variable Hilbert space, and show that their maximum possible quantum violation is by a factor 2(N−1)/2. I find quantum states for which this maximum violation is reached. I also show that the familiar displaced squeezed vacuum for a single optical mode, which has a positive Wigner function, can violate the inequality by a factor for odd . The arbitrarily large non-classicality means that realizations of the pseudo-spin measurements even in a single mode photon state might afford similar opportunities in quantum information tasks as entangled N qubit systems with large N.

Resource theory of quantum non-Gaussianity and Wigner negativity

We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state set --- i.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone --- the Wigner logarithmic negativity. We use the latter to assess the resource content of experimentally relevant states --- e.g., photon-added, photon-subtracted, cubic-phase, and cat states --- and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to sub-universal and universal quantum information processing over continuous variables.

Higher-order stochastic differential equations and the positive Wigner function

General higher-order stochastic processes that correspond to any diffusion-type tensor of higher than second order are obtained. The relationship of multivariate higher-order stochastic differential equations with tensor decomposition theory and tensor rank is explained. Techniques for generating the requisite complex higher-order noise are proved to exist either using polar coordinates and $\ensuremath{\gamma}$ distributions, or from products of Gaussian variates. This method is shown to allow the calculation of the dynamics of the Wigner function, after it is extended to a complex phase space. The results are illustrated physically through dynamical calculations of the positive Wigner distribution for three-mode parametric downconversion, widely used in quantum optics. The approach eliminates paradoxes arising from truncation of the higher derivative terms in Wigner function time evolution. Anomalous results of negative populations and vacuum scattering found in truncated Wigner quantum simulations in quantum optics and Bose-Einstein condensate dynamics are shown not to occur with this type of stochastic theory.

Possibility to Probe Negative Values of a Wigner Function in Scattering of a Coherent Superposition of Electronic Wave Packets by Atoms.

Within a plane-wave approximation in scattering, an incoming wave packet's Wigner function stays positive everywhere, which obscures such purely quantum phenomena as nonlocality and entanglement. With the advent of the electron microscopes with subnanometer-sized beams, one can enter a genuinely quantum regime where the latter effects become only moderately attenuated. Here we show how to probe negative values of the Wigner function in scattering of a coherent superposition of two Gaussian packets with a nonvanishing impact parameter between them (a Schrödinger's cat state) by atomic targets. For hydrogen in the ground 1s state, a small parameter of the problem, a ratio a/σ_{⊥} of the Bohr radius a to the beam width σ_{⊥}, is no longer vanishing. We predict an azimuthal asymmetry of the scattered electrons, which is found to be up to 10%, and argue that it can be reliably detected. The production of beams with the not-everywhere-positive Wigner functions and the probing of such quantum effects can open new perspectives for noninvasive electron microscopy, quantum tomography, particle physics, and so forth.

Discrete Wigner Function Derivation of the Aaronson–Gottesman Tableau Algorithm

The Gottesman–Knill theorem established that stabilizer states and Clifford operations can be efficiently simulated classically. For qudits with odd dimension three and greater, stabilizer states and Clifford operations have been found to correspond to positive discrete Wigner functions and dynamics. We present a discrete Wigner function-based simulation algorithm for odd-d qudits that has the same time and space complexity as the Aaronson–Gottesman algorithm for qubits. We show that the efficiency of both algorithms is due to harmonic evolution in the symplectic structure of discrete phase space. The differences between the Wigner function algorithm for odd-d and the Aaronson–Gottesman algorithm for qubits are likely due only to the fact that the Weyl–Heisenberg group is not in S U ( d ) for d = 2 and that qubits exhibit state-independent contextuality. This may provide a guide for extending the discrete Wigner function approach to qubits.

Contractivity properties of a quantum diffusion semigroup

We consider a quantum generalization of the classical heat equation and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

Revealing quantum correlation by negativity of the Wigner function

We analyze two two-mode continuous variable separable states with the same marginal states. We adopt the definition of classicality in the form of well-defined positive Wigner function describing the state and find that although the states possess positive local Wigner functions, they exhibit negative Wigner functions for the global states. Using the negativity of Wigner function as an indicator of nonclassicality, we show that despite these states possess different negativities of the Wigner function, they do not reveal this difference as phase space nonclassicalities such as negativity of the Mandel $Q$ parameter or quadrature squeezing. We then concentrate on quantum correlation of these states and show that quantum discord and local quantum uncertainty, as two well-defined measures of quantum correlation, manifest the difference between negativity of the Wigner functions. The non-Gaussianity of these states is also examined and show that the difference in behavior of their non-Gaussianity is the same as the difference between negativity of their Wigner functions. We also investigate the influence of correlation rank criterion and find that when the states can be produced locally from classical states, the Wigner functions can not reveal their quantum correlations.

Proposal for a macroscopic test of local realism with phase-space measurements

We propose a new test of local realism based on correlation measurements of continuum valued functions of positions and momenta, known as modular variables. The Wigner representation of these observables are bounded in phase space, and therefore, the associated inequality holds for any state described by a non-negative Wigner function. This agrees with Bell's remark that positive Wigner functions, serving as a valid probability distribution over local (hidden) phase space coordinates, do not reveal non-locality. We construct a class of entangled states resulting in a violation of the inequality, and thus truly demonstrate non-locality in phase space. The states can be realized through grating techniques in space-like separated interferometric setups. The non-locality is verified from the spatial correlation data, collected from the screens.

Non-Gaussian pure states and positive Wigner functions

Non-Gaussian correlations in a pure state are inextricably linked with certain nonclassical features, such as a non-positive-definite Wigner function. In a commonly used simulation technique in ultracold atoms and quantum optics, known as the truncated Wigner method, the quantum dynamics is mapped to stochastic trajectories in phase space, governed by a positive approximation to the true Wigner distribution. The question thus arises: How accurate is this approach in predicting truly nonclassical behavior? In this article, we benchmark the ability of the truncated Wigner phase-space method to reproduce the non-Gaussian statistics of the single-mode anharmonic oscillator. We find that the this method can reliably predict departures from Gaussian statistics over a wide range of particle numbers, whereas the positive-$P$ representation, which involves no approximations, is limited by rapidly growing statistical uncertainty. The truncated Wigner function, furthermore, is able to reproduce the non-Gaussian correlations while satisfying the condition for purity.

Quantum non-Gaussian Depth of Single-Photon States.

We introduce and experimentally explore the concept of the non-Gaussian depth of single-photon states with a positive Wigner function. The depth measures the robustness of a single-photon state against optical losses. The directly witnessed quantum non-Gaussianity withstands significant attenuation, exhibiting a depth of 18dB, while the nonclassicality remains unchanged. Quantum non-Gaussian depth is an experimentally approachable quantity that is much more robust than the negativity of the Wigner function. Furthermore, we use it to reveal significant differences between otherwise strongly nonclassical single-photon sources.