When is the Wigner function of multidimensional systems nonnegative?

Soto, Felipe Ruan; Claverie, P.

doi:10.1063/1.525607

Cited by 130 publications

(90 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence, including a negative Wigner function element is mandatory in order to design a CV sub-universal quantum circuit that cannot be efficiently simulated by a classical device. By virtue of the Hudson theorem [10], this necessarily corresponds to the use of non-Gaussian resources.…”

Section: Introductionmentioning

confidence: 99%

Continuous-variable sampling from photon-added or photon-subtracted squeezed states

et al. 2017

View full text Add to dashboard Cite

We introduce a new family of quantum circuits in continuous variables and we show that, relying on the widely accepted conjecture that the polynomial hierarchy of complexity classes does not collapse, their output probability distribution cannot be efficiently simulated by a classical computer. These circuits are composed of input photon-subtracted (or photon-added) squeezed states, passive linear optics evolution, and eight-port homodyne detection. We address the proof of hardness for the exact probability distribution of these quantum circuits by exploiting mappings onto different architectures of sub-universal quantum computers. We obtain both a worst-case and an average-case hardness result. Hardness of Boson Sampling with eight-port homodyne detection is obtained as the zero squeezing limit of our model. We conclude with a discussion on the relevance and interest of the present model in connection to experimental applications and classical simulations.

show abstract

Section: Introductionmentioning

confidence: 99%

Continuous-variable sampling from photon-added or photon-subtracted squeezed states

et al. 2017

View full text Add to dashboard Cite

show abstract

“…Indeed Hudson [4] showed that the only pure states ψ(x) for which the Wigner function is non-negative are Gaussian in x. This carries over to any number of dimensions [6], and also, for odd dimensions at least, to the formulation of discrete Wigner functions [7].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, if any Wigner function is convoluted, or smeared, by integration with respect to the Wigner function of the vacuum state, itself a gaussian function on phase space, then the smoothed function, called the Q-function (or Husimi function), is non-negative and corresponds to an ordering in Cohen's class different from that of Wigner and Weyl [9]. More generally, if any Wigner function is convoluted with respect to a Gaussian function which is itself the Wigner function of a pure coherent state, then the result is non-negative [6,9,10,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

GeneralizedQ-functions

Smith¹

2006

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

The modulus squared of a class of wave functions defined on phase space is used to define a generalized family of Q or Husimi functions. A parameter λ specifies orderings in a mapping from the operator |ψ σ| to the corresponding phase space wave function, where σ is a given fiducial vector. The choice λ = 0 specifies the Weyl mapping and the Q-function so obtained is the usual one when |σ is the vacuum state. More generally, any choice of λ in the range (−1, 1) corresponds to orderings varying between standard and anti-standard. For all such orderings the generalized Q-functions are non-negative by construction. They are shown to be proportional to the expectation of the system stateρ with respect to a generalized displaced squeezed state which depends on λ and position (p, q) in phase space. Thus, when a system has been prepared in the stateρ, a generalized Q-function is proportional to the probability of finding it in the generalized squeezed state. Any such Q-function can also be written as the smoothing of the Wigner function for the system stateρ by convolution with the Wigner function for the generalized squeezed state.

show abstract

“…All WDFs of higher-order functions have regions with negative values. 7 Due to these negative values the WDF is able to account for interference effects. Table 1: The WDF of some commonly used electric fields.The propagation distance along the optical axis is represented by z, the wavenumber by k and the two-dimensional spatial frequencies by the vector k.…”

mentioning

confidence: 99%

A Wigner-based ray-tracing method for imaging simulations

Mout

Wick

Bociort

et al. 2015

Optical Systems Design 2015: Computational Optics

View full text Add to dashboard Cite

The Wigner Distribution Function (WDF) forms an alternative representation of the optical field. It can be a valuable tool for understanding and classifying optical systems. Furthermore, it possesses properties that make it suitable for optical simulations: both the intensity and the angular spectrum can be easily obtained from the WDF and the WDF remains constant along the paths of paraxial geometrical rays. In this study we use these properties by implementing a numerical Wigner-Based Ray-Tracing method (WBRT) to simulate diffraction effects at apertures in free-space and in imaging systems. Both paraxial and non-paraxial systems are considered and the results are compared with numerical implementations of the Rayleigh-Sommerfeld and Fresnel diffraction integrals to investigate the limits of the applicability of this approach. The results of the different methods are in good agreement when simulating free-space diffraction or calculating point spread functions (PSFs) for aberration-free imaging systems, even at numerical apertures exceeding the paraxial regime. For imaging systems with aberrations, the PSFs of WBRT diverge from the results using diffraction integrals. For larger aberrations WBRT predicts negative intensities, suggesting that this model is unable to deal with aberrations.

show abstract

When is the Wigner function of multidimensional systems nonnegative?

Cited by 130 publications

References 5 publications

Continuous-variable sampling from photon-added or photon-subtracted squeezed states

Continuous-variable sampling from photon-added or photon-subtracted squeezed states

GeneralizedQ-functions

A Wigner-based ray-tracing method for imaging simulations

Contact Info

Product

Resources

About