What is the Wigner Function Closest to a Given Square Integrable Function?

Ben-Benjamin, J.; Cohen, Leon; Dias, Nuno Costa; Loughlin, Patrick J.; Prata, João Nuno

doi:10.1137/18m116633x

Cited by 6 publications

(7 citation statements)

References 48 publications

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“…In [6] the authors obtained an iterative method for obtaining the Wigner function closest to a given square integrable function on the phase space.…”

Section: Solving the Wigner-moyal Equation In An Interval With Profil...mentioning

confidence: 99%

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Boundaries and profiles in the Wigner formalism

Dias,

Prata

2021

Preprint

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We consider a quantum device contained in an interval in the context of the Weyl-Wigner formalism. This approach was originally suggested by Frensley, and is known to be plagued with several problems, such as non-physical and non-unique solutions. We show that some of these problems may be avoided if one writes the correct dynamical equation. This requires the (non-local) influence of the potential outside of the device and the inclusion of singular boundary potentials. We also discuss the problem of imposing boundary conditions on the Wigner function that mimic the effect of the external environment. We argue that these conditions have to be chosen with extreme care, as they may otherwise lead to non-physical solutions.

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“…In [6] the authors obtained an iterative method for obtaining the Wigner function closest to a given square integrable function on the phase space.…”

Section: Solving the Wigner-moyal Equation In An Interval With Profil...mentioning

confidence: 99%

“…In [6] we developed an iterative method, which allows us to obtain the optimal solution with any degree of accuracy.…”

Section: Solving the Wigner-moyal Equation In An Interval With Profil...mentioning

confidence: 99%

Boundaries and profiles in the Wigner formalism

Dias,

Prata

2021

Preprint

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“…At this point, it should be clear to the reader that the norm on L 2 r (Aff) is the most natural choice to consider. The analogous problem for the classical Wigner distribution has been recently investigated in [2]. Our approach will be different from the one taken in [2] as it will emphasize the quantization picture.…”

Section: Affine Wigner Approximationmentioning

confidence: 99%

“…Notice that (1.6) measures how far f is from being an affine Wigner distribution. The analogous problem in time-frequency analysis has been recently studied in [2]. For each symbol f ∈ L 2 r (Aff) there is a Hilbert-Schmidt operator A f on L 2 (R + , a −1 da) that is weakly defined by the relation…”

Section: Introductionmentioning

confidence: 99%

The Affine Wigner Distribution

Berge¹,

Berge²,

Luef³

2019

Preprint

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We examine the affine Wigner distribution from a quantization perspective with an emphasis on the underlying group structure. One of our main results expresses the scalogram as (affine) convolution of affine Wigner distributions. We strive to unite the literature on affine Wigner distributions and we provide the connection to the Mellin transform in a rigorous manner. Moreover, we present an affine ambiguity function and show how this can be used to illuminate properties of the affine Wigner distribution. In contrast with the usual Wigner distribution, we demonstrate that the affine Wigner distribution is never an analytic function.Our approach naturally leads to several applications, one of which is an approximation problem for the affine Wigner distribution. We show that the deviation for a symbol to be an affine Wigner distribution can be expressed purely in terms of intrinsic operator-related properties of the symbol. Finally, we present an affine positivity conjecture regarding the non-negativity of the affine Wigner distribution.

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“…We note that the problem of finding the state that minimizes a relevant Hilbert-Schmidt distance (potentially subject to constraints) is investigated in other contexts in Refs. [9,10].…”

Section: Introductionmentioning

confidence: 99%

The quantum low-rank approximation problem

Ezzell¹,

Holmes²,

Coles³

2022

Preprint

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We consider a quantum version of the famous low-rank approximation problem. Specifically, we consider the distance D(ρ, σ) between two normalized quantum states, ρ and σ, where the rank of σ is constrained to be at most R. For both the trace distance and Hilbert-Schmidt distance, we analytically solve for the optimal state σ that minimizes this distance. For the Hilbert-Schmidt distance, the unique optimal state is σ = τR + NR, where τR = ΠRρΠR is given by projecting ρ onto its R principal components with projector ΠR, and NR is a normalization factor given by NR = 1−Tr(τ R ) R ΠR. For the trace distance, this state is also optimal but not uniquely optimal, and we provide the full set of states that are optimal. We briefly discuss how our results have application for performing principal component analysis (PCA) via variational optimization on quantum computers.

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What is the Wigner Function Closest to a Given Square Integrable Function?

Cited by 6 publications

References 48 publications

Boundaries and profiles in the Wigner formalism

Boundaries and profiles in the Wigner formalism

The Affine Wigner Distribution

The quantum low-rank approximation problem

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