Characterization of informational completeness for covariant phase space observables

Kiukas, Jukka; Lahti, Pekka; Schultz, Jussi; Werner, R. F.

doi:10.1063/1.4754278

Cited by 22 publications

(53 citation statements)

References 26 publications

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“…A consequence of our results shows that this procedure will not always work. This connection and result has also been noted in [3,14].…”

Section: This Papersupporting

confidence: 84%

“…Weyl operator e 2πiωQ−ixP [14,15] Integrated Schrödinger representation Weyl quantization [15] Twisted Weyl symbol, Fourier-Wigner transform Weyl representation [15] Glauber-Sudarshan representation anti-Wick symbol [5], contravariant Berezin symbol, upper symbol [20], symbol for localization operator [1] Husimi representation Berezin transform [1], covariant Berezin symbol, lower symbol [20] Table 1. A dictionary relating the terminology in this paper to other common terminologies in mathematical physics.…”

Section: This Papermentioning

confidence: 99%

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Convolutions for Berezin quantization and Berezin-Lieb inequalities

Luef

Skrettingland

2018

Journal of Mathematical Physics

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Concepts and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, are identified as the appropriate setting for Berezin quantization and Berezin-Lieb inequalities. Based on this insight we provide a rigorous approach to generalized phase-space representation introduced by Klauder-Skagerstam and their variants of Berezin-Lieb inequalities in this setting. Hence our presentation of the results of Klauder-Skagerstam gives a more conceptual framework, which yields as a byproduct an interesting perspective on the connection between Berezin quantization and Weyl quantization. 1991 Mathematics Subject Classification. 47G30; 35S05; 46E35; 47B10.

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“…A consequence of our results shows that this procedure will not always work. This connection and result has also been noted in [3,14].…”

Section: This Papersupporting

confidence: 84%

Section: This Papermentioning

confidence: 99%

Convolutions for Berezin quantization and Berezin-Lieb inequalities

Luef

Skrettingland

2018

Journal of Mathematical Physics

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“…Werner [66] has proved a version of Wiener's Tauberian theorem for operators. The theorem was later generalized in [45], and more equivalent statements and a proof may be found in [45,55]. We state the relevant parts of the theorem for our purposes.…”

Section: Localization Operators and Spectrograms As Convolutionsmentioning

confidence: 94%

“…In this section we will approach the mixed-state localization operators χ Ω ⋆ S from another perspective, namely that of covariant positive operator valued measures (POVMs). This perspective has been ever-present when the convolutions of operators have been introduced and discussed in quantum physics [38,44,45,66], and we wish to show that it may be of interest also in time-frequency analysis. A POVM F gives two possible measures of the time-frequency content of a signal ψ in a domain Ω in the the time-frequency plane.…”

Section: Localization Operators and Positive Operator Valued Measuresmentioning

confidence: 98%

Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

Luef

Skrettingland

2019

J Fourier Anal Appl

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We study mixed-state localization operators from the perspective of Werner's operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore we provide a description of the Cohen class in terms of Werner's convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen's class.where π(z)ψ(t) = e 2πiωt ψ(t−x). In [55] we showed that this yields a natural class of Banach modules. There are two types of convolutions in this noncommutative setting: (i) The convolution between a function f ∈ L 1 (R 2d ) and a trace class operator S:(ii) the convolution between two trace class operators S and T is defined by1991 Mathematics Subject Classification. 47G30; 35S05; 46E35; 47B10. Key words and phrases. localization operators, Cohen class, uncertainty principle, phase retrieval, positive operator valued measures.F W S(z) = e −πix·ω tr(π(−z)S) for z ∈ R 2d . Note that the Fourier-Wigner transform and the spreading function differ only by a phase factor. The Fourier-Wigner transform has many properties analogous to those of the Fourier transform of functions [55,66]. In the case of rank-one operators these concepts of quantum harmonic analysis turn into wellknown objects from time-frequency analysis. Suppose ϕ 2 ⊗ ϕ 1 is the rank-one operator for ϕ 1 , ϕ 2 ∈ L 2 (R d ). Then we havewhich is a localization operator (or STFT-filter or 46]) and is denoted by A ϕ1,ϕ2f , and f is called the mask of the STFT-filter. Similarly, the convolution of two rank-one operators becomeswhereξ(x) = ξ(−x), which reduces for η = ψ and ψ = φ to the spectrogram [42]. The Fourier-Wigner transform of a rank-one operator is the ambiguity function. There is also a Hausdorff-Young inequality associated to the Fourier-Wigner transform [55,66], that in the rank-one case is the non-sharp Lieb's inequality for ambiguity functions [51]. Let us return to the objectives of this paper. Since localization operators are convolutions of a function and a rank-one operator, a natural extension of localization operators are operators of the form f ⋆ S for a trace-class operator S. The results of this paper indicate that these operators describe the time-frequency localization in various ways. For example we are interested in the amount of "spreading" in time and frequency that an operator performs on a function which we describe in form of bounds on the concentration of the spreading function, or equivalently ...

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“…1.70]. Zero-free Wigner distributions occur prominently in [26,28] in a similar context. It is therefore natural to ask for examples that satisfy Bayer's assumptions:…”

Section: Introductionmentioning

confidence: 98%

Zeros of the Wigner distribution and the short-time Fourier transform

2019

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We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson's theorem for the positivity of the Wigner distribution and to Hardy's uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W (f, 1 (0,1) ) always possesses a zero f ∈ S ∩ L p for p < ∞, but may be zero-free for f ∈ S ∩ L ∞ . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

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Characterization of informational completeness for covariant phase space observables

Cited by 22 publications

References 26 publications

Convolutions for Berezin quantization and Berezin-Lieb inequalities

Convolutions for Berezin quantization and Berezin-Lieb inequalities

Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

Zeros of the Wigner distribution and the short-time Fourier transform

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