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

Abstract

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 ...

On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

“…The results above also shed some light on results in [29], where we considered the question of recovering Ω from χ Ω ⋆ S. The approach in [29] was only concerned with establishing conditions on S for this to be possible, and offered no clue as to how Ω could be recovered. Theorem 1.2 shows that ρ S Ω , defined using a finite number of eigenfunctions of χ Ω ⋆ S, estimates χ Ω .The sharpness of the bounds is contained in theorem 1.3:…”

“…The following result relates the convolutions of operators to the standard convolutions of Weyl symbols. The statements follow by combining propositions 3.12 and 3.16 (5) in [29].…”

“…The main object of this paper is an in-depth treatment of the mixed-state localization operators from [29] and their associated time-frequency distributions from the perspective developed in [2,3], and to describe them we first recall some facts about quantum harmonic analysis [27,36]. Concretely, the convolution between two trace class operators and the convolution between a function and a trace class operator.…”

“…We have shown in [29] that Q belongs to Cohen's class if it is of the form Q S (ψ) =Š ⋆ (ψ ⊗ ψ), where S is a linear operator mapping the Schwartz class S(R 2d ) to the space of tempered distributions S ′ (R 2d ). In particular, density operators S provide distributions in Cohen's class.…”

“…Furthermore we have given the following characterization in [29]: S is a density operator if and only if Q S (ψ) is a positive function and R 2d Q S (ψ)(z)dz = ψ 2 2 for any ψ ∈ L 2 (R 2d ). Note that Q ϕ⊗ϕ (ψ) is the spectrogram |V ϕ ψ(z)| 2 and thus Q S is the correct generalization of the spectrogram.…”