Operator-stable quantum probability operators
type: Item
instance of: scholarly article
title: Operator-stable quantum probability operators
zbMATH Open document ID: 1022.46040
DOI: 10.1023/A:1011382802345
published in: Journal of Mathematical Sciences (New York)
publication date: 2002-06-16T00:00:00Z
review text: Let \(V(z)\), \(z\in {\mathbb R}^{2s}\), denote a projective unitary representation of \({\mathbb R}^{2s}\) satisfying the Weyl-Segal commutation relation. Given a probability operator (i.e. a positive operator with unit trace) \(T\) in the space \({\mathcal L}_1\) of nuclear operators on a separable infinite-dimensional Hilbert space \({\mathcal H}\), let \(\widehat{T}(z) = \operatorname {tr} TV(z)\), \(z\in {\mathbb R}^{2s}\). Define the convolution \(\star\) by setting \(\widehat{T_1\star T_2} = \widehat{T}_1 \cdot \widehat{T}_2\), and let \(T\circ \mu\) be defined for all probability operator \(T\) and probability measure \(\mu\) by \(\widehat{T\circ \mu} = \widehat{T}\cdot \widehat{\mu}\), where \(\widehat{\mu}\) denotes the classical Fourier transform, or characteristic function, of \(\mu\). Finally, given a positive diagonal matrix \(A\), let the linear transformation \(U_A\) of the space \({\mathcal L}_2\) of Hilbert-Schmidt operators on \({\mathcal H}\) be defined from \((\widehat{U_AT})(z) = \widehat{T}(A z)\), \(z\in {\mathbb R}^{2s}\). Let now \(\{ A_{kn}\), \(T_{kn}\}\) be a uniformly infinitesimal triangular array of admissible positive diagonal matrices and probability operators with \(A_{kn}=A_n\) and \(T_{kn}=T\), \(k=1,2,\ldots , n\), \(n\geq 1\). The author shows that if there exists a sequence \((c_n)_{n\geq 1}\) of vectors in \({\mathbb R}^{2s}\) such that the sequence \((\star_{k=1}^n U_{A_{kn}} T_{kn})\circ \delta_{c_n}\) converges to a probability operator \(S\) where \(\delta_{c_n}\) denotes the Dirac measure at \(c_n\), then \(S\) is Gaussian, i.e. \(\widehat{S}(z) = e^{-(1/2)\langle qz,z\rangle + i \langle z,z_0\rangle}\), where \(z_0\in {\mathbb R}^{2s}\) and \(q\) is a nonnegative selfadjoint operator on \({\mathbb R}^{2s}\). This yields a partial generalization of the non-commutative limit theorem of \textit{K. Urbanik} [Stud. Math. 78, 59--75 (1984; Zbl 0535.60014)] who considered diagonal matrices with constant coefficients.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 81S25
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 60A99
zbMATH DE Number: 1756310
zbMATH Keywords: noncommutative probability
zbMATH Keywords: limit theorems
zbMATH Keywords: Weyl-Segal commutation relation
author: Katarzyna Lubnauer
MaRDI profile type: Publication
full work available at URL: a:1011382802345
OpenAlex ID: W92587654
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description: scientific article; zbMATH DE number 1756310