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Theory of Probability and Its Applications

On the Lévy–Khinchin Formula in Noncommutative Probability Theory

Александр Семенович Холево

Published Dec 1, 1994• 3 citations
Publisher

Abstract

An important role in modern statistical quantum measurement theory is played by measures that assume values in a noncommutative algebra of transformations. This paper investigates convolution semigroups of such measures arising in connection with measurement processes that proceed continuously in time. The principal result is a noncommutative generalization of the Lévy–Khinchin formula, which describes the structure of the convolution semigroups in terms of their Fourier transforms.

DOI
10.1137/1138066

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An important role in modern statistical quantum measurement theory is played by measures that assume values in a noncommutative algebra of transformations. This paper investigates convolution semigroups of such measures arising in connection with measurement processes that proceed continuously in time. The principal result is a noncommutative generalization of the Lévy–Khinchin formula, which describes the structure of the convolution semigroups in terms of their Fourier transforms.
In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical theory of these equations has been developed in [11], where also “structural properties” of this class of equations have been studied. In the paper [11] it has been shown that such equations are connected with certain semigroups of linear operators and the form of the generator of semigroups related to such SDE’s has been established.
Previous article Next article A Representation of Lévy-Khinchine Type in Quantum Probability TheoryA. S. KholevoA. S. Kholevohttps://doi.org/10.1137/1132015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. S. Kholevo, Infinitely divisible measurements in quantum probability, Theory Probab. Appl., 32 (1986), 493–497 0623.60005 Google Scholar[2] A. Barchielli, , L. Lanz and , G. M. Prosperi, Statistics of continuous trajectories in quantum mechanics: operation-valued stochastic processes, Found. Phys., 13 (1983), 779–812 87b:81006 CrossrefGoogle Scholar[3] A. Barchielli and , G. Lupieri, Dilations of operation valued stochastic processesQuantum probability and applications, II (Heidelberg, 1984), Lecture Notes in Math., Vol. 1136, Springer, Berlin, 1985, 57–66 87g:46101 0572.60046 CrossrefGoogle Scholar[4] Erik Christensen and , David E. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J. London Math. Soc. (2), 20 (1979), 358–368 81f:46082 0448.46040 CrossrefGoogle Scholar[5] E. B. Davies, Quantum theory of open systems, Academic Press [Harcourt Brace Jovanovich Publishers], London, 1976x+171 58:8853 0388.46044 Google Scholar[6] Alain Guichardet, Symmetric Hilbert spaces and related topics, Springer-Verlag, Berlin, 1972v+197, Lecture Notes in Math., 26 58:12416 0265.43008 CrossrefGoogle Scholar[7] K. R. Parthasarathy and , K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Springer-Verlag, Berlin, 1972vi+107, Lecture Notes in Math., 272 58:29849 0237.43005 CrossrefGoogle Scholar[8] K. R. Parthasarathy, One parameter semigroups of completely positive maps on groups arising from quantum stochastic differential equations, 1985, Indian Statist. Inst., New Delhi, preprint Google Scholar[9] Shôichirô Sakai, C*-algebras and W*-algebras, Springer-Verlag, New York, 1971xii+253 56:1082 CrossrefGoogle Scholar[10] W. Forrest Stinespring, Positive functions on $C\sp *$-algebras, Proc. Amer. Math. Soc., 6 (1955), 211–216 16,1033b 0064.36703 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Generalized Bochner theorem and Levy-Khinchine representation for completely positive definite generalized Toeplitz kernelsIntegral Equations and Operator Theory, Vol. 27, No. 4 Cross Ref On the Lévy–Khinchin Formula in Noncommutative Probability TheoryA. S. Kholevo17 July 2006 | Theory of Probability & Its Applications, Vol. 38, No. 4AbstractPDF (1196 KB)Convolution Semigroups in Noncommutative Probability TheoryL. V. Denisov17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 3AbstractPDF (636 KB)Infinitely Divisible Markov Mappings in Quantum Probability TheoryL. V. Denisov17 July 2006 | Theory of Probability & Its Applications, Vol. 33, No. 2AbstractPDF (521 KB) Volume 32, Issue 1| 1988Theory of Probability & Its Applications History Submitted:01 April 1986Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132015Article page range:pp. 131-136ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics
A concept of quantum stochastic convolution cocycle is introduced and studied in two different contexts -- purely algebraic and operator space theoretic. A quantum stochastic convolution cocycle is a quantum stochastic process on a coalgebra satisfying the convolution cocycle relation and the initial condition given by the counit. The notion generalises that of quantum Levy process, which in turn is a noncommutative probability counterpart of classical Levy process on a group. Convolution cocycles arise as solutions of quantum stochastic differential equations. In turn every sufficiently regular cocycle satisfies an equation of that type. This is proved along with the corresponding existence and uniqueness of solutions for coalgebraic quantum stochastic differential equations. The stochastic generators of unital *-homomorphic cocycles are characterised in terms of structure maps on a *-bialgebra. This yields a simple proof of the Schurmann Reconstruction Theorem for a quantum Levy process; it also yields a topological version for a quantum Levy process on a C*-bialgebra. Precise characterisation of the stochastic generators of completely positive and contractive quantum stochastic convolution cocycles in the C*-algebraic context is given, leading to some dilation results. A few examples are presented and some interpretations offered for quantum stochastic convolution cocycles and their stochastic generators on different types of *-bialgebra.
We review the basic concepts of quantum probability and stochastics using the universal Itô B *-algebra approach. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The general Lévy process is defined in terms of the modular B *-Itô algebra, and the corresponding quantum stochastic master equation on the predual space of the W *-algebra is derived as a noncommutative version of the Zakai equation driven by the process. This is done by a noncommutative analog of the Girsanov transformation, which we introduce here in full generality.
We review the basic concepts of quantum stochastics using the universal Itô *-algebra approach.The main notions and results of classical and quantum stochastics are reformulated in this unifying approach.The thermal quantum Lévy process with commuting increments is defined in terms of the modular *-Itô algebra.The quasi-Markov quantum stochastic processes over increasing W*-algebras generated by a quantum Lévy process is characterized in terms of their quantum stochastic germs.The corresponding quantum stochastic master equation on the increasing predual L-spaces is derived as a noncommutative and non-Markov generalization of the Zakai equation driven by the dual quantum Lévy process.This is done by a noncommutative analog of the Girsanov transformation which we introduce here in full generality.