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Theory of Probability and Its Applications
A Representation of Lévy-Khinchine Type in Quantum Probability Theory Александр Семенович Холево
Published Jan 1, 1988 • 7 citations
Abstract 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
DOI 10.1137/1132015 Related papers Infinitely Divisible Measurements in Quantum Probability Theory
Александр Семенович Холево
Theory of Probability and Its Applications • Sep 1, 1987• 20 citations
Infinitely Divisible Markov Mappings in Quantum Probability Theory
L. V. Denisov
Theory of Probability and Its Applications • Jan 1, 1989• 17 citations
Measurement and Filtering of Quantum Diffusion II
V. P. Belavkin
Theory of Probability and Its Applications • Jan 1, 1995• 2 citations
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 ∗ C\sp * 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
Previous article Next article Infinitely Divisible Measurements in Quantum Probability TheoryA. S. KholevoA. S. Kholevohttps://doi.org/10.1137/1131063PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. Sakai, $C\sp*$-algebras and $W\sp*$-algebras, Springer-Verlag, New York, 1971xii+253 56:1082 0219.46042 CrossrefGoogle Scholar[2] E. B. Davies, Quantum theory of open systems, Academic Press [Harcourt Brace Jovanovich Publishers], London, 1976x+171 58:8853 0388.46044 Google Scholar[3] V. V. Sazonov and , V. N. Tutubalin, Probability distributions on topological groups, Theory Probab. Appl., 11 (1966), 1–45 10.1137/1111001 0171.38701 LinkGoogle Scholar[4] 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[5] 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[6] E. Schmidt, On multiplicative Lebesgue integration and families of evolution operators, Math. Scand., 29 (1971), 113–133 50:4885 0232.47024 Google Scholar[7] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys., 48 (1976), 119–130 54:1990 0343.47031 CrossrefGoogle Scholar[8] D. E. Evans and , J. T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A, (1977), v+104 58:8915 0365.46059 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Matrix product states and projected entangled pair states: Concepts, symmetries, theorems17 December 2021 | Reviews of Modern Physics, Vol. 93, No. 4 Cross Ref Generalized ansatz for continuous matrix product states7 May 2020 | Physical Review A, Vol. 101, No. 5 Cross Ref Continuum limits of matrix product states19 November 2018 | Physical Review B, Vol. 98, No. 17 Cross Ref Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation24 June 2017 | Entropy, Vol. 19, No. 7 Cross Ref Lie-semigroup structures for reachability and control of open quantum systems: kossakowski-lindblad generators form lie wedge to markovian channelsReports on Mathematical Physics, Vol. 64, No. 1-2 Cross Ref Lévy Processes and Continuous Quantum Measurements Cross Ref A composite Fourier‐wavelet transform and square‐integrable group representationsJournal of Mathematical Physics, Vol. 35, No. 8 Cross Ref 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 31, Issue 3| 1987Theory of Probability & Its Applications History Submitted:29 January 1986Published online:28 July 2006 InformationCopyright © 1987 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1131063Article page range:pp. 493-497ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics
Previous article Next article Infinitely Divisible Markov Mappings in Quantum Probability TheoryL. V. DenisovL. V. Denisovhttps://doi.org/10.1137/1133064PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Herbert Heyer, Probability measures on locally compact groups, Springer-Verlag, Berlin, 1977x+531, New York 58:18648 0376.60002 CrossrefGoogle Scholar[2] John Yuan, On the construction of one-parameter semigroups in topological semigroups, Pacific J. Math., 65 (1976), 285–292 54:10473 0337.22002 CrossrefGoogle Scholar[3] A. S. Kholevo, Infinitely divisible measurements in quantum probability theory, Theory Probab. Appl., 31 (1986), 493–496 10.1137/1131063 0623.60005 LinkGoogle Scholar[4] A. S. Kholevo, Representations of Levy-Khinchin type in quantum probability theory, Theory Probab. Appl., 32 (1987), 131–135 10.1137/1132015 LinkGoogle Scholar[5] J. F. C. Kingman, The imbedding problem for finite Markov chains, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 1 (1962), 14–24 26:3119 0118.13501 CrossrefGoogle Scholar[6] S. Johansen, A central limit theorem for finite semigroups and its application to the imbedding problem for finite state Markov chains, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 26 (1973), 171–190 48:9852 0247.60038 CrossrefGoogle Scholar[7] O. Bratteli and , D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vol. 1, Springer-Verlag, New York, 1989, vol. 2, 1981 Google Scholar[8] D. E. Evans and , J. T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A, (1977), v+104 58:8915 0365.46059 Google Scholar[9] Jun Tomiyama, On the projection of norm one in $W\sp{\ast}$-algebras, Proc. Japan Acad., 33 (1957), 608–612 20:2635 0081.11201 CrossrefGoogle Scholar[10] Richard V. Kadison, Isometries of operator algebras, Ann. Of Math. (2), 54 (1951), 325–338 13,256a 0045.06201 CrossrefGoogle Scholar[11] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys., 48 (1976), 119–130 54:1990 0343.47031 CrossrefGoogle Scholar[12] R. S. Philips, The adjoint semi-group, Pacific J. Math., 5 (1955), 269–283 17,64a CrossrefGoogle Scholar[13] P. Tondeur, Introduction to Lie Groups and Transformation Groups, Lect. Notes Math., Vol. 7, 1969 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Matrix product states and projected entangled pair states: Concepts, symmetries, theoremsReviews of Modern Physics, Vol. 93, No. 4 | 17 December 2021 Cross Ref Variational-state quantum metrologyNew Journal of Physics, Vol. 22, No. 8 | 20 August 2020 Cross Ref Generalized ansatz for continuous matrix product statesPhysical Review A, Vol. 101, No. 5 | 7 May 2020 Cross Ref Roots of completely positive mapsLinear Algebra and its Applications, Vol. 587 | 1 Feb 2020 Cross Ref Divisibility of qubit channels and dynamical mapsQuantum, Vol. 3 | 20 May 2019 Cross Ref Continuum limits of matrix product statesPhysical Review B, Vol. 98, No. 17 | 19 November 2018 Cross Ref The Complexity of Relating Quantum Channels to Master EquationsCommunications in Mathematical Physics, Vol. 310, No. 2 | 4 January 2012 Cross Ref Characterization of decoherence from an environmental perspectivePhysical Review A, Vol. 83, No. 4 | 18 April 2011 Cross Ref Lie-semigroup structures for reachability and control of open quantum systems: kossakowski-lindblad generators form lie wedge to markovian channelsReports on Mathematical Physics, Vol. 64, No. 1-2 | 1 Aug 2009 Cross Ref Dividing Quantum ChannelsCommunications in Mathematical Physics, Vol. 279, No. 1 | 1 February 2008 Cross Ref Volume 33, Issue 2| 1989Theory of Probability & Its Applications207-404 History Submitted:03 August 1987Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1133064Article page range:pp. 392-395ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics
Next article Measurement and Filtering of Quantum Diffusion IIV. P. BelavkinV. P. Belavkinhttps://doi.org/10.1137/1139026PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea Publishing Company, New York, 1984 Google Scholar[2] John von Neumann, Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, 1955xii+445, N.Y. 16,654a Google Scholar[3] A. S. Holevo, Probability and Stochastic Aspects of Quantum Theory, Princeton University Press, Princeton, N.J., 1955 Google Scholar[4] B. Misra and , E. C. G. Sudarshan, The Zeno's paradox in quantum theory, J. Mathematical Phys., 18 (1977), 756–763 10.1063/1.523304 55:5076 CrossrefGoogle Scholar[5] V. P. Belavkin, A stochastic calculus of quantum input-output processes and quantum nondemolition filteringCurrent problems in mathematics. Newest results, Vol. 36 (Russian), Itogi Nauki i Tekhniki, Akad. 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CrossrefGoogle Scholar Next article FiguresRelatedReferencesCited ByDetails Open-system many-body dynamics through interferometric measurements and feedbackPhysical Review A, Vol. 94, No. 5 | 17 November 2016 Cross Ref Continuous Quantum Hypothesis TestingPhysical Review Letters, Vol. 108, No. 17 | 1 April 2012 Cross Ref Volume 39, Issue 3| 1995Theory of Probability & Its Applications History Submitted:08 October 1991Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1139026Article page range:pp. 363-378ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics
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.
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