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Theory of Probability and Its Applications

Infinitely Divisible Markov Mappings in Quantum Probability Theory

L. V. Denisov

Published Jan 1, 1989• 17 citations
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Abstract

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

DOI
10.1137/1133064

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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∗W\sp{\ast}W\sp∗-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
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 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
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
Probability Measures in Terms of Creation, Annihilation and Neutral Operators (L Accardi et al.) Generating Function Methods for Orthogonal Polynomials and Jacobi-Szego Parameters (N Asai et al.) Multiquantum Markov Semigroups, Interacting Branching Processes and Nonlinear Kinetic Equations. Finite Dimensional Case (V Belavkin & C Williams) A Note on Vacuum-Adapted Semimartingales and Monotone Independence (A Belton) Regular Quantum Stochastic Cocycles have Exponential Product Systems (B V R Bhat & M Lindsay) Quantum Mechanics on the Circle Through Hopf q-Deformations of the Kinematical Algebra with Possible Applications to Levy Processes (V K Dobrev et al.) On Algebraic and Quantum Random Walks (D Ellinas) Dual Representations for the Schrodinger Algebra (P H Feinsilver & R Schott) A Limit Theorem for Conditionally Independent Beam Splittings (K H Fichtner et al.) On Quantum Logical Gates on a General Fock Space (W Feudenberg et al.) On an Argument of David Deutsch (R Gill) The Method of Double Product Integrals in Quantization of Lie Bialgebras (R Hudson) Asymptotics of Large Truncated Haar Unitary Matrices (J Reffy) Three Ways to Representations of Ba(E) (M Skeide) On Topological Entropy of Quotients and Extensions (J Zacharias) and other papers.
Previous article Next article Minimax Measurements in a General Statistical Decision TheoryN. A. BogomolovN. A. Bogomolovhttps://doi.org/10.1137/1126084PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Abraham Wald, Statistical Decision Functions, John Wiley & Sons Inc., New York, N. Y., 1950ix+179 12,193f 0040.36402 Google Scholar[2] C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York, 1976 Google Scholar[3] A. S. Kholevo, Investigations in the general theory of statistical decisions, Proc. Steklov Inst. Math., 124 (1976), , (In Russian.) 0497.62010 Google Scholar[4] M. Ozawa, Optimal measurements for general quantum systems, Rep. Math. Phys., 18 (1980), 11–28 10.1016/0034-4877(80)90036-1 84m:81026 0571.46051 CrossrefGoogle Scholar[5] N. N. Chentsov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, Providence, RI, 1981 Google Scholar[6] N. Bourbaki, General Topology : Topological Groups, Numbers, and the Groups and Spaces Related to Them, Addison-Wesley, Reading, MA, 1966 Google Scholar[7] V. S. Varadaraian, Measures on topological spaces, Amer. Math. Soc. Transl.(2), 48 (1965), 161–228 Google Scholar[8] Michael Reed and , Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York, 1972xvii+325 58:12429a 0242.46001 Google Scholar[9] E. Hewitt and , K. Ross, Abstract Harmonic Analysis, Vol. 1, Springer-Verlag, Berlin-New York, 1963 0830.43001 CrossrefGoogle Scholar[10] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950xi+304 11,504d 0040.16802 CrossrefGoogle Scholar[11] Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York, 1969ix+113 40:4776 0174.19001 Google Scholar[12] A. S. Holevo (Kholevo), Estimation of shift parameters of a quantum state, Rep. Math. Phys., 13 (1978), 379–399 10.1016/0034-4877(78)90065-4 80a:81060 0402.47006 CrossrefGoogle Scholar[13] Oscar Wesler, Invariance theory and a modified minimax principle, Ann. Math. Statist., 30 (1959), 1–20 21:5263 0089.14501 CrossrefGoogle Scholar[14] L. LeCam, An extension of Wald's theory of statistical decision functions, Ann. Math. Statist., 26 (1955), 69–81 16,730d 0064.38702 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Multiple-Phase Quantum Interferometry: Real and Apparent Gains of Measuring All the Phases Simultaneously27 January 2022 | Physical Review Letters, Vol. 128, No. 4 Cross Ref Minimax quantum state estimation under Bregman divergence4 March 2019 | Quantum, Vol. 3 Cross Ref Quantum Information Geometry and Quantum Estimation4 November 2016 Cross Ref Fourier Analytic Approach to Quantum Estimation of Group Action16 August 2016 | Communications in Mathematical Physics, Vol. 347, No. 1 Cross Ref Quantum Hypothesis Testing for Gaussian States: Quantum Analogues of χ2, t-, and F-Tests5 February 2013 | Communications in Mathematical Physics, Vol. 318, No. 2 Cross Ref Volume 26, Issue 4| 1982Theory of Probability & Its Applications History Submitted:10 November 1980Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1126084Article page range:pp. 787-795ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics
Extensions of Quantum Stochastic Calculus (S Attal) Quantum Ito Algebras: Axioms, Representations, Decompositions (V Belavkin) Free Probability for Probabilists (P Biane) Conditional Expectations on von Neumann Algebras (C Cecchini) Classical Probability Theory: An Outline of Stochastic Integrals and Diffusions (M Emery) Quantum Stochastic Differential Equations (F Fagnola) Canonical Commutation and Anticommutation Relations (M Fannes) Quantum and Classical Stochastic Calculus (A Holevo) An Introduction to Quantum Stochastic Calculus and Some of Its Applications (R Hudson) Stationary Processes in Quantum Probability (B Kummerer)