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Nagoya Mathematical Journal

A note on a formula of the Lévy-Khinchin type in quantum probability

Alberto Barchielli, Anna Maria Paganoni

Published Mar 1, 1996• 14 citations
Publisher

Abstract

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.

DOI
10.1017/s0027763000005511

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A Representation of Lévy-Khinchine Type in Quantum Probability Theory

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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

Theory of Probability and Its Applications•Jan 1, 1988•7 citations

On the Lévy–Khinchin Formula in Noncommutative Probability Theory

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Some Numerical Methods And Analysis of A Class of Quantum Stochastic Differential Equations

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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
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure [ 1 , 2 , 3 , 4 , 5 , 6 ], arisen inside the operational approach [ 1 , 7 ] to quantum mechanics, by using functional integrals [ 8 , 9 , 10 ], by using quantum stochastic differential equations [ 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 ] and by using classical stochastic differential equations (SDE’s) [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 4 , 19 , 20 , 21 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 6 ]. Various types of SDE’s are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both.
The classical stochastic differential equation given as dX(t,ω)=a(t,X) dt+b(t,X) dQ(t),X(t0)=X0,t∈[t0,T],dX(t, \omega) = a(t, X) \, dt + b(t, X) \, dQ(t), \quad X(t_0) = X_0, \quad t \in [t_0, T],dX(t,ω)=a(t,X)dt+b(t,X)dQ(t),X(t0​)=X0​,t∈[t0​,T], where the driving process Q(t) Q(t) Q(t) is a martingale, and a a a and b b b are adapted functionals has several numerical schemes developed for it, where each exhibits distinct stability and convergence properties depending on the nature of the driving process and the functional space of the solution. A generalization of this equation is the quantum stochastic differential equation introduced by Hudson and Parthasarathy, expressed as dX(t)=E(t,X(t)) dΛπ(t)+F(t,X(t)) dAf†(t)+G(t,X(t)) dAg(t)+H(t,X(t)) dt,dX(t) = E(t, X(t)) \, d\Lambda_\pi(t) + F(t, X(t)) \, dA_f^\dagger(t) + G(t, X(t)) \, dA_g(t) + H(t, X(t)) \, dt,dX(t)=E(t,X(t))dΛπ​(t)+F(t,X(t))dAf†​(t)+G(t,X(t))dAg​(t)+H(t,X(t))dt, where E,F,G, E, F, G, E,F,G, and H H H are stochastic operator-valued functions defined on a Hilbert space. In 1992, Ekhaquere reformulated this equation in an equivalent form involving the non-classical differential equation stated as; ddt⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0)=X0,t∈[t0,T],\frac{d}{dt} \langle \eta, X(t) \xi \rangle = P(t, X(t)) \langle \eta, \xi \rangle, \quad X(t_0) = X_0, \quad t \in [t_0, T],dtd​⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0​)=X0​,t∈[t0​,T], which represents an ordinary differential equation of non-classical type. The stochastic process X(t) X(t) X(t) acts as a densely defined linear operator on a tensor product of two Hilbert spaces, one of which is the Boson Fock space. The vectors η \eta η and ξ \xi ξ belong to a dense subset of this tensor product Hilbert space, and the mapping (η,ξ)↦P(t,X)(η,ξ)(\eta, \xi) \mapsto P(t, X)(\eta, \xi)(η,ξ)↦P(t,X)(η,ξ) defines a sesquilinear form for each fixed pair (t,X)(t, X)(t,X). We analyze the numerical approximation of the QSDE using its equivalent inner-product form. The numerical investigation employs three principal methods: the Lagrangian Quadrature Method, the Linear Multistep Scheme, and the Newton–Cotes Method. Each method is carefully designed to ensure the consistency, stability, and convergence of the approximate solution X(t)X(t)X(t) toward the exact stochastic process. These schemes are implemented using Python programming language, which provides a flexible and efficient computational environment for testing the proposed algorithms. Our computational results show that the developed schemes generalize Taylor-type approximations of Itô processes under weak convergence criteria. The proposed algorithms exhibit numerical stability and yield results that compare favorably with analytical benchmarks. Furthermore, the implementation avoids direct approximation of the driving quantum processes, thereby preserving the natural adaptation of the schemes to quantum dynamical models.