Abstract
Stochastic differential equations for processes with values in Hilbert spaces are now largely used in the quantum theory of open systems. In this work we present a class of such equations and discuss their main properties; moreover, we explain how they are derived from purely quantum mechanical models, where the dynamics is represented by a unitary evolution in a Hilbert space, and how they are related to the theory of continual measurements. An essential tool is an isomorphism between the bosonic Fock space and the Wiener space, which allows to connect certain quantum objects with probabilistic ones.
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References
Gisin, N.,Quantum measurements and stochastic processes, Phys. Rev. Lett.,52 (1984), 1657–1660.
Belavkin, V.P.,Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes, in A. Blaquière (ed.),Modelling and Control of Systems, Lecture Notes in Control and Information Sciences,121, Springer, Berlin, (1988), 245–265.
Belavkin, V.P.,Quantum continual measurements and a posteriori collapse on CCR, Commun. Math. Phys.,146 (1992), 611–635.
Gisin, N. andPercival, I.C.,The quantum-state diffusion model applied to open systems, J. Phys. A: Math. Gen.,25 (1992), 5677–5691.
Carmichael, H.J.,An Open System Approach to Quantum Optics, Lect. Notes Phys.,m 18, Springer, Berlin, (1993).
Barchielli, A.,On the quantum theory of measurements continuous in time, Rep. Math. Phys.,33 (1993), 21–34.
Barchielli, A., Lanz, L. andProsperi, G.M.,Statistics of continuous trajectories in quantum mechanics: Operation-valued stochastic processes, Found. Phys.,13 (1983), 779–812.
Barchielli, A. andHolevo, A.S.,Constructing quantum measurement processes via classical stochastic calculus, Stochastic Process. Appl.,58 (1995), 293–317.
Barchielli, A. andPaganoni, A.M.,Detection theory in quantum optics: stochastic representation, Quantum Semiclass. Opt.,8 (1996), 133–156.
Da Prato, G. andZabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, (1992).
Barchielli, A., Paganoni, A.M. andZucca, F.,On stochastic differential equations and semigroups of probability operators in quantum probability, to appear in Stochastic Process. Appl.
Holevo, A.S.,On dissipative stochastic equations in a Hilbert space, Probab. Theory Relat. Fields,104 (1996), 483–500.
Barchielli, A.,Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics, Quantum Opt.,2 (1990), 423–441.
Parthasarathy, K.R., An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, (1992).
Hudson, R.L. andParthasarathy, K.R.,Quantum Ito's formula and stochastic evolutions, Commun. Math. Phys.,93 (1984), 301–323.
Frigerio, A.,Covariant Markov dilations of quantum dynamical semigroups, Publ. RIMS Kyoto Univ.,21 (1985), 657–675.
Gardiner, C.W. andCollet, M.J.,Input and output in damped quantum systems: quantum stochastic differential equations and the master equation, Phys. Rev. A,31 (1985), 3761–3774.
Barchielli, A. andLupieri, G.,Quantum stochastic calculus, operation valued stochastic processes, and continual measurements in quantum mechanics, J. Math. Phys.,26 (1985), 2222–2230.
Frigerio, A.,Some applications of quantum probability to stochastic differential equations in Hilbert space, in G. Da Prato, L. Tubaro (eds.),Stochastic Partial Differential Equations and Applications II, Lect. Notes Math.,1390 Springer, Berlin, (1989), 77–90.
Barchielli, A.,Quantum stochastic calculus, measurements continuous in time, and heterodyne detection in quantum optics, in H. D. Doebner, W. Scherer, F. Schroeck Jr. (eds.),Classical and Quantum Systems, Foundations and Symmetries, Proceedings of the II International Wigner Symposium (World Scientific, Singapore, 1993), 488–491.
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Conferenza tenuta da A. Barchielli il 18 novembre 1996
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Barchielli, A., Zucca, F. On a class of stochastic differential equations used in quantum optics. Seminario Mat. e. Fis. di Milano 66, 355–376 (1996). https://doi.org/10.1007/BF02925365
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DOI: https://doi.org/10.1007/BF02925365