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Des matrices de Pauli aux bruits quantiques

Abstract : Since its first definition by Hudson and Parthasarathy in 1984, quantum stochastic integration has become a powerful tool to describe evolutions in quantum physics. Yet many questions remain open, in particular in in the field of integral representability of operators. The recent definition by Attal of a completely explicit method of approximation of the usual Fock space by a discrete-time analogue justifies the interest of a good knowledge of quantum stochastic integration in this discrete-time framework to apply this approximation procedure. In this thesis we rigorously define such a stochastic calculus and derive a characterization of those operators that admit representations in the form of integrals or of Maassen-Meyer kernel operators, with fully explicit formulas for both type of representations.These results in turn allow us to precise the link between discrete and continuous-time quantum stochastic calculus and in particular to prove that the quantum Ito formula for composition of integrals is a consequence of the commutation relations between specifif operators, for example the Pauli matrices. We then apply those results to obtain, in usual Fock space, a characterization of the operators that are representable as quantum stochastic integrals among the physically important families of second quantization and differential second quantization operators. Then we use the formerly developed techniques to obtain results for the convergence of solutions of difference equations to solutions of quantum stochastic differential equations. These results allow us to prove that any evolution obtained in a quantum setup by means of repeated interactions is fully determined by a quantum Langevin equation. This quantum Langevin equation describes the coupling between a ``small system'' and a ``reservoir'', this reservoir and the limit equation being obtained in a fully explicit way from the repeated interaction. These results yield in particular a rigourous description of continuous measurement and of ``coarse graining'' approximations in quantum optics.
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Contributor : Arlette Guttin-Lombard <>
Submitted on : Monday, December 22, 2003 - 2:56:33 PM
Last modification on : Wednesday, November 4, 2020 - 1:59:44 PM
Long-term archiving on: : Wednesday, September 12, 2012 - 12:10:34 PM


  • HAL Id : tel-00004050, version 1



Yan Pautrat. Des matrices de Pauli aux bruits quantiques. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2003. Français. ⟨tel-00004050⟩



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