The Mathematical Institute, University of Oxford, Eprints Archive

A matrix formulation of quantum stochastic calculus

Belton, A. C. R. (1998) A matrix formulation of quantum stochastic calculus. PhD thesis, University of Oxford.

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Abstract

We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness.

This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion.

A new type of adaptedness, known as $\Omega$-adaptedness, is defined. We show that quantum stochastic integrals of $\Omega$-adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy:

$U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+
G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $

where the coefficients are time-dependent, bounded, $\Omega$-adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz.

$(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$

where $W$ is unitary and $K$ self-adjoint, are necessary and
sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.

Item Type:Thesis (PhD)
Subjects:O - Z > Quantum theory
D - G > Functional analysis
O - Z > Probability theory and stochastic processes
Research Groups:Functional Analysis Group
ID Code:29
Deposited By:Eprints Administrator
Deposited On:09 Mar 2004
Last Modified:20 Jul 2009 14:18

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