Theoretical Statistics and Mathematics Unit, ISI Delhi

Quantum Stochastic Calculus and Quantum Gaussian Processes

by K. R. Parthasarathy

n this lecture we present a brief outline of boson Fock space
stochastic calculus based on the creation, conservation and annihilation
operators of free field theory, as given in the 1984 paper of Hudson
and
Parthasarathy \cite{9}. We show how a part of this architecture yields
Gaussian fields stationary under a group action. Then we introduce the notion
of semigroups of quasifree completely positive maps on the algebra of all
bounded operators in the boson Fock space $\Gamma (\mathbb{C}^n)$ over
$\mathbb{C}^n.$ These semigroups are not strongly continuous but their preduals
map Gaussian states to Gaussian states. They were first introduced and their
generators were shown to be of the Lindblad type by Vanheuverzwijn \cite{19}.
They were recently investigated in the context of quantum information theory by
Heinosaari, Holevo and Wolf \cite{7}. Here we present the exact noisy
Schr\"odinger equation which dilates such a semigroup to a quantum Gaussian
Markov process.

isid/ms/2002/10 [fulltext]

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