The world of quantum noise and dual output processes

Viacheslav Belavkin, University of Nottingham

Defining the dual output to a noncommutative input Ito algebra as its commutant in the GNS representation, we develop the dual output quantum stochastic calculus based on the dual co- and contra-variant tables. Using a noncommutative Radon-Nikodym type Theorem we prove that any normal state on a kernel algebra of multiple quantum stochastic integrals generated by the dual Ito algebra can be represented by a positive quantum stochastic martingale as a density-kernel of a multiple quantum stochastic integral with respect to the input quantum noise algebra. We illustrate this input-output quantum Ito formalism by an example of a quantum Cameron-Martin transformation.