Stochastic processes associated with quantum systems

C. V. Heer

Un Cig Ji, Kalyan B. Sinha

We develop the quantum stochastic calculus associated with the square of annihilation and creation processes. We study the quantum stochastic integral and quantum stochastic differential equation associated with the square of annihilation and creation processes. Under a sufficient condition the solution of the quantum stochastic differential equation becomes unitary and then the unitary solution gives a unitary dilation of an evolution system of which the Hamiltonian is depending on time.

Palle E. T. Jørgensen, Feng Tian

We address a new factorization theory, and conditioning analysis, in the context of noncommutative stochastic systems. Our focus is that of transition matrices/kernels as they arise in diverse contexts, including in quantum theory. Our initial positive definite operator valued kernels [Formula: see text] are considered as generators of both a noncommutative systems; and, in different contexts, as a noncommutative covariance operators, or a covariance kernel. We address the following two questions: For the first, we study an associated system of path spaces. For the second, we consider associated stochastic process (typically Gaussian) fitting a framework introduced by Arveson [J. Funct. Anal. 66, 173 (1986)]. In the general context, our analysis of the associated stochastic processes we introduce new infinite-dimensional systems of polymorphisms.

Stanley Gudder

R. F. Streater

Alberto Barchielli, G. Lupieri

The physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process (OVSP). In this article, it is shown how the formalism of quantum stochastic calculus of Hudson and Parthasarathy allows, in a simple way, for constructing a large class of OVSP’s that in particular contains the quantum counting processes of Davies and Srinivas and continual ‘‘Gaussian’’ measurements. This result is obtained by means of a stochastic dilation of the OVSP’s: at the level of the enlarged system probabilities turn out to be expressed in terms of projection valued measures associated with certain time-dependent, commuting, self-adjoint operators.