Positive Definite Germs of Quantum Stochastic Processes

TL;DR

This paper introduces a new concept of stochastic germs for quantum processes, characterizes their differentials for positive definite processes, and provides a representation theorem for quantum stochastic evolution equations.

Contribution

It presents a novel notion of stochastic germs, characterizes differentials for PD processes, and proves a representation theorem for quantum stochastic evolutions.

Findings

Characterization of stochastic differentials for PD quantum processes.

Representation theorem for germ-matrix pseudo-Hilbert dilation.

General form of quantum stochastic evolution equations derived.

Abstract

A new notion of stochastic germs for quantum processes is introduced and a characterisation of the stochastic differentials for positive definite (PD) processes is found in terms of their germs for arbitrary Ito algebra. A representation theorem, giving the pseudo-Hilbert dilation for the germ-matrix of the differential, is proved. This suggests the general form of quantum stochastic evolution equations with respect to the Poisson (jumps), Wiener (diffusion) or general quantum noise.