Quantum Stochastic Positive Evolutions: Characterization, Construction, Dilation

TL;DR

This paper characterizes and constructs quantum stochastic positive evolutions, providing a comprehensive framework for their dilation, including the unitary quantum stochastic dilation of flows with unbounded generators.

Contribution

It introduces a general form for quantum stochastic evolutions and constructs their dilations, advancing understanding of quantum stochastic processes with unbounded generators.

Findings

Characterization of unbounded stochastic generators of quantum CP flows

Construction of quantum stochastic adapted evolutions for various noises

Dilations of stochastic CP and dissipative equations over operator algebras

Abstract

A characterization of the unbounded stochastic generators of quantum completely positive flows is given. This suggests the general form of quantum stochastic adapted evolutions with respect to the Wiener (diffusion), Poisson (jumps), or general Quantum Noise. The corresponding irreversible Heisenberg evolution in terms of stochastic completely positive (CP) maps is constructed. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L(H) is discovered, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators. A unitary quantum stochastic cocycle, dilating the subfiltering CP flows over L(H), is reconstructed.