Some Quantum Dynamical Semi-groups with Quantum Stochastic Dilation

TL;DR

This paper investigates quantum dynamical semi-groups derived from unbounded Lindbladians on UHF algebras, demonstrating the existence of unitary solutions to related quantum stochastic differential equations and establishing their conservativity.

Contribution

It introduces a novel class of unbounded Lindbladians on UHF algebras and proves the existence of associated unitary solutions to quantum stochastic differential equations.

Findings

Existence of unitary solutions to Hudson-Parthasarathy equations for these Lindbladians

Construction of minimal semi-groups from homomorphic co-cycles

Demonstration of conservativity of the associated quantum stochastic flows

Abstract

We consider the GNS Hilbert space $\mathcal{H}$ of a uniformly hyper-finite $C^*$- algebra and study a class of unbounded Lindbladian arises from commutators. Exploring the local structure of UHF algebra, we have shown that the associated Hudson-Parthasarathy type quantum stochastic differential equation admits a unitary solution. The vacuum expectation of homomorphic co-cycle, implemented by the Hudson-Parthasarathy flow, is conservative and gives the minimal semi-group associated with the formal Lindbladian. We also associate conservative minimal semi-groups to another class of Lindbladian by solving the corresponding Evan-Hudson equation.