An algebraic construction of quantum flows with unbounded generators

TL;DR

This paper develops a method to construct quantum stochastic cocycles with unbounded generators, extending classical process constructions to quantum systems under specific algebraic and growth conditions.

Contribution

It introduces a new algebraic framework for constructing quantum flows with unbounded generators, generalizing classical Feller process constructions to quantum dynamical semigroups.

Findings

Constructed quantum flows for unbounded generators satisfying specific invariance and structure conditions.

Verified the construction in cases including classical random walks, quantum exclusion processes, and flows on non-commutative spaces.

Extended the classical theory of Feller processes to a quantum algebraic setting.

Abstract

It is shown how to construct *-homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C* algebras; this generalises the construction of a classical Feller process and semigroup from a given generator. The construction is possible provided the generator satisfies an invariance property for some dense subalgebra A_0 of the C* algebra A and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for A_0, must satisfy a growth condition. Furthermore, it is assumed that either the subalgebra A_0 is generated by isometries and A is universal, or A_0 contains its square roots. These conditions are verified in four cases: classical random walks on discrete groups, Rebolledo's symmetric quantum exclusion processes and flows on the non-commutative torus and the universal rotation algebra.