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Feedback interconnection of open quantum systems: A small gain theorem

This paper examines the stability of quantum feedback networks. We introduce a novel characterization, in terms of equivalence classes of operators, that may be used to describe open quantum systems. In this characterization, equivalence classes of operators are shown to be elements of a Banach space such that the norm of an operator is analogous to the root mean square expectation value of the operator. This characterization is ideally suited to describing quantum feedback systems and enables us to prove the uniqueness and existence of solutions for feedback systems. In this context, we prove a quantum version of the small-gain theorem that is a generalisation of the classical small gain theorem.