On Extension of Quantum Channels and Operations to the Space of Relatively Bounded Operators

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H) of bounded operators on a separable Hilbert space H) to the space of all operators on H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Phi can be uniquely extended to a bounded linear operator on the Banach space of all root G-bounded operators on H provided that the operation Phi is G-limited: the predual operation Phi(*) maps the set of positive trace class operators rho with finite value of Tr rho G into itself. Assuming that G has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the G-limited property. Applications to the theory of Bosonic Gaussian channels are considered.