On extension of quantum channels and operations to the space of relatively bounded operators

TL;DR

This paper investigates conditions under which quantum operations can be extended to relatively bounded operators, establishing a unique extension criterion linked to the $G$-limited property, with applications to Bosonic Gaussian channels.

Contribution

It introduces a criterion for extending quantum operations to relatively bounded operators and proves the equivalence with the $G$-limited property for a broad class of operations.

Findings

Quantum operations can be uniquely extended under the $G$-limited condition.

The $G$-limited property is necessary and sufficient for extension in certain cases.

Applications to Bosonic Gaussian channels demonstrate practical relevance.

Abstract

We analyse possibility to extend a quantum operation (sub-unital normal CP 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 $\sqrt{G}$-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 $\mathrm{Tr}\rho G$ 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.