On completion of the cone of CP linear maps with respect to the energy-constrained diamond norm

Abstract

For a given positive operator $G$ we consider the cones of linear maps between Banach spaces of trace class operators characterized by the Stinespring-like representation with $\sqrt{G}$-bounded and $\sqrt{G}$-infinitesimal operators correspondingly. We prove the completeness of both cones w.r.t. the energy-constrained diamond norm induced by $G$ (as an energy observable) and the coincidence of the second cone with the completion of the cone of CP linear maps w.r.t. this norm. We show that the sets of quantum channels and quantum operations are complete w.r.t. the energy-constrained diamond norm for any energy observable $G$. Some properties of the maps belonging to the introduced cones are described. In particular, the corresponding generalization of the Kretschmann-Schlingemann-Werner theorem is obtained. We also give a nonconstructive description of the completion of the set of all Hermitian-preserving completely bounded linear maps w.r.t. the ECD norm.

26 pages, any comments are welcome

References in corpus (3)

• Quantum-limited loss sensing: Multiparameter estimation and Bures distance between loss channels
• Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities
• Operator E-norms and their use

Cited by in corpus (2)

• Operator E-norms and their use
• On quantum operations of photon subtraction and photon addition