On Completion of the Cone of Completely Positive 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 √ G-bounded and √ 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.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.

Energy-Сonstrained Diamond Norms and Quantum Dynamical Semigroups

“…the ECD norm. So, the implication (i) ⇒ (iii) directly follows from the results of Section 5 in [25]. The implication (iii) ⇒ (ii) is trivial.…”

“…It is clear that (25) is substantially stronger than (24). On the other hand, the property (25) is weaker than the differentiability of the semigroup Φ t w.r.t.…”

“…Remark 4. Proposition 3 in [24] does not imply that (25) follows from (24), since the family of superoperators (Φ t − Id A )/t may not be bounded. This conclusion is confirmed by the examples considered in Section 3.…”

“…Proof. It is clear that (25) implies that the superoperator S belongs to the completion of the set of all Hermitian preserving completely bounded superoperators w.r.t. the ECD norm.…”

On Quantum Operations of Photon Subtraction and Photon Addition

“…Recently, the quantum operations have been extended to the space of relatively bounded operators [24][25][26], where the bound is related with the system Hamiltonian. In the case of the one-mode electromagnetic radiation, the Hamiltonian is essentially the photon number operator and reads H = a † a.…”