Decomposition of 2G(A, B) maps as differences of 3G(A, B) maps

Establish whether every Hermitian-preserving completely bounded linear map Φ ∈ 2G(A, B), with G a positive discrete unbounded operator on HA, can be represented as Φ = Γ1 − Γ2 for some Γ1, Γ2 ∈ 3G(A, B), where 3G(A, B) consists of maps admitting Stinespring-like representations with VG-infinitesimal representing operators.

Background

In the standard (diamond norm) setting, any Hermitian-preserving completely bounded map can be written as a difference of two completely positive maps. The authors extend the framework to the energy-constrained diamond norm by introducing 2G(A, B) and the cone 3G(A, B). They prove that for discrete G, the positive cone in 2G(A, B) coincides with 3G(A, B), and 2G(A, B) is the completion of the Hermitian-preserving completely bounded maps with respect to the ECD norm.

Motivated by these results, the authors conjecture an analogue of the classical decomposition: that every map in 2G(A, B) can be expressed as the difference of two maps from 3G(A, B), which would extend the decomposition property to the energy-constrained setting.