On complementary channels and the additivity problem

We explore complementarity between output and environment of a quantum channel (or, more generally, CP map), making an observation that the output purity characteristics for complementary CP maps coincide. Hence, validity of the mutiplicativity/additivity conjecture for a class of CP maps implies its validity for complementary maps. The class of CP maps complementary to entanglement-breaking ones is described and is shown to contain diagonal CP maps as a proper subclass, resulting in new class of CP maps (channels) for which the multiplicativity/additivity holds. Covariant and Gaussian channels are discussed briefly in this context. In what followsHA,HB, . . . will denote (finite dimensional) Hilbert spaces of quantum systems A,B, . . . .M (H) denotes the algebra of all operators,S (H)− the convex set of density operators (states) and P (H) = extS (H)− the set of pure states (one-dimensional projections) in H. For a natural d, Hd denotes the Hilbert space of d−dimensional complex vectors, and Md – the algebra of all complex d× d−matrices. Given three finite spaces HA,HB, HC and a linear operator V : HA → HB ⊗HC , the relation ΦB(ρ) = TrHCV ρV , ΦC(ρ) = TrHBV ρV ; ρ ∈ M (HA) (1) defines two completely positive (CP) maps ΦB : M (HA) → M (HB) , ΦC : M (HA) → M (HC) , which will be called mutually complementary. If V is an isometry, both maps are trace preserving (TP) i.e. channels. The name