On Strong Subadditivity of the Entanglement of Formation

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation EF and the conjugate function E of the entanglement function E(ρ) = S(TrA ρ). We then consider the conjectured strong superadditivity of the entanglement of formation EF (ρ) ≥ EF (ρI) + EF (ρII), where ρI and ρII are the reductions of ρ to the different Hilbert space copies, and prove that it is equivalent with subadditivity of E. Furthermore, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for quantum filtering operations, when purity is measured by Schatten p-norms for p tending to 1.