Strong superadditivity conjecture holds for the quantum depolarizing channel in any dimension

Given a quantum channel Φ in a Hilbert space H, set HΦ(ρ)=minρav=ρΣj=1kπjS(Φ(ρj)), where ρav=Σj=1kπjρj, the minimum is taken over all probability distributions π=(πj) and states ρj in H, and S(ρ)=-Trρ log ρ is the von Neumann entropy of a state ρ. The strong superadditivity conjecture states that HΦxΨ(ρ)≥HΦ(TrK(ρ))+HΨ(TrH(ρ)) for two channels Φ and Ψ in Hilbert spaces H and K, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in any dimensions