The strong superadditivity conjecture holds for the quantum depolarizing channel in any dimension

Given a quantum channel $\mathrm{\Phi }$ in a Hilbert space $H$ put $\hat H_{\Phi}(\rho)=\min \limits_{\rho_{av}=\rho}\Sigma_{j=1}^{k}\pi_{j}S(\Phi (\rho_{j}))$,where$\rho_{av}=\Sigma_{j=1}^{k}\pi_{j}\rho_{j}$, the minimum is taken over all probability distributions $\pi =\{{\pi }_j\}$ and states ${\rho }_j$ in $H$, $S(\rho )=-Tr\rho \log \rho$ is the von Neumann entropy of a state $\rho$. The strong superadditivity conjecture states that $\hat H_{\Phi \otimes \Psi}(\rho)\ge \hat H_{\Phi}(Tr_{K}(\rho))+\hat H_{\Psi}(Tr_{H}(\rho))$fortwochannels$\Phi $and$\Psi $inHilbertspaces$H$and$K$, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in any dimensions.