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Given a quantum channel $\Phi $ in a Hilbert space $H$ put $\hatH_{\Phi}(\rho)=\min \limits_{\rho_{av}=\rho}\Sigma_{j=1}^{k}\pi_{j}S(\Phi(\rho_{j}))$, where $\rho_{av}=\Sigma_{j=1}^{k}\pi_{j}\rho_{j}$, the minimum istaken 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 astate $\rho$. The strong superadditivity conjecture states that $\hat H_{\Phi\otimes \Psi}(\rho)\ge \hat H_{\Phi}(Tr_{K}(\rho))+\hat H_{\Psi}(Tr_{H}(\rho))$for two channels $\Phi $ and $\Psi $ in Hilbert spaces $H$ and $K$,respectively. We have proved the strong superadditivity conjecture for thequantum depolarizing channel in any dimensions.

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