On strong superadditivity for a class of quantum channels

Abstract

Given a quantum channel $\Phi$ in a Hilbert space $H$ put ${\hat{H}}_{\Phi }(\rho )=\underset{{\rho }_{av}=\rho }{\min }{\Sigma }_{j=1}^k{\pi }_{j}S(\Phi ({\rho }_j))$, 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 ${\hat{H}}_{\Phi \otimes \Psi }(\rho )\ge {\hat{H}}_{\Phi }(T{r}_K(\rho ))+{\hat{H}}_{\Psi }(T{r}_H(\rho ))$ for two channels $\Phi$ and $\Psi$ in Hilbert spaces $H$ and $K$, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in prime dimensions. The estimation of the quantity ${\hat{H}}_{\Phi \otimes \Psi }(\rho )$ for the special class of Weyl channels $\Phi$ of the form $\Phi =\Xi \circ {\Phi }_{dep}$, where ${\Phi }_{dep}$ is the quantum depolarizing channel and $\Xi$ is the phase damping is given.

revtex, 4 pages