An Entropy Inequality

Let $S(\rho)=- Tr (\rho \log\rho)$ be the von Neumann entropy of an $N$-dimensional quantum state $\rho$ and $e_2(\rho)$ the second elementary symmetric polynomial of the eigenvalues of $\rho$. We prove the inequality $S(\rho) \le c(N) \sqrt{e_2(\rho)}$ where $c(N)=\log(N) \sqrt{\frac{2N}{N-1}}$. This generalizes an inequality given by Fuchs and Graaf \cite{fuchsgraaf} for the case of one qubit, i.e., N=2. Equality is achieved if and only if $\rho$ is either a pure or the maximally mixed state. This inequality delivers new bounds for quantities of interest in quantum information theory, such as upper bounds for the minimum output entropy and the entanglement of formation as well as a lower bound for the Holevo channel capacity.