Strict concavity of Von Neumann entropy
= Strict concavity of Von Neumann entropy
For $0<b<1$ and distinct <density operators>, the <Von Neumann entropy> of $\rho=b\rho_1+(1-b)\rho_2$ is strictly larger than the corresponding average. The difference is $bD(\rho_1\|\rho)+(1-b)D(\rho_2\|\rho)$, positive by <nonnegativity of quantum relative entropy> unless both states coincide. On the closed interval $0\le b\le1$, equality holds exactly when $b=0$, $b=1$, or $\rho_1=\rho_2$.