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I am looking over some old course notes. First, Von Neumann entropy is defined.

The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv -\mbox{tr}(p\mbox{ ln}p)$.

Then the following claim is left unproven.

$S(\rho)\geq -\mbox{ln }\lambda_1$ where $\lambda_1$ is the largest eigenvalue of $\rho$.

I am sure this is true, but I can't seem to prove it.

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up vote 2 down vote accepted

Let $\{\lambda_i\ge 0\}$ be all the eigenvalues of $\rho$. Then $\sum_i \lambda_i=\mathrm{tr}\rho =1$ and $S(\rho)=-\sum_i \lambda_i\ln \lambda_i$. Since $ \lambda_1$ is the largest eigenvalue, $S(\rho)\ge -\sum_i \lambda_i\ln \lambda_1=-\ln \lambda_1$.

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