The Schatten $p$-norm $\|M\|_p$ of a (finite-dimensional) matrix $M$ is defined as $\|M\|_p=\left(\sum_i s_i^p\right)^{1/p}$ where $s_i$ are singular values of $M$.
Let $P,Q$ be positive semidefinite matrices. Show that $$\|P-Q\|_1\le 2\sqrt{\min\left({\rm rank}(P),{\rm rank}(Q)\right)}\ \|P-Q\|_2.$$
How about the case that $P,Q$ are Hermitian matrices?