For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices?

If not, there is a weaker statement that I am interested in: Is it true that the inequality $\sigma_n( \sqrt{A} - \sqrt{B}) \leqslant \sigma_n(|A-B|^{\frac{1}{2}})$, where by $\sigma_n$ I denote the $n$-th singular value (they are listed in decreasing order), holds?

If even this fails, maybe the following is true: $\sigma_{2n}(\sqrt{A} - \sqrt{B}) \leqslant \sigma_{n} (|A-B|^{\frac{1}{2}})$?

I suspect that analogous inequalities should hold for any $0<r<1$.