I don't have the answer to your question, but I can give you the following:
$$\|\sqrt A-\sqrt B\|_\infty\le\sqrt{\|A-B\|_\infty\,}\,,$$
where the $\infty$-Schatten norm is nothing but the operator norm.

A proof for this claim can be found [here][1].


  [1]: https://dspace.library.uvic.ca/bitstream/handle/1828/1506/DM%20442%20IR.pdf?sequence=1