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