Hello again @fedja, I am using this fact in a proof in my paper. I would like to include your proof. I think it's a nice technique and people in the quantum community would benefit from it. Let me know if this is okay. This is how it's currently written (attribution is red)- drive.proton.me/urls/KJXWX0T9QW#oV0DwK4fkGBA

Hi again @fedja, what exactly did you have in mind when you said that you can bound $\Sigma_2 \leq 2K \Sigma_1$ because I was writing this out for myself and I can only do something $O(K^2)$

Thank you, this is great. This answer is still pretty non-trivial to me and those who work with me. I do remember looking at this vector space decomposition but obviously could not go too far. I have two soft questions: 1. if I use this result in my research, how would you like to be cited (you can dm me). 2. if you do indeed think this is a simple problem, has a similar technique been used before and if so where?

You get $ \Pi (1+\sqrt{\epsilon})\sigma \Pi \geq \Pi \rho \Pi$, how do you transform this to $(1+\sqrt{\epsilon})\sigma \geq \Pi \rho \Pi $? It doesn't seem to me that $\Pi$ and $\sigma$ commute. This seems to me to be the same argument as Theorem 2.2 in my notes (github.com/goforashutosh/CloseStatesImplyNiceProjector/blob/…)

@IosifPinelis In the code, $\sigma$ is WLOG assumed to be diagonal, so I basically search over all possible diagonal projectors (projectors over all combinations of the eigenspace of $\sigma$) and see if they satisfy the conditions. This is done in a slightly more efficient manner