If the quantum states (finite dimensional matrices which are positive semidefinite and have unit trace) $\rho$ and $\sigma$ are close to each other in the trace norm, $\Vert \rho - \sigma \Vert_1 \leq \epsilon$ [$\Vert X\Vert_1 = \text{tr}((X^\dagger X)^{1/2}$)] for some small $\epsilon >0$, does there exist a projector $\Pi$, such that $\Pi \rho \Pi \leq (1+g_1(\epsilon)) \sigma$ (the operator inequality $A \leq B$ represents that the operator $B-A$ is positive semidefinite) and $\text{tr}(\Pi \sigma) \geq 1- g_2(\epsilon)$ for some small functions $g_1(\epsilon)$ and $g_2(\epsilon)$ (i.e., both $g_1(\epsilon)$ and $g_2(\epsilon)$ tend to 0 as $\epsilon$ tends to 0)? Note $g_1$ and $g_2$ should be independent of the dimensions of the matrices. 

This is true for classical states (matrices which commute) and also true when we also sandwich $\sigma$ in the operator inequality (proofs are [here][1]). 

This also seems to be true for randomly selected matrices in small dimensions (program and description are [here][2]; the Jupyter notebook also contains some more observations).

So far, I have not been able to come up with a way which avoids having a projector on $\sigma$ as well. It seems that using the assumption $\Vert \sqrt{\rho}- \sqrt{\sigma} \Vert_2 \leq \epsilon$, which is equivalent (upto the exponent of $\epsilon$) to the original assumption, is far more easier, since there does not seem to be a lot of ways one can manipulate the trace norm. I have been trying to use a strengthened version of Gresgorin Theorem (Corollary 6.1.6 of [Horn and Johnson Matrix Analysis][3]) but no luck so far. 

Any help or ideas are appreciated. 


  [1]: https://github.com/goforashutosh/CloseStatesImplyNiceProjector/blob/master/ClassicalProof.pdf
  [2]: https://github.com/goforashutosh/CloseStatesImplyNiceProjector/blob/master/CloseStatesImplyNiceProj.ipynb
  [3]: https://www.amazon.ca/Matrix-Analysis-Roger-Horn/dp/0521548233