Let the [trace norm][1] of $X$ be

$$\Vert X\Vert_1 := \operatorname{tr} \left(\,(X^\dagger X)^{1/2}\right)$$

and let the operator inequality $A \leq B$ denote that the operator $B-A$ is positive semidefinite.

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If the **quantum states** (finite-dimensional Hermitian positive semidefinite matrices that have unit trace) $\rho$ and $\sigma$ are close to each other in the [trace norm][1], $\Vert \rho - \sigma \Vert_1 \leq \epsilon$, for some small $\epsilon > 0$, does there exist a projector $\Pi$, such that 

$$ \begin{aligned} \Pi \rho \Pi &\leq (1+g_1(\epsilon)) \sigma \\ \operatorname{tr} (\Pi \sigma) &\geq 1- g_2(\epsilon) \end{aligned} $$

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 $\epsilon \to 0$? Note $g_1$ and $g_2$ should be independent of the dimensions of the matrices. Some observations:

1. This is true for classical states (matrices which commute) and also true when we also sandwich $\sigma$ in the operator inequality (proofs are [here][2]). 

2. It seems that this is also true for randomly selected matrices in small dimensions (program and description are [here][3]; 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 & Johnson's 2nd edition of [Matrix Analysis][4]) but no luck so far. 

Any help or ideas are appreciated. 


  [1]: https://en.wikipedia.org/wiki/Matrix_norm#Schatten_norms
  [2]: https://github.com/goforashutosh/CloseStatesImplyNiceProjector/blob/master/ClassicalProof.pdf
  [3]: https://github.com/goforashutosh/CloseStatesImplyNiceProjector/blob/master/CloseStatesImplyNiceProj.ipynb
  [4]: https://www.cambridge.org/us/universitypress/subjects/mathematics/algebra/matrix-analysis-2nd-edition