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Updated and edited question:

Let $N_{\delta}:P(\mathcal{H}_A)\rightarrow P(\mathcal{H}_B)$ be a completely positive trace nonincreasing map from the set of positive semidefinite operators in $\mathcal{H}_A$ to positive semidefinite operators in $\mathcal{H}_B$. For any $\rho\in P(\mathcal{H}_{A})$, we want $\sigma = N_{\delta}(\rho)$ such that all its positive eigenvalues are greater than $\delta$ and all remamining eigenvalues are zero.

Can such a map $N_\delta$ be constructed?

Based on a previous version of the question, the comment of @Rammus and the answer of @NorbertSchuch suggested using a projection operation $P$ into the eigenspace with eigenvalues larger than $\delta$. However, the required projector depends on the input state $\rho$, which I would like to avoid.

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    $\begingroup$ The TNICP map that projects onto the positive eigenspace of your density matrix should do the trick I think. $\endgroup$
    – Rammus
    Jun 9 '20 at 21:38
  • $\begingroup$ With your edit, it makes no sense. The way you write it the map should neither depend on rho nor on $\delta$. Or is this what you really mean? Please write a formal statement (which means: a formula, with clear statements on what the map can depend.) -- For instance, a formal statement would read: "For every delta, there is a CP map E such that for all rho, XYZ holds." $\endgroup$ Jun 9 '20 at 21:51
  • $\begingroup$ Much better! I guess what is missing is that you want that $\sigma=P\rho P^\dagger$, where $P$ is the projector onto the eigenvalues of $\rho$ larger than $\delta$. $\endgroup$ Jun 9 '20 at 22:17
  • $\begingroup$ @Rammus and Norbert Schuch, thank you - your answers to both this and the old question were helpful! $\endgroup$ Jun 9 '20 at 22:20
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Answer to the new version:

No, such a map cannot exist. CP maps are linear, i.e., $$ \mathcal E(\lambda\rho) = \lambda \mathcal E(\rho)\ . $$ Thus, you can easily see that for $\lambda\ne 1$, this is inconsistent with your definition.


Answer to the original version:

Sure, just project onto the eigenspace which you want to keep: $$ \rho\mapsto P\rho P^\dagger $$ with $P$ the orthogonal projector onto the eigenspace.

This is in Kraus form, and thus CP.

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  • $\begingroup$ Sorry, I should have clarified in the question but I meant a CP map that works for any input $\rho$, not just a specific one. That is, it always checks if any eigenvalues are smaller than $\delta$ and sets them to zero if this is so. Is that possible? $\endgroup$ Jun 9 '20 at 21:38
  • $\begingroup$ @user1936752 Linearity should instantly imply that "no". (For instance, just apply the map to $10\rho$ instead of $\rho$!) But before I take the effort to write this in an answer please ask this in a clean and formal way in the question. $\endgroup$ Jun 9 '20 at 21:52

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