# Quantum operation to get rid of small but nonzero eigenvalues

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.

• The TNICP map that projects onto the positive eigenspace of your density matrix should do the trick I think. Jun 9 '20 at 21:38
• 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." Jun 9 '20 at 21:51
• 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 $\delta$. Jun 9 '20 at 22:17
• @Rammus and Norbert Schuch, thank you - your answers to both this and the old question were helpful! Jun 9 '20 at 22:20

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.
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.
• 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? Jun 9 '20 at 21:38
• @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. Jun 9 '20 at 21:52