Timeline for Why is $\Phi\otimes \operatorname{Id}_n$ being positive on maximally entangled states sufficient to know that $\Phi$ is CP?
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| Jun 1, 2022 at 1:37 | comment | added | Evangeline A. K. McDowell | @NorbertSchuch Actually something that always bugs me is whether this theorem implies 2-positivity is enough to prove CP, since your proof essentially says that if it's 2-positive then it's true for $k > 2$. I have never seen this connection stated clearly. | |
| May 10, 2022 at 9:02 | vote | accept | glS♦ | ||
| May 10, 2022 at 7:17 | comment | added | Norbert Schuch | @gls ... or my 4 using the SVD of M :) In fact, this is one of the advanced questions I like asking in oral exams, if everything goes smoothly ;) (Will have to remember to delete this comment! :-o) | |
| May 10, 2022 at 7:07 | comment | added | glS♦ | sure. I just like this equation because it gives a direct explicit relation answering the question: "how does the action of $\Phi\otimes\operatorname{Id}_k$ on a generic state $|\chi\rangle$ relate to the action on maximally entangled states?". It's essentially your point 4, yes, observing that the matrix $D$ is just the diagonal matrix whose elements are the Schmidt coefficients | |
| May 9, 2022 at 22:54 | comment | added | Norbert Schuch | If D is singular this is a one-way relation. But isn't this precisely my point 4? | |
| May 9, 2022 at 21:47 | comment | added | glS♦ | ah, nice, this is the kind of thing I was hoping for. One could also make a trivial modification to the argument to make it "positive" I think: $$(\Phi\otimes\operatorname{Id}_k)(|\chi\rangle\!\langle\chi|)=(I\otimes D)[(\Phi\otimes\operatorname{Id}_n)(|\Omega\rangle\!\langle\Omega|)](I\otimes D)$$ with $D$ diagonal matrix with the Schmidt coefficients of $\chi$, so that we get a direct relation between positivity on $|\Omega\rangle$ and positivity on arbitrary (pure) states | |
| May 8, 2022 at 23:53 | history | edited | Norbert Schuch | CC BY-SA 4.0 |
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| May 8, 2022 at 23:38 | history | edited | Norbert Schuch | CC BY-SA 4.0 |
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| May 8, 2022 at 23:33 | history | answered | Norbert Schuch | CC BY-SA 4.0 |