Talk:Choi's theorem on completely positive maps

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Problems[edit]

The current proof of Choi's theorem has some problems (though it is essentially correct).

The notation for the Choi matrix, (Φ(E_ij))_ij is a bit confusing. It needs to mean (in matrix-of-matrix notation) the matrix whose ijth entry is Φ(E_ij), but I'm not convinced this is the most obvious interpretation of (Φ(E_ij))_ij. (In the notation of the start of the page it would be $\sum _{ij}E_{ij}\otimes \Phi (E_{ij})$ .)

It also isn't quite made clear the identifications of $\mathbb {C} ^{n}\otimes \mathbb {C} ^{n}\otimes \mathbb {C} ^{m}\otimes \mathbb {C} ^{m}$ with $\mathbb {C} ^{n\times n}\otimes \mathbb {C} ^{m\times m}$ and $\mathbb {C} ^{nm\times nm}$ , and how these relate to the direct sum decomposition used in the proof.

There are various other minor problems: $\Phi \otimes I$ should be $I\otimes \Phi$ to be consistent with the rest of the page. The summing index m clashes with the use of m as the dimension of the one of the spaces. The eigenvalues are non-negative, not positive. v_i P_m should say P_m v_i. P_i should lie in C^m×nm.

If no-one objects, I'll tidy all this up. Alex Selby (talk) 06:14, 18 November 2010 (UTC)[reply]

Now done. Alex Selby (talk) 20:29, 24 November 2010 (UTC)[reply]

Someone should please double-check the reference to Choi's paper. I checked the journal for that issue and those pages and it is not there. If I am not mistaken Choi's proof can in fact be found in Can. J. Math, Vol. XXIV, No. 3, 1972, pp. 520-529. M cuffa (talk) 17:26, 21 January 2012 (UTC)[reply]

Positivity notions[edit]

It would be useful to relate it to other notions of positivity. For example, is there an analogous notion and theorem for real symmetric matrices? 178.38.90.0 (talk) 20:12, 5 May 2015 (UTC)[reply]