Talk:Choi's theorem on completely positive maps

Problems

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)

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

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)

Positivity notions

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)