Landau and Streater proved that a set of Kraus operators, Ai, is extremal if and only if the set


$\{A\_{k}^{\dagger}A\_{l}\}\_{k,l \ldots N}$ 


are linearly independent. I have seen very convincing arguments both for and against. You can even see two PDFs of Mathematica notebooks "proving" *both* answers here: <http://quantummoxie.wordpress.com/2010/01/28/a-quirky-mathematical-problem-in-need-of-explanation/>


What is missing from these proofs?



Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel?
Hi Jon. Actually, the requirement that the set $A\_{i}^{\dagger}A\_{j} \oplus A\_{j}A\_{i}^{\dagger}$ be linearly independent is specifically for extremal *unital* channels. If the requirement on unitality is relaxed, Landau and Streater showed that only the set $A\_{i}^{\dagger}A\_{j}$ need be linearly independent.


jc: I see your point, but then can you tell me what is wrong with my Mathematica code? In other words, there must be something wrong with it if it returns a "True" for linear independence. Maybe someone has a better way to check these in Mathematica.


