In my(very little) understanding of convexity, I am unable to understand how can a set of superchannel(that takes channels in system A to channels in system B) be convex.
We call a function f to be convex if:
f(\sum_{i} a_i x_i )
A convex set is a set which is closed under convex combinations of its elements. This is different from the condition you wrote for the convex function.
Let S be a set. To show its convexity, you need to show that any convex combination of S also belong to S.
I suggest you to try showing the convexity of the set of density matrices. Then you will be able to easily show the convexity of quantum superchannels.
oh yes, thank you Junaid Ur Rehman! I was mixing the two definitions in head. Eventually, I was able to prove that the set of superchannels is closed and convex.
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