Understanding POVMs and Quantum Information in Corollary Proofs

[aaaa202]

Hi everyone I need help understanding the lines on the attached picture.

In the proof of corollary 14 how does it follow from corollary 13 that:

Λ(X) = ∑li><il Tr[K_ijXK_ij^†]

I think what has been done is take a partial trace over the system H_B, but what exactly is the effect of this? It says that it is when the effect is not on the outcome measurement, so does the system carry information about the outcome of the measure while the system H_A contains information about something else.

 

[fresh_42]

Don't forget that $H_B = ℂ$ is one-dimensional in Cor. 14.