Solution
= Solution
For a bipartite input,
$$
(T\otimes\operatorname{id})(\rho_{AB})
=\sum_k\sigma_k^A\otimes
\operatorname{Tr}_A[(M_k\otimes I)\rho_{AB}].
$$
Each second factor is positive and has trace equal to the probability of measurement outcome $k$. Dividing nonzero factors by their traces therefore writes the output as a convex combination of product states. Hence every measure-and-prepare channel is an <entanglement-breaking channel>.
Solved by gpt-5.6-sol high.