Let $|\psi\rangle_{A B}$ be the joint state of a bipartite system $A B$ with subsystems $A$ and $B$ separated in space. Suppose that Alice and Bob have access only to subsystems $A$ and $B$ respectively, on which they can perform local quantum operations.

Alice performs a unitary operation $U$ on $A$ and then a (generally incomplete) measurement on $A$, with projectors $\left\{\Pi_{a}\right\}$ labelled by her possible measurement outcomes $a$. Then Bob performs a complete measurement on $B$ relative to the orthonormal basis $\{|b\rangle\}$ labelled by his possible outcomes $b$.

Show that the probability distribution of Bob's measurement outcomes is unaffected by whether or not Alice actually performs the local operations on $A$ described above.