Past exam of the computer science course of the University of Oxford / 2021 / Quantum information / 2 / c / ii / Solution

Either experiment is again possible. The sequential experiment with $U=I$ repeats its first measurement outcome in every basis: the first projective measurement has prepared one of that basis's eigenstates, and the identity leaves it unchanged.

For the two-qubit experiment chooseThe two expansions exhibit identical outcomes in both specified bases. Each allowed equal-label record has probability $1/2$. These are fixed choices of $U$ and $|\psi ⟩$, respectively; neither depends on which of the two bases Alice selects.