= Solution
\b[Only the two-<qubit> experiment is possible.] Choose the <spin singlet state>
$$
|\psi\rangle=|\Psi^-\rangle=\frac{|01\rangle-|10\rangle}{\sqrt2}.
$$
It has <eigenvalue> $-1$ under each of $X\otimes X$, $Y\otimes Y$ and $Z\otimes Z$. Thus measuring either <qubit> in the same Pauli basis as the other always gives opposite <eigenvalue> labels. Each opposite-label pair occurs with <probability> $1/2$. More generally, <collective-unitary covariance of the two-qubit singlet> ensures opposite outcomes in any common orthonormal <qubit> basis.
For the sequential experiment, the first outcome is uniformly distributed and the <Lüders rule> prepares the corresponding <eigenstate>. Certain opposite outcomes in the <computational basis> force the fixed <unitary matrix> to have the form
$$
U=\begin{pmatrix}0&a\\b&0\end{pmatrix},\qquad |a|=|b|=1.
$$
Certain opposite <Hadamard basis> outcomes require $\langle+|U|+\rangle=(a+b)/2=0$, hence $b=-a$. Therefore $U=iaY$. But this operator preserves the two <Pauli Y eigenstate> rays: it gives the same Y-basis label on the second measurement, rather than the opposite one. This contradicts the third set of observations.
Equivalently, flipping all three bases would require $U^\dagger XU=-X$, $U^\dagger YU=-Y$, and $U^\dagger ZU=-Z$. Conjugating $XY=iZ$ in these two ways gives both $iZ$ and $-iZ$, an impossibility. This <three-axis obstruction to unitary qubit orthogonalization> is a finite test of the obstruction behind the <universal NOT gate>.
\Image[/past-exam-of-the-computer-science-course-of-the-university-of-oxford/2021/quantum-information-sequential-and-spatial.png]
{title=Sequential qubit correlations and singlet correlations differ in the third basis}
{description=Original exact predictions, not measured data. With a maximally mixed initial qubit, the sequential experiment using U=Y has opposite Z and X results but equal Y results. A singlet pair has opposite results in all three bases. Correlation +1 means equal labels and −1 means opposite labels.}
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