= Solution
At $\theta=0$ the four <eigenstates> are the computational product basis, with irrelevant signs on two vectors. Alice and Bob measure their own system qubits in $Z$. These local <projective measurements> preserve each product <eigenstate>; their pair of records identifies the global outcome after <local operations and classical communication>.
At $\theta=\pi/4$ the four states are the <Bell states>. They are simultaneous <eigenstates> of the commuting <Pauli operators> $Z_AZ_B$ and $X_AX_B$: $\Phi^+,\Phi^-,\Psi^+,\Psi^-$ have respective pairs $(+,+),(+,-),(-,+),(-,-)$. Their nonlocal parity measurements can be performed without directly distinguishing the local system spins.
Use the first shared <Bell state> pair to perform the <entanglement-assisted nondemolition parity measurement> of $Z_AZ_B$. Use the second shared pair for $X_AX_B$: both parties apply a local <Hadamard gate> to their system qubit, execute the same local system-to-meter <CNOT gates> and $z$-meter measurements, then undo the <Hadamard gates>. This measures $X_AX_B$ because $HZH=X$. The two system parity projectors commute, since anticommutation at both sites cancels:
$$
[Z_AZ_B,X_AX_B]=0,\qquad
P_{z,x}=\frac14(I+zZ_AZ_B)(I+xX_AX_B),\quad z,x\in\{\pm1\}.
$$
Each $P_{z,x}$ is the corresponding rank-one <Bell state> projector. Every complete tuple of four local meter records has system <Kraus operator> $P_{z,x}/2$ for its two parities. Summing the four record tuples compatible with $(z,x)$ gives the ideal outcome map $\rho\mapsto P_{z,x}\rho P_{z,x}$. \b[The protocol is a Bell-state nondemolition measurement]: an input <Bell state> is preserved, while an arbitrary input is projected onto the reported <Bell state> with the <Born rule> probability.
The local circuits need no adaptive communication between the laboratories, so both parties can finish inside the specified time window. Global identification of $(z,x)$ still requires later <local operations and classical communication>. This endpoint protocol respects <quantum no-signalling>, unlike the hypothetical intermediate-angle instrument in part (i).
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