= Solution
Assume the proposed device distinguishes the four displayed <eigenstates>, as an ideal rank-one <projective measurement>. This assumption matters: if all four had the same eigenvalue, the identity <observable> would admit that eigenbasis but its <Lüders rule> measurement would do nothing and could not signal. The printed eigenvectors alone do not exclude this degeneracy.
For the complete resolving measurement, let $c=\cos\theta$, $s=\sin\theta$, and $\Pi_j$ be the four rank-one projectors. If the nonlocal outcome is ignored, the <nonselective projective measurement> channel is $\mathcal D_\theta(\rho)=\sum_j\Pi_j\rho\Pi_j$. Each listed state's local <reduced density matrix> is diagonal in $Z_A$, with $Z_A$ expectation $\cos2\theta$ for either plus state and $-\cos2\theta$ for either minus state. Consequently
$$
\langle Z_A\rangle_{\rm out}=\cos2\theta\,\operatorname{Tr}(C_\theta\rho),\qquad
C_\theta=\Pi_{\Phi^+}-\Pi_{\Phi^-}+\Pi_{\Psi^+}-\Pi_{\Psi^-}.
$$
Compute $C_\theta$ in its even and odd two-dimensional blocks: both have diagonal entries $\cos2\theta,-\cos2\theta$ and off-diagonal entries $\sin2\theta$. Thus
$$
C_\theta=\cos2\theta\,Z_A\otimes I+\sin2\theta\,X_A\otimes X_B,
$$
and
$$
\langle Z_A\rangle_{\rm out}=\cos^2(2\theta)\langle Z_A\rangle_{\rm in}
+\sin2\theta\cos2\theta\langle X_A\otimes X_B\rangle_{\rm in}.
$$
To signal, prepare Alice in $|+\rangle$ and Bob initially in $|+\rangle$. Bob encodes a bit by either doing nothing or applying a local <Pauli Z gate>, which changes his state to $|-\rangle$. Alice's input <reduced density matrix> is identical in both cases, but after the hypothetical instantaneous measurement her local $Z_A$ expectation is
$$
\boxed{\langle Z_A\rangle_{\rm out}=\pm\sin2\theta\cos2\theta=\pm\tfrac12\sin4\theta}.
$$
The two probabilities for Alice's outcome $0$ are $(1\pm\sin2\theta\cos2\theta)/2$, so their difference is $\sin2\theta\cos2\theta$. It is strictly positive for $0<\theta<\pi/4$. Repeated trials let Alice infer Bob's bit while their operations are still spacelike, violating <quantum no-signalling> and relativistic causality. The <relativistic causality constraint on an ideal nonlocal measurement> therefore permits only
$$
\boxed{\theta=0\quad\text{or}\quad\theta=\pi/4}
$$
within the specified interval. The argument requires no rapid communication of the hypothetical nonlocal outcome: Alice reads her own changed local statistics. It rules out the full ideal instrument, not merely the later classical comparison of locally obtained records.
Back to article page
