= Solution
The opposite $x$-axis <pure states> have projectors $(I\pm\sigma_x)/2$. Averaging them gives
$$
\boxed{\rho_C=\frac12\frac{I+\sigma_x}{2}+\frac12\frac{I-\sigma_x}{2}=\frac I2=\rho_B.}
$$
Therefore \b[C and B cannot be distinguished by any measurement on the atoms alone]. Every <POVM> outcome has the same <probability> $\operatorname{Tr}(E\rho)$ for the two preparations; this is the operational equivalence of different <quantum state ensembles>. Accessible preparation labels or additional correlations would be extra information. C can be distinguished from A by the $x$-axis <spin measurement> in part (c).
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