= Solution
The <projective measurement> probabilities are $P(\sigma_j=\pm1)=\operatorname{Tr}[\rho(I\pm\sigma_j)/2]$. Both <density operators> have zero $z$-component of their <Bloch vector>, so
$$
P_A(\sigma_z=\pm1)=P_B(\sigma_z=\pm1)=\frac12.
$$
Their full $\sigma_z$ outcome distributions, not merely their mean values, coincide. Along $x$, however,
$$
\boxed{P_A(\sigma_x=+1)=1,\quad P_A(\sigma_x=-1)=0;\qquad
P_B(\sigma_x=\pm1)=\frac12.}
$$
Thus repeated <spin measurement> along $x$ distinguishes the preparations statistically. A single negative outcome excludes A, while a single positive outcome is compatible with either; under B the <probability> of $N$ independent positive outcomes is $2^{-N}$.
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