that's not how I remembered the problem there, though. The task was encoding info only in the choice of measurement basis, right? So Bob performs some PVM $\{\Pi^b_\alpha\}_\alpha$, and passes along the resulting state (but not the outcome). Alice never gets the measurement outcome, so she gets $\sum_\alpha (I\otimes \Pi^b_\alpha)\rho(I\otimes \Pi^b_\alpha)$, from which she wants to estimate $b$.
So the task is to find $b$ from the probability distribution $p(a|b)=\sum_\alpha \langle \mu(a), (I\otimes \Pi^b_\alpha)\rho(I\otimes \Pi^b_\alpha)\rangle$ maybe?