= POVM does not determine the post-measurement state
{c}
A <POVM> fixes outcome probabilities, but a <quantum instrument> additionally fixes the conditional output state. For computational-basis effects $E_y=|y\rangle\langle y|$, the operators $M_y=|y\rangle\langle y|$ leave the measured eigenstate as output, whereas $N_y=|0\rangle\langle y|$ reset every outcome to $|0\rangle$. Both have $M_y^\dagger M_y=N_y^\dagger N_y=E_y$. Thus a specification of effects alone cannot select a unique joint state that retains the measured quantum system.
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