 POVM does not determine the post-measurement state

A POVM fixes outcome probabilities, but a quantum instrument additionally fixes the conditional output state. For computational-basis effects $E_y=|y⟩⟨y|$, the operators $M_y=|y⟩⟨y|$ leave the measured eigenstate as output, whereas $N_y=|0⟩⟨y|$ reset every outcome to $|0⟩$. Both have $M_y^{†}{M}_y=N_y^{†}{N}_y=E_y$. Thus a specification of effects alone cannot select a unique joint state that retains the measured quantum system.