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only PNRD which is the most faithful to the measurement performed by the APD corre- sponds to the photon number n = 0. In terms of retrodictive probabilities (22), this means that the most probable state in which the system was prepared, before the measurement giving the result ’off’, is the vacuum state. For a low efficiency η → 0, all the photon-number states become equally probable. We can understand this since the measurement is not at all projective ( π$_{off}$ ≃ 0) for such efficiencies. 

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For the measurement corresponding to the result ’on’, the situation is more delicate since the POVM element corresponding to this result leads to divergences in the retrodiction of the pre-measurement state. Indeed, we can only determine an asymptotic equivalent for this pre-measurement state: 

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where D is in fact the dimension of the Hilbert space. The fidelity between this measurement 

