Thus, in the predictive approach [III B], the study of an apparatus is restricted to non- destructive measurements. Indeed, it needs a kind of QPT for determining the measure- ment operators ˆ M$_{n,µ}$ . Such a approach also depends on the initial state of the measured system, and cannot lead to a study of the apparatus as an implementation of tests checking propositions. Furthermore, different measurement processes { ˆ M$_{n,µ}$ } µ can check one same proposition, since the decomposition of a given POVM element (9) is not unique. In other words, there is no general rule to design a measuring device with a given POVM, underlying the importance of an experimental determination of POVM elements describing an appara- tus [4]. Therefore, the predictive approach is more adapted for studying the measurement effects on a system, instead to really translate its behavior in terms of states. The retrodictive approach [III C] seems to be more close to such a goal. Indeed, from a given result, we make predictions about the preparations of the system before its interaction with the apparatus checking the property corresponding to this result. The main tool of this approach is the retrodicted pre-measurement state, that we propagate backward in time for predicting these preparations as depicted on Fig. 1. We can therefore determine in which kind of states the system was prepared for leading to such a result. Actually, the pre-measurement states of an apparatus can directly be reconstructed with the same tools as a QST. This experimental procedure is pictured on Fig. 3 for optical detectors. In such a experiment, we can probe the behavior of an optical detector with a statistical mixture of coherent states ˆ ρ [?] = ∑ m p$_{m}$ | α$_{m}$ 〉〈 α$_{m}$ | , obtained by amplitude and phase mod- ulations randomly performed on each light pulse. As a first step, the QST of this mixture could be necessary and leads to its Cholesky decomposition, ˆ ρ [?] = ˆ σ $^{†}$ˆ σ . We must have det { ˆ ρ $^{[?]}$} ̸ = 0 for generating the proposition operators: With these operators, we realize an usual QST of the state ˆ ρ$_{n}$ giving the conditional prob- abilities Pr ( m | n ) = Tr { ˆ ρ$_{n}$ ˆ Λ$_{m}$ } , measured directly in the experiment. For this purpose, we replace the POVM elements describing the measurements in a QST method by these proposition operators (11). We determine then the state ˆ ρ$_{n}$ giving the closest probabilities 9 ˆ Λ$_{m}$ = ( ˆ σ − $^{1}$) † p$_{m}$ | α$_{m}$ 〉〈 α$_{m}$ | ˆ σ − $^{1}$. (11) B. ’State’ translation of measurements