content_image/363324.jpg content_image/363321.jpg
system was prepared in the state ˆ ρ$_{m}$ , is then given by: 

<img src='content_image/363324.jpg'>

This result constitutes the Born’s rule, in which the proposition operator ˆ Π$_{n}$ is nothing else than the POVM element corresponding to the result ’n’. In the framework of states and propositions, the POVM elements should be understood as propositions about the state of the system interacting with the measuring device. These operators also describe the behavior of responses of this measuring device, and the corresponding propositions are simply labeled by the responses of the apparatus performing their tests. 

Moreover, any proposition can be reduced to more simple propositions corresponding to projectors on a larger Hilbert space. This is the Neumark’s extension [11] which can often be interpreted as a noise influence on ideal measurements. Such a modeling is in fact common in quantum optics. An imperfect optical measurement can be viewed as an ideal measurement onto the signal field, which has interacted with an appropriate noise field. 

## C. Retrodictive approach 

Contrary to the previous one, this approach is less usual and leads to difficulties linked to an inappropriate use of its tools (see for example [12]). We start with an example in order to ”demystify” such a approach. 

We consider a perfect photon counter able to discern the number of absorbed photons. When such a device displays n counts, the proposition about the measured system simply corresponds to the projector ˆ P$_{n}$ = | n 〉〈 n | . The pre-measurement state, which is sufficient to describe this result, is then this photon number state ˆ P$_{n}$ . This one should allow us to predict the states in which the measured system was prepared, before its interaction with the apparatus giving the result on which we base such a retrodiction. 

The link between the proposition checked by the apparatus and the pre-measurement state is obvious in this ideal case. We will generalize it by using the generalization of Gleason’s theorem [II]. Indeed, we can already write an expression for the retrodictive probability. The probability of preparing the measured system in a given state ˆ ρ$_{m}$ , when we have the result ’n’, can be written as 

<img src='content_image/363321.jpg'>

