Quantum Properties of Measurements
A recent work shows that the properties of a measurement are not revealed by the POVM element corresponding to the measurement, but by its pre-measurement state. This one is the main tool of the retrodictive approach of quantum physics in which we make predictions about state preparations leading to a measurement result. We show, that this state simply corresponds to the normalized POVM element:
We can make predictions about preparations leading to the result 'n' by using an expression similar to Born's rule:
in which is a hermitian and positive operator corresponding to a proposition about the state of the measured system just after its preparation in some a state . Such an approach allows us to determine in which kind of states the system was prepared for leading to the result 'n'.
Thus, the non-classicality of a measurement corresponds to the non-classicality of its pre-measurement state, for which such a notion can be measured by different signatures of non-classicality. The projective character of a measurement can be measured by its projectivity which is the purity of its pre-measurement state:
The measurement is projective when its pre-measurement state is a pure quantum state . Thus, the corresponding POVM element is given by:
where is in fact the detection efficiency of the state, since Born's rule leads to . Therefore, the measurement can be projective but non-ideal, which is an important distinction with the usual definition of projective measurements.
Read more about this topic: POVM
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