POVM - Quantum Properties of Measurements

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:

\hat{\rho}_{retr}^{}=\frac{\hat{\Pi}_{n}}{\mathrm{Tr}\lbrace\hat{\Pi}_{n}\rbrace}.

We can make predictions about preparations leading to the result 'n' by using an expression similar to Born's rule:

\mathrm{Pr}\left(m\vert n\right)=\mathrm{Tr}\lbrace\hat{\rho}_{retr}^{}\hat{\Theta}_{m}\rbrace,

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:

\pi_{n}=\mathrm{Tr}\left}\right)^{2}\right].

The measurement is projective when its pre-measurement state is a pure quantum state . Thus, the corresponding POVM element is given by:

\hat{\Pi}_{n}=\eta_{n}\vert\psi_{n}\rangle\langle\psi_{n}\vert,

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.

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