Fidelity of Quantum States - Fidelity of Quantum Measurements

Fidelity of Quantum Measurements

The fidelity of a measurement with a projective measurement is defined as the overlap between their pre-measurement states:

$\mathcal{F}_{n}\left(\psi_{tar}\right)=\langle\psi_{tar}\vert\hat{\rho}_{retr}^{}\vert\psi_{tar}\rangle,$

where and are respectively the pre-measurement state corresponding to the result "n" and the target state in which we would like measuring the system before its interaction with the measurement apparatus.

The pre-measurement state is the main tool of the retrodictive approach of quantum physics in which we make predictions about state preparations leading to a certain measurement result. In such an approach, this fidelity has an interesting meaning: this is nothing else than the retrodictive probability of preparing the system in the target state when we read the result "n". Thus, when a measurement is sufficiently faithful, the most probable state in which the system was prepared before the measurement giving the result "n" is this target state .

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