nt - p h / 03 08 13 7 v 1 2 6 A ug 2 00 3 What is the value of an observable between pre - and postselection ?

Hall’s recent derivation of an exact uncertainty relation [Phys. Rev. A64, 052103 (2001)] is revisited. It is found that the Bayes estimator of an observable between preand postselection equals the real part of the weak value. The quadratic loss function equals the expectation of the squared imaginary part of the weak value. Hall recently solved the problem of finding the most efficient estimator for an observable on basis of a measurement of another, incompatible observable [1]. As a consequence, he also found an “exact uncertainty relation”. In this short note, we revisit Hall’s derivation and provide a new interpretation of the results. In particular, we demonstrate that the weak value of an observable is the most efficient estimator of an observable between preselection and postselection. This result is consistent with the fact that weak values are measured in weak measurements where the interaction between the system and the measurement apparatus is weak [2]. Also, we show that the loss function, which expresses the uncertainty in the estimate, equals the squared imaginary part of the weak value. This relation has been called an exact uncertainty relation [1]. It was demonstrated by Aharonov et al. [2] that if a “weak measurement” is performed of an observable â between preselection of a state | ψ〉 and postselection of a state | b〉, then what is measured is the “ weak value” of the observable, aw(b) = 〈b | â | ψ〉 〈b | ψ〉 . (1) More precisely, the expectation value of the meter reading equals the real part of the weak value. e-mail: [email protected]