Nonperturbative theory of weak pre- and post-selected measurements

This paper starts with a brief review of the topic of strong and weak preand post-selected (PPS) quantum measurements, as well as weak values, and afterwards presents original work. In particular, we develop a nonperturbative theory of weak PPS measurements of an arbitrary system with an arbitrary meter, for arbitrary initial states of the system and the meter. New and simple analytical formulas are obtained for the average and the distribution of the meter pointer variable. These formulas hold to all orders in the weak value. In the case of a mixed preselected state, in addition to the standard weak value, an associatedweak value is required to describeweak PPSmeasurements. In the linear regime, the theory provides the generalized Aharonov–Albert–Vaidman formula. Moreover, we reveal two new regimes of weak PPSmeasurements: the strongly-nonlinear regime and the inverted region (the regime with a very large weak value), where the system-dependent contribution to the pointer deflection decreaseswith increasing themeasurement strength. The optimal conditions for weak PPSmeasurements are obtained in the strongly-nonlinear regime, where the magnitude of the average pointer deflection is equal or close to the maximum. This maximum is independent of the measurement strength, being typically of the order of the pointer uncertainty. In the optimal regime, the small parameter of the theory is comparable to the overlap of the preand post-selected states. We show that the amplification coefficient in the weak PPS measurements is generally a product of two qualitatively different factors. The effects of the free system and meter Hamiltonians are discussed. We also estimate the size of the ensemble required for a measurement and identify optimal and efficient meters for weakmeasurements. Exact solutions are obtained for a certain class of the measured observables. These solutions are used for numerical calculations, the results of which agree with the theory. Moreover, the theory is extended to allow for a completely general post-selection measurement. We also discuss timesymmetry properties of PPS measurements of any strength and the relation between PPS and standard (not post-selected) measurements. © 2012 Elsevier B.V. All rights reserved.