Nonperturbative theory of weak pre- and post-selected measurements

Details Nonperturbative theory of weak pre- and post-selected measurements Nonperturbative theory of weak pre- and post-selected measurements

2011-09-28

arxiv.org/abs/1109.6315v1

Physics

Quantum Physics

92 pages, 13 figures

Scientific paper

This paper starts with a review of the topic of strong and weak pre- and post-selected (PPS) 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 associated weak value is required to describe weak PPS measurements. In the linear regime, the theory provides the generalized Aharonov-Albert-Vaidman formula. Moreover, we reveal two new regimes of weak PPS measurements: 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 decreases with increasing the measurement strength. The optimal conditions for weak PPS measurements 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 pre- and 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 making a measurement. Exact solutions for a qubit coupled to several types of meters are obtained and used for numerical calculations which agree with the theory.

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