Weak value amplification considered harmful

Liang Xu, Lijian Zhang

Christopher Ferrie, Joshua Combes

We show using statistically rigorous arguments that the technique of weak value amplification (WVA) does not perform better than standard statistical techniques for the tasks of single parameter estimation and signal detection. Specifically we prove that post-selection, a necessary ingredient for WVA, decreases estimation accuracy and, moreover, arranging for anomalously large weak values is a suboptimal strategy. In doing so, we explicitly provide the optimal estimator, which in turn allows us to identify the optimal experimental arrangement to be the one in which all outcomes have equal weak values (all as small as possible) and the initial state of the meter is the maximal eigenvalue of the square of the system observable. Finally, we give precise quantitative conditions for when weak measurement (measurements without post-selection or anomalously large weak values) can mitigate the effect of uncharacterized technical noise in estimation.

Yutaka Shikano

Yuichiro Mori

The novel notion of a physical quantity called 'weak value', which was proposed by Aharonov et al. in 1988, has attracted much attention in the fields of quantum foundations and quantum measurement.In particular, the fact that through proper state selections the weak value can be made arbitrarily large, exceeding the conventional bound set by the maximum eigenvalue of the physical observable, has been widely regarded as a promising source for precision measurement.Indeed, in recent years we have seen increasingly many reports of successful applications of the weak value in photonic systems with various instances of observables amplified effectively.However, it has also been recognized that there are two major obstacles in realizing this weak value amplification (WVA).First, it is unclear under what conditions the WVA becomes effective due to the conflict between the ambiguity caused by the state selections and the amplification effect.Second, the required state selections may not be available depending on the systems one is dealing with.In order to expand the applicability of the WVA, one must solve these problems properly.In this dissertation, we first review the basics of precision measurement and the WVA and then present our method of evaluation for the WVA considering the measurement uncertainty including the ambiguity of the WVA.We applied our method of evaluation for the two celebrated experiments to confirm the basic validity of the results of the two experiments.In addition, as a new application of the WVA, we consider the time evolution of unstable particles to find that the decay time distribution is generally characterized by the weak value of the Hamiltonian and that the state selections can indeed be performed by measuring the particles produced during the decay.Our result indicates that the WVA may also be applicable to the system of unstable particles.

Josiah Sinclair, David Spierings, Aharon Brodutch +1 more

Wei-Tao Liu, Julián Martínez-Rincón, John C. Howell

We discuss interferometric parameter estimation of the amplitude, instead of the phase, via weak value amplification. The considered weak interaction introduces modulation on the amplitude of the wave function; therefore, the two-party state experiences a nonunitary evolution. With the same pre- and postselection states as those of original weak value amplification, a much larger anomalous amplification factor can be attained. The shift in the intensity profile at the dark port and the signal-to-noise ratio for the parameter of interest get further amplified by the weak value, compared to phase-based weak value amplification. Although the nonunitary evolution introduces loss, more information about the parameter of interest can be extracted. Thus the weak value amplification is extended to measurement of amplitude and nonunitary processes. We also discuss possible applications of this idea for precisely measuring tiny rotations or enhancing the image contrast of transparent objects.