Weak-value-amplification analysis beyond the Aharonov-Albert-Vaidman limit

The weak-value (WV) measurement proposed by Aharonov, Albert, and Vaidman (AAV) has attracted a great deal of interest in connection with quantum metrology. In this paper, we extend the analysis beyond the AAV limit and obtain a few main results. (i) We obtain a nonperturbative result for the signal-to-noise ratio (SNR). In contrast to the AAV prediction, we find that the SNR asymptotically gets worse when the AAV WV A(w) becomes large, i.e., in the case g vertical bar A(w)vertical bar(2)>> 1, where g is the measurement strength. (ii) With the increase of g (but also small), we find that the SNR is comparable to the result under the AAV limit, while both can reach-actually the former can slightly exceed-the SNR of the standard measurement. However, with a further increase of g, the WV technique will become less efficient than the standard measurement, despite the fact that the postselection probability is increased. (iii) We find that the Fisher information can characterize the estimate precision qualitatively well as the SNR, yet the difference will become more prominent with the increase of g. (iv) We carry out analytic expressions of the SNR in the presence of technical noises and illustrate the particular advantage of the imaginary WV measurement. The nonperturbative result of the SNR manifests a favorable range of the noise strength and allows an optimal determination.