Merit of amplification by weak measurement in view of measurement uncertainty

Aharonov’s weak value is a physical quantity obtainable by weak measurement, which admits amplification through state selections. The amplification is expected to be useful for precision measurement, but it is achieved at the expense of statistical deterioration due to the selections, which brings the question as to whether it can provide a truly superior means over the conventional measurement. We approach this problem by taking measurement uncertainty into account, and present a probabilistic evaluation of its impact to the final result of the measurement in weak measurement. By examining the significance condition for detecting the coupling g, we show that the trade-off relation between the amplification effect and the statistical deterioration does permit a finite range of usable amplification where the superiority is ensured. Apart from the Gaussian state models employed for demonstration, our argument is mathematically rigorous and general; it is free from approximation and valid for arbitrary observables A and couplings g. © 2014, Chapman University.