M. Anholm, D. Ashery, Spencer Behling, Benjamin Fenker, D. Melconian, M. Mehlman, John Behr, Alexandre Gorelov, K. Olchanski, Claire Preston, Claire Warner, G. Gwinner
The presence of loss limits the precision of an approach to phase measurement using maximally entangled states, also referred to as NOON states. A calculation using a simple beam-splitter model of loss shows that, for all nonzero values L of the loss, phase measurement precision degrades with increasing number N of entangled photons for N sufficiently large. For L above a critical value of approximately 0.785, phase measurement precision degrades with increasing N for all values of N. For L near zero, phase measurement precision improves with increasing N down to a limiting precision of approximately 1.018Lradians, attained at N approximately equal to 2.218∕L, and degrades as N increases beyond this value. Phase measurement precision with multiple measurements and a fixed total number of photons NT is also examined. For L above a critical value of approximately 0.586, the ratio of phase measurement precision attainable with NOON states to that attainable by conventional methods using unentangled coherent states degrades with increasing N, the number of entangled photons employed in a single measurement, for all values of N. For L near zero this ratio is optimized by using approximately N=1.279∕L entangled photons in each measurement, yielding a precision of approximately 1.340L∕NTradians.
In this study, we investigate whether the weak measurement can enhance the estimation precision, when coupling between system and measurement apparatus is extremely weak. Our result shows that the weak measurement could enhance the estimation precision, when the noise is introduced during interaction process. By calculating and comparing the Fisher information under weak measurement and traditional quantum measurement scheme, we quantify the precision-improvement by the weak measurement and find the optimal post-selection protocol. Furthermore, we argue that there exists a threshold for the noise strength that the quantum weak measurement renders a higher precision compared to the traditional measurement.