Loss-induced limits to phase measurement precision with maximally entangled states

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 1.018L radians, 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 N-T 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 1.340 root L/N-T radians.