ua nt - p h / 06 12 08 0 v 4 8 M ar 2 00 7 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 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 1.340 √ L/NT radians. ∗This work was sponsored by the Air Force under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the U.S. Government.