Loss-induced limits to phase measurement precision with maximally entangled states
Mark A. Rubin, Sumanth Kaushik · Physical Review A · 2007
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 $1.018L\phantom{\rule{0.3em}{0ex}}\text{radians}$, 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 $1.340\sqrt{L∕{N}_{T}}\phantom{\rule{0.3em}{0ex}}\text{radians}$.