Maximum information states for coherent scattering measurements

The use of coherent light for precision measurements has been a key driving force for numerous research directions, ranging from biomedical optics to semiconductor manufacturing. Recent work demonstrates that the precision of such measurements can be significantly improved by tailoring the spatial profile of light fields used for estimating an observable system parameter. These advances naturally raise the intriguing question of which states of light can provide the ultimate measurement precision. Here, we introduce a general approach to determine the optimal coherent states of light for estimating any given parameter, regardless of the complexity of the system. Our analysis reveals that the light fields delivering the ultimate measurement precision are eigenstates of a Hermitian operator which quantifies the Fisher information based on the system's scattering matrix. To illustrate this concept, we experimentally show that these maximum information states can probe the phase or the position of an object that is hidden by a disordered medium with a precision improved by an order of magnitude as compared to unoptimized states. Our results enable optimally precise measurements in arbitrarily complex systems, thus establishing a new benchmark for metrology and imaging applications.