The use of coherent light for precision measurements has been a key driving force for numerous research directions, ranging from biomedical optics^1,2 to semiconductor manufacturing³. Recent work demonstrates that the precision of such measurements can be substantially improved by tailoring the spatial profile of light fields used for estimating an observable system parameter^{4,5,6,7,8,9,10}. These advances naturally raise the intriguing question of which states of light can provide the ultimate measurement precision¹¹. 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 that quantifies the Fisher information from the system’s scattering matrix¹². 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 compared with unoptimized states. Our results enable optimally precise measurements in arbitrarily complex systems, thus establishing a new benchmark for metrology and imaging applications^3,13.

使用相干光进行精密测量一直是众多研究方向的关键驱动力，从生物医学光学^1,2到半导体制造³。最近的工作表明，通过调整用于估计可观察系统参数^{4、5、6、7、8、9、10}的光场的空间分布，可以显着提高此类测量的精度。这些进步自然提出了一个有趣的问题，即哪种光状态可以提供最终的测量精度¹¹. 在这里，我们介绍了一种通用方法来确定用于估计任何给定参数的最佳光相干状态，而不管系统的复杂性如何。我们的分析表明，提供最终测量精度的光场是 Hermitian 算子的本征态，该算子从系统的散射矩阵¹²中量化了 Fisher 信息。为了说明这个概念，我们通过实验表明，这些最大信息状态可以探测被无序介质隐藏的物体的相位或位置，与未优化状态相比，精度提高了一个数量级。我们的结果可以在任意复杂的系统中实现最佳精确测量，从而为计量和成像应用建立新的基准^3,13.