Quantum-limited imaging using diffractive optical neural networks

Imaging systems are fundamentally limited by diffraction and the shot noise associated with the quantum nature of light. While quantum metrology has successfully improved the resolution of simple tasks like point-source separation, extending these principles to reconstruct arbitrary, complex objects remains a challenge. This paper addresses how to design a practical measurement apparatus that reaches the fundamental quantum precision limits for general imaging.

The authors formulate general incoherent imaging as a multiparameter quantum estimation problem by parameterizing objects through their band-limited spatial-frequency amplitudes. They use semidefinite programming to compute the Nagaoka-Hayashi Cramér-Rao bound (NHCRB), which defines the precision limit for separable, single-copy measurements. To physically realize this, they employ a diffractive optical neural network (DONN)—a series of trainable phase masks—trained via gradient descent to minimize the variance of the estimated amplitudes. This approach allows the measurement basis to emerge naturally from the training process without requiring prior knowledge of the specific scene.

The study shows that for single-parameter estimation, the NHCRB coincides with the ultimate quantum Cramér-Rao bound (QCRB). When estimating multiple parameters, the NHCRB becomes the relevant, tighter limit due to the non-commutativity of the optimal observables. The authors demonstrate that a DONN with sufficient depth can saturate this NHCRB across the entire transmitted spatial-frequency band. In simulations of complex 2D objects, including a diatom fragment and an atomic lattice, the DONN-based architecture consistently outperforms direct imaging by providing more precise measurements of high-frequency components, thereby enabling superior image reconstruction.