Quantum-Limited Distance Estimation in Three-Dimensional Optical Superresolution

In conventional optical imaging, the Rayleigh criterion limits the ability to distinguish two closely spaced point sources. While quantum superresolution has shown that this limit is not fundamental, the specific quantum limits for estimating the distance between two incoherent sources in a three-dimensional (3D) environment remain poorly understood. This paper investigates how the spatial structure of the point-spread function (PSF) and the orientation of the sources influence the ultimate estimation precision.

The authors derive the quantum Fisher information matrix (QFIM) for the distance between two incoherent point sources in a 3D spatially invariant imaging system. They introduce a second-order displacement-response tensor, $K$, which characterizes how the imaging system encodes spatial displacements into the optical field. By analyzing the eigensystem of this tensor, the authors determine the optimal and worst-case orientations for source displacement relative to the imaging system's principal response axes.

The study demonstrates that distance information remains finite even in the sub-Rayleigh regime, meaning the "Rayleigh curse" can be overcome. The precision of distance estimation is governed by the geometry of the PSF: specifically, the eigenvalues of the response tensor $K$ dictate the sensitivity. The authors show that for anisotropic imaging systems, the estimation precision is highly dependent on the orientation of the source displacement. Consequently, one can optimize resolution by rotating the imaging system to align its principal response direction with the source displacement vector. Furthermore, the authors show that reflection symmetries of the PSF provide a straightforward method for identifying these optimal principal axes. For the specific case of a 3D Gaussian PSF, the response tensor is directly proportional to the inverse of the spatial covariance matrix, providing a clear geometric interpretation of the quantum limit.