We establish a simple method to assess the quantum Fisher information required for resolving two incoherent point sources with an imaging system. The resulting Cramér-Rao bound shows that the standard Rayleigh limit can be surpassed by suitable coherent measurements. We explicitly find these optimal strategies and present a realization for Gaussian and slit apertures. This involves a projection onto the optimal bases that is accomplished with digital holographic techniques and is compared with a CCD position measurement. Our experimental results unequivocally confirm unprecedented sub-Rayleigh precision.

We establish the multiparameter quantum Cramér-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cramér-Rao bound is independent of the source separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost important in realistic scenarios, such as observational astronomy.The time-honored Rayleigh criterion [1] specifies the minimum separation between two incoherent optical sources using a linear imaging system. As a matter of fact, it is the size of the point spread function [2] that determines the resolution: two points closer than the PSF width will be difficult to resolve due to the substantial overlap of their images.Thus far, this Rayleigh criterion has been considered as a fundamental limit. Resolution can only be improved either by reducing the wavelength or by building higher numericalaperture optics, thereby making the PSF narrower. Nonetheless, outstanding methods have been developed lately that can break the Rayleigh limit under special circumstances [3][4][5][6][7][8][9][10][11][12]. Though promising, these techniques are involved and require careful control of the source, which is not always possible, especially in astronomical applications.Despite being very intuitive, the common derivation of the Rayleigh limit is heuristic and it is deeply rooted in classical optical technology [13]. Recently, inspired by ideas of quantum information, Tsang and coworkers [14][15][16][17] have revisited this problem using the Fisher information and the associated Cramér-Rao lower bound (CRLB) to quantify how well the separation between two point sources can be estimated. When only the intensity at the image is measured (the basis of all the conventional techniques), the Fisher information falls to zero as the separation between the sources decreases and the CRLB diverges accordingly; this is known as the Rayleigh curse [14]. However, when the Fisher information of the complete field is calculated, it stays constant and so does the CRLB, revealing that the Rayleigh limit is not essential to the problem.These remarkable predictions prompted a series of experimental implementations [18][19][20] and further generalizations [21][22][23][24][25], including the related question of source localization [26][27][28]. All this previous work has focused on the estimation of the separation, taking for granted a highly symmetric configuration with identical sources. In this Letter, we approach the issue in a more realistic scenario, where both sources may have unequal intensities. This involves the si...

By projecting onto complex optical mode profiles, it is possible to estimate arbitrarily small separations between objects with quantum-limited precision, free of uncertainty arising from overlapping intensity profiles. Here we extend these techniques to the time-frequency domain using mode-selective sum-frequency generation with shaped ultrafast pulses. We experimentally resolve temporal and spectral separations between incoherent mixtures of single-photon level signals ten times smaller than their optical bandwidths with a tenfold improvement in precision over the intensity-only Cramér-Rao bound.

We establish the conditions to attain the ultimate resolution predicted by quantum estimation theory for the case of two incoherent point sources using a linear imaging system. The solution is closely related to the spatial symmetries of the detection scheme. In particular, for real symmetric point spread functions, any complete set of projections with definite parity achieves the goal.

We propose a scheme for the reconstruction of the quantum state without a priori knowledge about the measurement setup. Using the data pattern approach, we develop an iterative procedure for obtaining information about the measurement which is sufficient for an estimation of a particular signal state. The method is illustrated with the examples of reconstruction with on/off detection and quantum homodyne tomography.