Discriminating two spatially separated sources is one of the most fundamental problems in imaging. Recent research based on quantum parameter estimation theory shows that the resolution limit of two incoherent point sources given by Rayleigh can be broken. However, in realistic optical systems, there often exists coherence in the imaging light field, and there have been efforts to analyze the optical resolution in the presence of partial coherence. Nevertheless, how the degree of coherence between two point sources affects the resolution has not been fully understood. Here, we analyze the quantum-limited resolution of two partially coherent point sources by explicitly relating the state after evolution through the optical systems to the coherence of the sources. In particular, we consider the situation in which coherence varies with the separation. We propose a feasible experiment scheme to realize the nearly optimal measurement, which adaptively chooses the binary spatial-mode demultiplexing measurement and direct imaging. Our results will have wide applications in imaging involving coherence of light.

Quantum stochastic phase estimation has many applications in the precise measurement of various physical parameters. Similar to the estimation of a constant phase, there is a standard quantum limit for stochastic phase estimation, which can be obtained with the Mach–Zehnder interferometer and coherent input state. Recently, it has been shown that the stochastic standard quantum limit can be surpassed with nonclassical resources such as squeezed light. However, practical methods to achieve quantum enhancement in the stochastic phase estimation remain largely unexplored. Here we propose a method utilizing the SU(1,1) interferometer and coherent input states to estimate a stochastic optical phase. As an example, we investigate the Ornstein–Uhlenback stochastic phase. We analyze the performance of this method for three key estimation problems: prediction, tracking, and smoothing. The results show significant reduction of the mean square error compared with the Mach–Zehnder interferometer under the same photon number flux inside the interferometers. In particular, we show that the method with the SU(1,1) interferometer can achieve fundamental quantum scaling, achieve stochastic Heisenberg scaling, and surpass the precision of the canonical measurement.

Laser communication using photons should consider not only the transmission environment’s effects, but also the performance of the single-photon detector used and the photon number distribution. Photon communication based on the superconducting nanowire single-photon detector (SNSPD) is a new technology that addresses the current sensitivity limitations at the level of single photons in deep space communication. The communication’s bit error rate (BER) is limited by dark noise in the space environment and the photon number distribution with a traditional single-pixel SNSPD, which is unable to resolve the photon number distribution. In this work, an enhanced photon communication method was proposed based on the photon number resolving function of four-pixel array SNSPDs. A simulated picture transmission was carried out, and the error rate in this counting mode can be reduced by 2 orders of magnitude when compared with classical optical communication. However, in the communication mode using photon-enhanced counting, the four-pixel response amplitude for counting was found to restrain the communication rate, and this counting mode is extremely dependent on the incident light intensity through experiments, which limits the sensitivity and speed of the SNSPD array’s performance advantage. Therefore, a BER theoretical calculation model for laser communication was presented using the Bayesian estimation algorithm in order to analyze the selection of counting methods for information acquisition under different light intensities and to make better use of the SNSPD array’s high sensitivity and speed and thus to obtain a lower BER. The counting method and theoretical model proposed in this work refer to array SNSPDs in the deep space field.