Experimental adaptive Bayesian estimation for a linear function of distributed phases in photonic quantum networks

The bipartite and multipartite entanglement resources of quantum networks can enhance sensitivity for estimating distributed parameters beyond the classical limits. Recent experimental studies on distributed parameter estimation based on quantum networks have achieved high precision beyond the shot-noise limit (SNL) within certain portions of the parameter space. Towards a realistic distributed parameter estimation scenario, a next key issue is how to achieve the high precision parameter independently with limited measurement data. In this work, we present a photonic experiment employing an efficient Bayesian method to estimate a linear function of four spatially distributed unknown phases. For arbitrary true phase values, our experiment shows the capability of achieving high sensitivities beyond the SNL in a post-selected regime using a restricted amount of measurement rounds. Our work gives a start for the experimental study of distributed adaptive Bayesian quantum estimation. Additionally, this method holds promising utility for more intricate or universal tasks associated with sensing distributed parameters in quantum networks.