Noisy distributed sensing in the Bayesian regime

We consider non-local sensing of scalar signals with specific spatial dependence in the Bayesian regime. We design schemes that allow one to achieve optimal scaling and are immune to noise sources with a different spatial dependence than the signal. This is achieved by using a sensor array of spatially separated sensors and constructing a multi-dimensional decoherence free subspace. While in the Fisher regime with sharp prior and multiple measurements only the spectral range Delta is important, in single-shot sensing with broad prior the number of available energy levelsLis crucial. We study the influence ofLand Delta also in intermediate scenarios, and show that these quantities can be optimized separately in our setting. This provides us with a flexible scheme that can be adapted to different situations, and is by construction insensitive to given noise sources.