  Experiments with quantum gas microscopes have started to explore the
antiferromagnetic phase of the two-dimensional Fermi-Hubbard model and effects
of doping with holes away from half filling. In this work we show how direct
measurements of the system averaged two-spin density matrix and its full
counting statistics can be used to identify different correlated magnetic
phases with or without long-range order. We discuss examples of phases which
are potentially realized in the Hubbard model close to half filling, including
antiferrromagnetically ordered insulators and metals, as well as insulating
spin-liquids and metals with topological order. For these candidate states we
predict the doping- and temperature dependence of local correlators, which can
be directly measured in current experiments.
