Obtaining conclusive information from incomplete experimental quantum tomography

We demonstrate that incomplete quantum tomography can give conclusive information in experimental realizations. We divide the state space into a union of multiple disjoint subsets and determine conclusively which of the subsets a system, prepared in completely unknown state, belongs to. In other words we construct and solve membership problems. Our membership problems are partitions of the state space into a union of four disjoint sets formed by fixing two maximally entangled reference states and boundary values of fidelity function "radius" between the reference states and the unknown preparation. We study the necessary and sufficient conditions of the measurements which solve these membership problems conclusively. We construct and experimentally implement such informationally incomplete measurement on two-photon polarization states with combined one-qubit measurements, and solve the membership problem in example cases.