  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.
