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We address the problem whether a given set of expectation values iscompatible with the first and second moments of the generic spin operators of asystem with total spin j. Those operators appear as the Stokes operator inquantum optics, as well as the total angular momentum operators in the atomicensemble literature. We link this problem to a particular extension problem forbipartite qubit states; this problem is closely related to the symmetricextension problem that has recently drawn much attention in different contextsof the quantum information literature. We are able to provide operational,approximate solutions for every large spin numbers, and in fact the solutionbecomes exact in the limiting case of infinite spin numbers. Solutions for lowspin numbers are formulated in terms of a hyperplane characterization, similarto entanglement witnesses, that can be efficiently solved with semidefiniteprogramming.

Truncated su(2) moment problem for spin and polarization states