If I understand your answer correctly, you are considering something like an ensemble of states? In that case, of course you can simply count the branches into which the state may split. I suppose its equivalent to checking some thermodynamic function of the ensemble, that will tell you about the degrees of freedom. But I was actually interested in the quantum mechanics of a single state of a single system in a single instance. For example you measure some property of the system such that it will be in state that is degenerate, say the energy of the electron in the H-Atom, such that its spin-state will be degenerate. Now can you apply some operator on that state function that gives you eigenvalue 2 and leaves the state function as it is. In this case its obviously the spin-multiplicity, in other cases it will be the angular momentum. But what about the general case? Are there any operators in general that can do the job?