Necessity of negative Wigner function for tunneling

We consider in what sense quantum tunneling is associated with nonclassical probabilistic behavior. We use the Wigner function quasiprobability picture. We give a definition of tunneling that allows us to say whether in a given scenario there is tunneling or not. We prove that this can only happen if the Wigner function is negative and/or a certain measurement operator which we call the tunneling rate operator has a negative Wigner function. We also investigate tunneling in postquantum theories.