}\left(t_P\over T_{\rm max}-t\right)^{1/3}$$ with $T_{\rm max}$ the lifetime of the black hole (how long it takes to evaporate) and the subscript $12$ denotes that it is the transition frequency between the two states of the system. Although this model sounds simple-minded it just underlies the robustness of the calculation: it just needs that the black hole have discrete energy levels characterized by a separation determined by the temperature of the black hole. It is general enough to be implemented either assuming the Bekenstein spectrum of area or the spectrum stemming from loop quantum gravity [@BaCaRo]. We assume that we start with the black hole in a pure state which is a superposition of different energy eigenstates (there is no reason to assume that the black hole is exactly in an energy eigenstate, which would imply a stationary state with no radiation being emitted; as soon as one takes into account the broadening of lines due to interaction one has to consider a superposition of states within the same broadened level with a time dependent separation with a similar behavior). Therefore the density matrix has off-diagonal elements.

A detailed calculation [@piombino] for the evolution of the density matrix shows that, $$|\rho_{12}(T_{\rm 