  Piechocinska provided proofs for both the classical and quantum cases. A brief sketch of Piechocinska's quantum proof is possible. First simply define a function  Γ(i, f, m, n)= ln pi −  ln pf − β(En − Em),  where pi is the occupation probability of an initial state i of the system, pf the occupation probability of a final state f, beta the dispersion parameter of the canonical distribution of the heat bath, and Em and En the energies of eigenstates of the heat bath. No particular physical significance is attached to this function. However, from the requirement that the overall evolution be unitary, it can be shown ⟨ e−Γ ⟩=1  and so by  convexity ⟨ Γ ⟩ ≥ 0. The mean heat generated in the heat bath can then easily be shown to  satisfy ⟨ Q ⟩ ≥ −ΔH k T ln 2  where  ΔH =  ∑ i pi log pi  −  ∑ f pf log pf  is the change in Shannon information over the states of the system during the operation. If there are assumed to be two equiprobable input states, the requirement that the operation be a resetting operation and leave the system determinately in a specific output state leads to ΔH = −1 and the usual expression of Landauer's principle is obtained.