obvious for Landauer erasures since all possible accessible final states do not span the whole of the Hilbert space of AB, because any initial pure state of AB is mapped to a final product state where A is in | ψ$_{0}$ 〉$_{A}$ state[24]. As U$_{AB}$ is not surjective, it cannot possibly be unitary. Such proper isometries can in principle be realized by a Hamiltonian evolution, provided that ideal tools like perfectly impenetrable barriers are used for making parts of the Hilbert space inaccessible. It is assumed that the bath is in canonical thermal equilibrium state before the transfor- mation and thus where β = 1 /k$_{B}$T is the inverse temperature and Z$_{B}$ is the corresponding partition function. It is appropriate to also assume that the bath is sufficiently small so that ρ$_{B}$ is manageable. We will say that B is a finite system if Z$_{B}$ is finite for all temperatures ( Z$_{B}$ = ∞ ). In this case, B has finite thermodynamical properties. Note that a single Hydrogen atom or a single free particle placed in an infinite space has an infinite partition function and therefore these are not finite systems by the current definition. But, if the atom or the particle is placed inside a large but finite box, then they will become finite systems. Note also that any realistic system, which has a finite number of particles constrained inside a finite volume, will be classified as finite by this definition. Infinite systems have individually vanishing level occupation probabilities and consequently the analysis of such systems present some difficulties. This is the main reason for using finite systems in this article. Let E$_{B}$ = tr ρ$_{B}$H$_{B}$ be the equilibrium energy of the bath. Since there is no structural change in the bath, the heat dumped to the bath during the realization is equal to the change in the average energy, Δ E$_{B}$ , between times t$_{1}$ and t$_{2}$ . It can be expressed as where Q is the HTO of this realization, which is defined as A number of comments have to be made in order to comprehend the real nature of the HTO. First, Q is a hermitian operator defined on the state space H$_{A}$ of the device A. Its 7 Q = tr$_{B}$ [ ( 1$_{A}$ ⊗ ρ$_{B}$ ) U † $_{AB}$( 1$_{A}$ ⊗ H$_{B}$ ) U$_{AB}$ ] − E$_{B}$ 1$_{A}$ . (9) Δ E$_{B}$ = tr( ρ ′ B − ρ$_{B}$ ) H$_{B}$ , (8) = tr$_{AB}$ ( ρ$_{A}$ ⊗ ρ$_{B}$ ) ( U † $_{AB}$( 1$_{A}$ ⊗ H$_{B}$ ) U$_{AB}$ ) − E$_{B}$ = tr$_{A}$( ρ$_{A}$Q ) ρ$_{B}$ = 1 Z$_{B}$ exp ( − βH$_{B}$ ) , (7)