For discussing the energetics of the aforementioned transformation, the time-dependent Hamiltonian H$_{AB}$ ( t ) of the composite system AB has to be discussed. The transformation process is assumed to be carried out over a finite time interval, say between times t$_{i}$ and t$_{f}$ , during which subsystems A and B couple and H$_{AB}$ ( t ) is time dependent. Outside this time interval, however, it is assumed that the two subsystems are uncoupled and have time- independent Hamiltonians. Thus, we have where H$_{A}$ and H ′ A are the Hamiltonians of the device before and after the transformation respectively and H$_{B}$ is the bath Hamiltonian. The Hamiltonian of the device is not relevant to the main subject of the current article and hence there are no restrictions on H$_{A}$ and H ′ $_{A}$. However, initial and final Hamiltonians of the bath are taken to be identical. This is also operationally reasonable for realistic quantum operations where the environment corresponds to B; a quantum operation on a device may perhaps structurally change the device (so that initial and final Hamiltonians may differ) but it merely couples with the environment without changing it structurally. Finally, no assumptions are made about the strength of the coupling and the Hamiltonian H$_{AB}$ ( t ) during the time interval t$_{i}$ ≤ t ≤ t$_{f}$ . The transformation U$_{AB}$ is the time-development operator corresponding to the Hamilto- nian H$_{AB}$ ( t ) between two times t$_{1}$ and t$_{2}$ where t$_{1}$ ≤ t$_{i}$ and t$_{f}$ ≤ t$_{2}$ . As indicated in Eqs. (5) and (6), it is assumed that A and B do not interact with each other outside the process time interval, ( t$_{i}$, t$_{f}$ ). For this reason, no special values are needed to be assumed about t$_{1}$$_{,}$$_{2}$ . However, the results of the current article can also be extended to cases where there is a weak coupling between A and B outside the process time interval. In this case, t$_{2}$ − t$_{f}$ should be greater than the thermal-equilibration time scales of the bath. Note that for all realistic processes the operator U$_{AB}$ is unitary, i.e., it satisfies U † $_{AB}$U$_{AB}$ = U$_{AB}$U † AB = 1$_{AB}$ . However, for the derivation of the results in this article, this requirement appears to be unnecessarily too restrictive; we only need to assume that U$_{AB}$ is an isometry, i.e., only U † $_{AB}$U$_{AB}$ = 1$_{AB}$ is assumed to be satisfied. Therefore, we allow the possibility that the transformation U$_{AB}$ is not surjective. The generalization of unitary transformations to isometries also serves another purpose. There are some quantum operations, like Landauer erasures, which can only be realized by proper isometries instead of unitaries. This is quite 6 H$_{AB}$ ( t ) = H$_{A}$ + H$_{B}$ for t < t$_{i}$ and (5) H$_{AB}$ ( t ) = H ′ A + H$_{B}$ for t > t$_{f}$ , (6)