When the agent is equipped with Thermal Operations, they can apply any unitary operation which preserves the energy of system and environment exactly, Eq. (2.4). The idea behind this requirement is that system and environment can be considered as a global, isolated system, and, according to the principle of energy conservation, the energy of such isolated system needs to be preserved during its evolution. Furthermore, the fact that energy is conserved allows us to precisely quantify the transfers occurring between system and environment, or between any other partition we might additionally consider. Notice that if energy were not conserved, we could also interpret any change in this quantity as an exchange with an additional system that we have not included yet into our description, that would act as a sink/source of energy.
One might question whether considering system and environment as an isolated system is a physically motivated assumption. For it to be a reasonable assumption, we need to include in our description a big enough portion of the environment surrounding the system, so that the interactions with the remaining environment are negligible compared to the energy scale of the global system under examination. This is the case, for example, of any system with local interactions, since the energy of the bulk scales like the volume of the system, whereas the energy on the boundary scales like its area. Otherwise, we can simply consider the entirety of
the surrounding environment, up to the point in which there is nothing else the global system can interact with, since we are essentially considering the whole universe.
In order to describe the interactions between system and environment, the formalism of Thermal Operations makes use of the unitary representation. An alternative description is given in terms of interaction Hamiltonians, which can be either time-dependent or -independent. Since this latter description is commonly used to describe processes occurring in a laboratory, it is worth investigating its connection with Thermal Operations, and understanding in which situations an interacting Hamiltonian can be linked to an energy-preserving unitary operation. A comparison of these two approaches can be found in Ref. [7, Supplemental Material]. The easiest example consists in the one in which the interaction Hamiltonian Hint ∈ B (HS⊗ HB) commutes with the total Hamiltonian of system and environment, that is [Hint, HS+ HB] = 0. In this scenario, the strength of the interaction can be arbitrary, and the coupling can be time-dependent or -independent, but the resulting unitary evolution still commutes with the total Hamiltonian. An example of such interaction Hamiltonian can be found in the (perfectly resonant) Jaynes-Cummings model [125,126]. This model describes the interaction between a two-level system inside a cavity, and a single mode of the electromagnetic field in that cavity. In this picture, the system absorbs a photon of the field to get excited, and emits a photon while decaying. If the energy gap of the system is equal to the energy of the absorbed/emitted photons (that is, when the field is perfectly resonant), the interaction Hamiltonian commutes with the Hamiltonian of system and radiation.
Another situation that can be approximatively described with Thermal Operations is the one in which system and environment are weakly coupled. In this case, the energy scale of the interaction Hamiltonian is negligible compared to the energy scale of the Hamiltonian of system and environment, and therefore these two operators (almost) commute. In classical thermodynamics, where the main system is macroscopic, the weak coupling assumption is often satisfied, and Thermal Operations would therefore apply to this scenario. However, when microscopic systems are considered, they can be strongly coupled with the environment. Our formalism is still able to describe this situation, if for example we slowly bring system and environment in contact, we make them interact (even strongly) and slowly separate them. If this process is slow enough, we find that due to the adiabatic theorem [127,128] the transformation
preserves all the eigenstates of the Hamiltonian of system and thermal reservoir, and therefore the evolution can be described by an energy-preserving unitary operation.
So far, we have seen that Thermal Operations can be used to describe situations in which the interaction Hamiltonian commutes with the total Hamiltonian, or where the interaction coupling is either weak or changes very slowly in time. We still need to consider the case in which the interaction between system and environment is strong and undergoes a sudden quench. This situation cannot be described with Thermal Operations unless we add a bit more structure to our model. If the operation changes the energy of the system, but does not introduce any coherence in the energy eigenbasis, then the transformation can be implemented with Thermal Operations by adding a battery to the framework, see Sec. 2.4. If, instead, the operation also introduces coherence in the energy eigenbasis, then we need to add to the picture a “control system”, i.e., an additional system able to coherently compensate for the energy change in system and environment due to their interaction. Within the framework of Thermal Operations, this system is known as a coherence reservoir, that we describe in more details in Sec. 2.2.5.