Refined Definitions of Heat and Work in Quantum Thermodynamics B. Ahmadi, 1, 2, ∗ S. Salimi, 1 and A. S. Khorashad 1 1 Department of Physics, University of Kurdistan, P.O.Box 66177-15175, Sanandaj, Iran 2 International Centre for Theory of Quantum Technologies, University of Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland (Dated: July 14, 2021) In this paper, unambiguous redefinitions of heat and work are presented for quantum thermo- dynamic systems. We will use genuine reasoning based on which Clausius originally defined work and heat in establishing thermodynamics. The change in the energy which is accompanied by a change in the entropy is identified as heat, while any change in the energy which does not lead to a change in the entropy is known as work. It will be seen that quantum coherence does not allow all the energy exchanged between two quantum systems to be only of the heat form. Several examples will also be discussed. Finally, it will be shown that these refined definitions will strongly affect the entropy production of quantum thermodynamic processes giving new insight into the irreversibility of quantum processes. I. INTRODUCTION In the last few decades we have been witnessing a constantly growing interest in understanding thermody- namic phenomena at the quantum scale [1–7]. Novel fundamental questions arise, such as: how do the laws of thermodynamics emerge in this regime? How can the concepts of heat and work be extended from classical thermodynamics to the quantum realm? How are ther- modynamic processes affected by the presence of quan- tum coherence and entanglement? Extending work and heat from classical thermodynamics to quantum thermo- dynamics has been one of the major issues in the liter- ature. As is discussed in the following some difficulties appear in identifying work and heat properly that need to be taken care of. In classical thermodynamics a change in the energy of a system is divided into two parts: heat and work [8–11], dE A = dQ A + dW A , (1) where dQ A is the heat absorbed by system A and dW A the work performed on system A. Eq. (1) is referred to as the first law of thermodynamics. Heat is defined as the energy in transit, between two systems, which is accom- panied by a change in the entropy of the system [8–11]. And work is defined as the energy in transit which does not lead to any change in the entropy of the system. Heat can only be transferred to the system of interest from another system (environment) through some interaction, while work can be done on the system in two ways: by an external force (field) or by another system via interaction [8–11] and since interactions are not generally under the control of the observer therefore some ambiguities may arise in distinguishing work from heat in both classical and quantum setups. For instance, consider a classical gas A (system of interest) in contact with another clas- sical gas B with a membrane separating them (see Fig. * Electronic address: b.ahmadi19@gmail.com (1)). The total system AB is insulated against heat from the surroundings. Work is done on system A through the external force, F ext , and heat can be transferred to system A through the membrane. If the membrane is movable, work can also be done on system A by system B via the membrane. This means that the exchanged energy between the two systems can be of both heat and work forms, i.e., dE exc = dW exc + dQ exc . FIG. 1: (Color online) A classical gas A (system of interest) is in contact with another gas B (environment). The total system AB is insulated against heat from the surroundings. If the pressures of the gases are different from each other and the membrane is movable then work can be done on system A by system B through the membrane. Thus for system A one has dW A = dW ext + dW exc , (2) dQ A = dQ exc . (3) Since the displacement of the membrane is not controlled, dW exc cannot be easily distinguished from dQ exc , there- fore some ambiguities may arise in identifying work and heat. One may claim that dW exc is not of importance to the observer therefore there is no point in distinguish- ing dW exc from dQ exc . But, as we will see in the fol- lowing, distinguishing dW exc from dQ exc becomes crucial when investigating the entropy production and the irre- versibility of a thermodynamic process [8–11]. In classi- cal systems one usually fixes the membrane not to move hence all the exchanged energy is of the heat form, i.e., dE exc = dQ exc . However, in the quantum version of the above example usually there is no way to control arXiv:1912.01983v6 [quant-ph] 13 Jul 2021