**1. Introduction**

Entropy is one of the central quantities in thermodynamics and, without its precise evaluation, one cannot predict what new phenomena are to be expected in the thermodynamics of a device. In quantum theory, entropy is defined as a nonlinear function of the density matrix, i.e., *S* = −Tr*ρ*<sup>ˆ</sup> ln *ρ*ˆ, in the units of the Boltzmann constant *kB*. The mere nonlinearity indicates that entropy is not physically observable because, by definition, observables are linear in the density matrix. Let us further describe this statement. Here, we do not assume that the density matrix is a physical quantity. The reason is that evaluating all components of a many-body density matrix requires many repetitions of the same experiment with the same initial state. Not only is this difficult but also the fact that measurement changes quantum states prevents exact evaluation. A physical quantity, such as energy or charge, can be measured in the lab in real time and can be defined in quantum theory to linearly depend on the density matrix. This is not true for entropy and therefore we cannot assume it is a physical quantity directly measurable in the lab.

In fact, the precise time evolution of entropy is still an open problem and has not been properly addressed in the literature [1–3]. A consistent theory of quantum thermodynamics can only be

achieved after finding nontrivial relations between the quantum of information and physics. In recent years, exquisite mesoscopic scale control over quantum states has led technology to the quantum realm. This has motivated exploring new phenomena such as exponential speed up in computation as well as power extraction from quantum coherence [4–8]. Recently, there have been attempts to implement quantum versions of heat engines using superconducting qubits [9]. However, recent developments in realizing quantum heat engines, such as in References [10–12], rely on semiclassical stochastic entropy production after discretizing energy. A long-lasting question is how the superposition of states transfers heat and how much entropy is produced as the result of such a transfer.

A quantum heat engine (QHE) is a system with several discrete quantum states and, similar to a common heat engine, is connected to several environments kept at different temperatures. In fact, a number of large heat baths in these engines share some degrees of freedom quantum mechanically. Such a system is supposed to transfer heat according to the laws of quantum mechanics. The motivation for research in QHE originates from differences they may controllably make on the efficiency and output powers. Let us consider the example of two heat baths *A* and *B*, both coupled through a quantum system *q* that contains discrete energies and allows for the superposition of states with long coherence time. Let us clarify that, in this paper, we study the flow of thermodynamic Renyi and von Neumann entropies between the heat baths and quantum system *q*. Therefore, other entropies are beyond the scope of this paper. This quantum system coupled to the two large heat baths is in fact a physical quantum system that is energetically coupled to the reservoirs and allows for stationary flow of heat as well as a flow of thermodynamic entropy from one reservoir to another. We will see in the next section that, similar to physical quantities such as energy and charge, the total entropy of a closed system is a conserved quantity and does not change in time. However, internally, entropy can flow from one subsystem to another. Therefore, sub-entropies may change in time and this change may indicate a change in the energy transfer. Some important questions one may ask are: *Does a quantum superposition change entropy?* This is one of the questions that we will address in this almost pedagogical review paper and we will furthermore describe how the information content in entropy can be meaningful in physics.

In a typical engine made of reservoirs *A*, *B* and an intermediate quantum system *q* with discrete energy levels, the change of entropy in one of the reservoirs, say *B*, between the time 0 and *t* is *SB* (*t*) − *SB* (0) = −Tr {*ρ* (*t*)ln *ρB* (*t*)} − Tr*B* # *ρ*eq *B* ln *ρ*eq *B* \$, where in the first term we have safely replaced one of the two partial density matrices with the total density matrix, and accordingly replaced the partial trace with total one. The conservation of entropy tells us that the total entropy maintains its initial value at the separable compound state *ρ* (0) = *ρq* (0) *ρ*eq *A ρ*eq *B* , i.e., −Tr {*ρ* (*t*)ln *ρ* (*t*)} = <sup>−</sup>Tr*q* 2 *ρq* (0)ln *ρq* (0) 3 − ∑*<sup>i</sup>*=*A*,*<sup>B</sup>* Tr*i* # *ρ*eq *i* ln *ρ*eq *i* \$. After a few lines of algebra one can find that the change of entropy at the reservoir is *SB* (*t*) − *SB* (0) = *SB ρ* (*t*)||*ρ*eq *A ρB* (*t*) *ρq* (0) + ∑*<sup>i</sup>*=*q*,*<sup>A</sup>* Tr*i* {(*ρi* (*t*) − *ρi* (0))ln *ρi* (0)}, with *S* (*ρ*||*ρ* ) ≡ Tr {*ρ* ln *ρ*} − Tr {*ρ* ln *ρ* } being the relative entropy. Since relative entropy is a positive number [13] and equals zero only for identical density matrices *ρ* = *ρ*, the first part of the entropy flow is positive and irreversible. This satisfies the classical laws of thermodynamics. We will show that, in contrast to what has been so far presented in the literature [14], the second term in the entropy flow is *not* heat transfer—the average change of energy at the two times *QB* ≡ *H* (0)*B* − *H* (*t*)*<sup>B</sup>*. Instead, it is the difference of incoherent and coherent heat transfers [15], i.e., (*QB*,incoh (*t*) − *QB*,coh (*t*)) − (*QB*,incoh (0) − *QB*,coh (0)). This is the new result that heavily modifies the flow of entropy in some quantum heat engines and leads to some recent new physics [16–19].

In this review paper, we look at some of the simplest and most important quantum heat engines. Depending on the external drive or internal degeneracy, the exact evaluation of entropy is indeed very different from what has been presented in the literature so far. We will describe how to precisely evaluate entropy and its flow by using a replica trick that properly allows for the mathematically involved nonlinearity. We introduce a new class of correlations that allow information transfers and are different from physical correlations. For equilibrium systems, these informational correlations satisfy a

generalized form of Kubo–Martin–Schwinger (KMS) relation [20,21]. This part of the analysis will be presented in a self-contained fashion after reviewing some of the classical and quantum definitions of entropy and introducing our replica trick for evaluating the time evolution of generalized Keldysh contours. We describe a short protocol for evaluating Keldysh diagrams and in some examples perform the evaluation of a number of diagrams. We present results of example quantum devices such as a two-level quantum heat engine, a photocell, as well as a resonator, each one mediating heat transfer between two large heat baths. Finally, we briefly report on the new correspondence that makes entropy flow directly measurably in the lab by monitoring physical quantities, i.e., the statistics of energy transfer.
