**Generalized KMS**

In the presence of replicas, similarly, the generalized correlations are defined. For the case in which there are *M* replicas in total and between creation and annihilations there are *N* replicas with 0 ≤ *N* ≤ *M*, the generalized correlation function is defined as

$$S\_{mn,pq}^{N,M}(\tau) \equiv \frac{\text{Tr}\_B\left(\mathcal{X}\_{mn}^{(B)}\left(0\right)\mathcal{R}\_B^N \mathcal{X}\_{pq}^{(B)}\left(\tau\right)\mathcal{R}\_B^{M-N}\right)}{\text{Tr}\_B\left(\hat{\mathcal{R}}\_B^M\right)}.\tag{23}$$

Similarly, one can show that

$$\frac{\int\_{0}^{\infty} d\tau \text{Tr}\_{B}\left\{\mathcal{X}\_{mn}^{(B)}(0)\mathcal{R}\_{B}^{N}\mathcal{X}\_{pq}^{(B)}\left(\tau\right)\mathcal{R}\_{B}^{M-N}\mathcal{e}^{i\omega\tau}\right\}}{\text{Tr}\_{B}\left(\hat{\mathcal{R}}\_{B}^{M}\right)} = \frac{S\_{mn,pq}^{N,M\left(B\right)}\left(\omega\right)}{2} + i\Pi\_{mn,pq}^{N,M\left(B\right)}\left(\omega\right),\tag{24}$$

with the definition <sup>Π</sup>*N*,*<sup>M</sup>* (*B*) *mn*,*pq* (*ω*) ≡ (*i*/2*π*) *<sup>d</sup>νSN*,*<sup>M</sup>* (*B*) *mn*,*pq* (*ν*) / (*ω* − *<sup>ν</sup>*). One can also check from definitions that, for any heat bath, the following identities: *SN*,*<sup>M</sup> mn*,*pq* (−*<sup>ω</sup>*) = *SM*−*N*,*<sup>M</sup> pq*,*mn* (*ω*), Π*N*,*<sup>M</sup> mn*,*pq* (−*<sup>ω</sup>*) = −Π*M*−*N*,*<sup>M</sup> pq*,*mn* (*ω*), and *<sup>χ</sup>*˜*mn*,*pq* (−*<sup>ω</sup>*) = −*χ*˜ *pq*,*mn* (*ω*).

Fourier transformation of this generalized correlation will define the frequency-dependent generalized correlation and, following the same mathematics as above, one can show at equilibrium thermal bath of temperature *TB* that all correlation functions can be determined through a generalized KMS relation:

$$S\_{mn,pq}^{N\_rM(B)}\left(\omega\right) = n\_B \left(\frac{\omega}{T\_B}\right) \tilde{\chi}\_{mn,pq}^{(B)}\left(\omega\right) e^{N\frac{\omega}{k\_B T\_B}}.\tag{25}$$

Further details can be found in [34]. 

Using these definitions as well as Equation (25), the sum of diagrams (a)–(d) in Figure 8 can be further simplified to

$$\begin{split} \sum\_{m,k=0,1(m\neq k)} \mathcal{R}\_{mm} \left\{ -\left(\frac{1}{2} S\_{km,mk}^{3,3} \left(\omega\_{dr}\eta\_{mk}\right) + i \Pi\_{km,mk}^{3,3} \left(\omega\_{dr}\eta\_{mk}\right)\right) \right. \\ \left. - \left(\frac{1}{2} S\_{mk,km}^{0,3} \left(\omega\_{dr}\eta\_{km}\right) + i \Pi\_{mk,km}^{0,3} \left(\omega\_{dr}\eta\_{km}\right)\right) \right\}, \\ \sum\_{m,k=0,1(m\neq k)} \mathcal{R}\_{kk} \left\{ + \left(\frac{1}{2} S\_{mk,km}^{1,3} \left(\omega\_{dr}\eta\_{km}\right) + i \Pi\_{mk,km}^{1,3} \left(\omega\_{dr}\eta\_{km}\right)\right) \right. \\ \left. + \left(\frac{1}{2} S\_{km,mk}^{2,3} \left(\omega\_{dr}\eta\_{mk}\right) + i \Pi\_{km,mk}^{2,3} \left(\omega\_{dr}\eta\_{mk}\right)\right) \right\}, \\ \sum\_{m,k=0,1(m\neq k)} - S\_{mk,km}^{0,3} \left(\omega\_{dr}\eta\_{km}\right) \mathcal{R}\_{mm} + S\_{mk,km}^{1,3} \left(\omega\_{dr}\eta\_{mk}\right) \mathcal{R}\_{kk}. \end{split} \tag{26}$$

In total, there are *M* number of terms similar to the last line in Equation (26) associated with similar diagrams at *M* worlds. It is important to notice that these self-replica correlated terms are determined in fact only by physical correlations and they make already known results for the flow of von Neumann entropy in the heat bath [37]. To see this more in more detail, one can expand the summation and use the KMS relation and its generalized version in Equation (25). After generalizing the result for *M* replicas, taking derivative with respect to *M* and analytically continuing the result to *M* → 1, the incoherent part of flow in von Neumann entropy is

$$\left. \frac{dS^{(B)}}{dt} \right|\_{\text{incoherent}} = -\frac{1}{T\_B} \left( \Gamma^{(B)}\_{\uparrow} p\_0 - \Gamma^{(B)}\_{\downarrow} p\_1 \right),\tag{27}$$

with Γ(*B*) ↑ ≡ *χ*˜ (*nB* (*<sup>ω</sup>dr*/*TB*) + 1) and Γ(*B*) ↓ ≡ *χ*˜*nB* (*<sup>ω</sup>dr*/*TB*), *χ*˜ ≡ *χ*˜10,01, and *pn* ≡ *Rnn*. These are only self-interacting replicas, which are incomplete as they ignore the following diagrams.

The new diagrams are the cross-world interactions. As discussed previously, cross-world diagrams cannot transfer physical quantities as they rely on the fact that entropy depends nonlinearly on the density matrix and therefore it is not a physical observable quantity. Some of these types of diagrams are shown in Figure 9—for the case that one interaction takes place in the leftmost replica and the second interaction in the middle replica, thus leaving the third replica intact.

**Figure 9.** Cross-replica interacting diagrams for a quantum system and a heat bath.

$$\begin{split} \mathcal{B}(\boldsymbol{\varepsilon}) := \left. - \int\_{0}^{\infty} d\tau \text{Tr}\_{B} \left\{ \sum\_{m,n,l} \mathcal{X}\_{mk}^{(B)}(t') \mathcal{R}\_{B} \mathcal{R}\_{ml} \mathcal{X}\_{nl}^{(B)} \left(t' - \tau \right) \mathcal{R}\_{B} \mathcal{R}\_{nl} e^{-i\omega\_{dr}\eta\_{m}\tau} \delta\_{E\_{nl}E\_{lm}} \mathcal{R}\_{B} \right\} / \text{Tr}\_{B} \left( \mathcal{R}\_{B}^{3} \right), \quad \text{for } \phi \in \mathcal{V}(\boldsymbol{\varepsilon}) \\ \mathcal{R}(\boldsymbol{\varepsilon}) := \left\{ \int\_{0}^{\infty} d\tau \mathcal{R}(\boldsymbol{\overline{B}})\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}\right)} \text{Tr}\_{B} \left( \mathcal{R}\_{\left(\boldsymbol{\varepsilon}$$

$$\begin{pmatrix} \begin{pmatrix} f \end{pmatrix} : & -\int\_{0}^{\infty} d\tau \mathrm{Tr}\_{B} \left\{ \sum\_{m,n,k,l} \mathscr{X}\_{mk}^{(B)}(t'-\tau) \mathscr{R}\_{B} \mathscr{R}\_{mk} \mathscr{X}\_{nl}^{(B)} \begin{pmatrix} t' \end{pmatrix} \mathscr{R}\_{B} \mathscr{R}\_{nl} e^{-i\omega\_{d\tau}\eta\_{mk}\tau} \delta\_{E\_{mk}E\_{ln}} \mathscr{R}\_{B} \right\} / \mathrm{Tr}\_{B} \left( \mathscr{R}\_{B}^{3} \right) \end{pmatrix}$$

$$\mathbf{f}(\mathbf{g}):\qquad\int\_{0}^{\infty} d\tau \mathrm{Tr}\_{\mathcal{B}}\left\{\sum\_{m,n,k,l} \hat{\mathcal{X}}\_{mk}^{(\mathcal{B})}(t') \hat{\mathcal{R}}\_{\mathcal{B}} \hat{\mathcal{R}}\_{mk} \hat{\mathcal{R}}\_{\mathcal{B}} \hat{\mathcal{R}}\_{ln} \hat{\mathcal{X}}\_{ln}^{(\mathcal{B})}\left(t'-\tau\right) \boldsymbol{\varepsilon}^{-i\omega\_{\mathcal{U}}\eta\_{\mathcal{U}}\tau} \boldsymbol{\delta}\_{\mathcal{E}\_{\mathcal{U}m}\mathcal{E}\_{\mathcal{U}m}} \hat{\mathcal{R}}\_{\mathcal{B}}\right\}/\mathrm{Tr}\_{\mathcal{B}}\left(\hat{\mathcal{R}}\_{\mathcal{B}}^{3}\right),\qquad(13.59)$$

$$\begin{aligned} \mathcal{E}(h) := \int\_0^\infty d\tau \mathrm{Tr}\_B \left\{ \sum\_{m,n,k,l} \hat{\mathcal{X}}\_{mk}^{(B)}(t'-\tau) \hat{\mathcal{R}}\_B \hat{\mathcal{R}}\_{mk} \hat{\mathcal{R}}\_B \hat{\mathcal{R}}\_{ln} \hat{\mathcal{X}}\_{ln}^{(B)} \left(t'\right) e^{-i\omega\_{dr}\eta\_{mk}\tau} \delta\_{\mathcal{E}\_{mk}\mathcal{E}\_{nl}} \hat{\mathcal{R}}\_B \right\} / \mathrm{Tr}\_B \left(\hat{\mathcal{R}}\_B^3 \right), \end{aligned}$$

$$(i):\qquad\int\_{0}^{\infty}d\tau\mathrm{Tr}\_{B}\left\{\sum\_{m,n,k,l}\mathcal{R}\_{B}\mathcal{R}\_{km}\mathcal{X}\_{km}^{(B)}(t')\mathcal{X}\_{nl}^{(B)}\left(t'-\tau\right)\mathcal{R}\_{B}\mathcal{R}\_{nl}e^{-i\omega\_{dr}\eta\_{nl}\tau}\delta\_{E\_{nl}E\_{mk}}\mathcal{R}\_{B}\right\}/\mathrm{Tr}\_{B}\left(\mathcal{R}\_{B}^{3}\right),\qquad(ii)$$

$$\begin{aligned} (j) := & \int\_0^\infty d\tau \,\mathrm{Tr}\_B \left\{ \sum\_{m,n,k,l} \mathcal{R}\_B \mathcal{R}\_{km} \mathcal{X}\_{km}^{(B)}(t'-\tau) \mathcal{X}\_{nl}^{(B)} \left(t'\right) \,\mathcal{R}\_B \mathcal{R}\_{nl} e^{-i\omega\_{dr}\eta\_{km}\tau} \delta\_{E\_{km}E\_{ln}} \mathcal{R}\_B \right\} / \,\mathrm{Tr}\_B \left(\mathcal{R}\_B^3\right), \end{aligned}$$

$$
\begin{pmatrix} (k): & -\int\_{0}^{\infty} d\tau \text{Tr}\_{B} \left\{ \sum\_{m,n,k,l} \hat{\mathcal{R}}\_{B} \hat{\mathcal{R}}\_{km} \hat{\mathcal{N}}\_{km}^{(B)}(t') \hat{\mathcal{R}}\_{B} \hat{\mathcal{R}}\_{ln} \hat{\mathcal{N}}\_{nl}^{(B)} \left(t' - \tau\right) e^{-i\omega\_{d\tau}\eta\_{ln}\tau} \delta\_{E\_{ln}E\_{mk}} \hat{\mathcal{R}}\_{B} \right\} / \text{Tr}\_{B} \left(\hat{\mathcal{R}}\_{B}^{3}\right), \qquad (10.105)
$$

$$
\tau \text{Tr}\_{\tau\eta\_{n}} \left\{ \sum\_{m,n,k,l} \hat{\mathcal{R}}\_{m} \hat{\mathcal{R}}\_{nl} \hat{\mathcal{R}}\_{mk} \hat{\mathcal{R}}\_{nl} \hat{\mathcal{R}}\_{nl} \hat{\mathcal{R}}\_{nl} \hat{\mathcal{R}}\_{nl} \hat{\mathcal{R}}\_{nl} \right\} / \text{Tr}\_{B} \left(\hat{\mathcal{R}}\_{B}^{3}\right), \qquad (10.106)
$$

$$\begin{pmatrix} \begin{pmatrix} I \end{pmatrix} : \ & -\int\_{0}^{\infty} d\tau \mathrm{Tr}\_{B} \left\{ \sum\_{m,n,k,l} \mathcal{R}\_{B} \mathcal{R}\_{km} \mathcal{R}\_{km}^{(B)} (t' - \tau) \mathcal{R}\_{B} \mathcal{R}\_{ln} \mathcal{R}\_{ln}^{(B)} \left( t' \right) e^{-i\omega\_{B\tau} \eta\_{km} \tau} \delta\_{E\_{km} E\_{nl}} \mathcal{R}\_{B} \right\} / \mathrm{Tr}\_{B} \left( \mathcal{R}\_{B}^{3} \right) \end{pmatrix}$$

where we used the following identity *ei<sup>ω</sup>dr*(*<sup>η</sup>mn*+*ηpq* )*t* = *<sup>δ</sup>Emn*,*Eqp*

One can evaluate all diagrams associated with a general number of replicas using the above example. After carefully analyzing all diagrams and proper simplifications—see [34]—the flow of Renyi entropy *dSM*/*dt* in the heat bath *B* can be found, and consequently the so-called coherent part of entanglement (von Neumann) entropy can be found as follows:

$$\left. \frac{dS^{(B)}}{dt} \right|\_{\text{coherent}} = -\frac{\Gamma\_{\downarrow}^{(B)} - \Gamma\_{\uparrow}^{(B)}}{T\_B} \left| R\_{01} \right|^2. \tag{28}$$

 .

This is the new part of the entropy flow that comes from the generalized KMS correlations. We call this part the coherent part because it is nonzero for degenerate states or equivalently a two-level system driven by their detuning frequency.

Therefore, the entanglement entropy flow is naturally separated into two parts and therefore it is equal to the sum between the two parts:

$$\begin{array}{rcl} \frac{dS^{(\mathcal{B})}}{dt} & = & \frac{dS^{(\mathcal{B})}}{dt} \Bigg|\_{\text{incoherent}} + \frac{dS^{(\mathcal{B})}}{dt} \Bigg|\_{\text{coherent}} \\ & = & -\frac{1}{T\_B} \left( \Gamma\_\uparrow p\_0 - \Gamma\_\downarrow p\_1 \right) - \frac{\Gamma\_\downarrow - \Gamma\_\uparrow}{T\_B} \left| R\_{01} \right|^2 \end{array} \tag{29}$$

in which the first term on the second line is what in textbooks has so far been mistakenly taken as total entropy flow.

As we can see, Equation (29) is not directly related to energy flow—which here corresponds to the incoherent part instead of a finite flow that depends on the quantum coherence (*<sup>R</sup>*01)2.

Consider that the two-level system with energy difference *ωo* is driven at the same frequency, i.e., *H* = Ω cos(*<sup>ω</sup>o<sup>t</sup>*) and weakly coupled to two heat reservoirs at temperatures *TA* and *TB*. From Equation (1) of Ref. [34], one can find the following time evolution equations for the density matrix and setting them to zero determines the stationary solutions:

$$\begin{array}{rcl} \frac{dR\_{11}}{dt} &=& -\frac{i\Omega}{2} \left( R\_{01} - R\_{10} \right) - \Gamma\_{\downarrow} R\_{11} + \Gamma\_{\uparrow} R\_{00} = 0, \\\frac{dR\_{01}}{dt} &=& -\frac{i\Omega}{2} \left( R\_{11} - R\_{00} \right) - \frac{1}{2} \left( \Gamma\_{\downarrow} + \Gamma\_{\uparrow} \right) R\_{01} = 0, \quad R\_{00} + R\_{11} = 1, \end{array}$$

which finds the stationary ground state population *R*00 = (<sup>Γ</sup>↓(<sup>Γ</sup>↓ + <sup>Γ</sup>↑) + <sup>Ω</sup><sup>2</sup>)/((<sup>Γ</sup>↓ + <sup>Γ</sup>↑)<sup>2</sup> + 2Ω<sup>2</sup>) and the stationary off-diagonal density matrix element *R*10 = −*i*Ω(<sup>1</sup> − <sup>2</sup>*R*00)/(<sup>Γ</sup>↓ + <sup>Γ</sup>↑), with Γ↓ ≡ Γ(*A*) ↓ + Γ(*B*) ↓ and Γ↑ ≡ Γ(*A*) ↑ + Γ(*B*) ↑ . By considering that *B* is a probe environment with zero temperature, substituting all solutions in Equation (28), the incoherent and coherent parts of entropy flow in the probe environment have been plotted in Figure 10 for different driving amplitudes and *<sup>ω</sup>*0/*TA*.

**Figure 10.** Entropy production in a probe bath that is kept at zero temperature and is coupled to a two-level system depicted in Figure 7. The entropy is the sum of two parts: the incoherent and the coherent parts. (**a**) the incoherent part of entropy is nothing new and can be determined by standard correlations. It is positive by the convention that entropy enters from a higher temperature bath (via the two-level system); (**b**) the coherent part of entropy is a previously unknown part as it comes from the informational correlations between different replicas. This part depends quadratically on the off diagonal density. Quite nontrivially, this part of entropy is negative and summing it with the incoherent part will result in a positive flow ye<sup>t</sup> with much smaller magnitude for entropy at small driving amplitudes.

### 3.6.2. Example 2: Entropy in a Four-Level Quantum Photovoltaic Cell

Scovil and Schulz–DuBois first introduced a model of a quantum heat engine (SSDB heat engine) in which a single three-level atom, consisting of a ground and two excited states, is in contact with two heat baths [38,39]. A large enough difference between the heat bath temperatures can create population inversion between the two excited states and a coherent light output. One hot photon is absorbed and one cold photon is emitted; therefore, a laser photon is produced. The SSDB heat engine model gives a clear demonstration of the quantum thermodynamics. However, we notice that some detailed

properties of this lasing heat engine, e.g., the threshold behavior and the statistics of the output light, are still not well studied. There are a number of applications for the model, such as light-harvesting biocells, photovoltaic cells, etc.

Since then, the model has been modified to describe other systems such as light-harvesting biocells, photovoltaic cells, etc.

Recently, in Ref. [40], one of us studied the entropy flow using the replica trick for a 4-level photovoltaic cell with two degenerate ground states and two excited states, see Figure 11. This heat engine was first proposed by Schully in [11] and recently studied in many further details by Schully and others [17,41].

**Figure 11.** A four-level doubly degenerate photovoltaic cell.

After finding all extended Keldysh diagrams for an arbitrary Renyi degree *M*, evaluating all self-interacting and cross-interacting diagrams and simplifying the results, the von Neumann entropy flow in heat bath *A* becomes [40]:

$$
\frac{dS}{dt}\Big|\_{A} = -\frac{1}{T\_A} \Big\{ \gamma p\_4 - \omega\_A \bar{\chi}\_{42} \hbar \left(\frac{\omega\_A}{T\_A}\right) p\_2 - \omega\_A \bar{\chi}\_{41} \hbar \left(\frac{\omega\_A}{T\_A}\right) p\_1
$$

$$
$$

$$
$$

The first two lines can be found using physical correlations. The last line, however, which plays an essential role in the entropy evaluation, can be obtained only through informational correlations. Here, the state probabilities are *px* ≡ *Rxx* with *x* being 1, 2, 3, 4 and depending on the characteristics of all heat baths. The dynamical response function is *χ*˜*α<sup>i</sup>* ≡ *<sup>χ</sup>*˜*i<sup>α</sup>*,*<sup>α</sup><sup>i</sup>*(*<sup>ω</sup>iα*) with *i* = 1, 2 and *α* = 3, 4, and *χ*˜1*<sup>α</sup>*,*α*<sup>2</sup> = √*χ*˜*α*1*χ*˜*α*2. Moreover, *γ* ≡ ∑*<sup>i</sup>*=1,2 [*n*¯ (*<sup>ω</sup>A*/*TA*) + 1] *<sup>ω</sup>Aχ*˜3*<sup>i</sup>*.

In order to evaluate the stationary value of the entropy flow in this heat bath, we must solve the quantum master equation for the density matrix time evolution. This can be found in Ref. [40]. The solution is such that the coupling between the environment and the quantum system introduces decoherence in quantum states. Energy exchange between the heat bath and a quantum system introduces a limited coherence time, namely *τ*1, for quantum state probabilities. The phase of a quantum state can fluctuate and, depending on environmental noise, the lifetime of quantum state can be limited to *τ*2. These two coherence times affect all elements of the density matrix. From solving the quantum Bloch equation, one can see that the only stationary solution in the off-diagonal part is the imaginary part of *R*12 whose real part of exponential decay due to dephasing is: Im*R*12 ∼ exp(*t*/*<sup>τ</sup>*2).

One can substitute the stationary solution of the density matrix in Equation (30) and the flow of entropy in the heat bath changes depending on the dephasing time—see Figure 2a,b in [40]. In fact, increasing the dephasing time will increase the contribution of the coherent part of the entropy flow, i.e., information correlations. This will reduce the total entropy flow in the heat bath, which will equivalently increase the output power in this photovoltaic cell.

### 3.6.3. Example 3: Entropy in a Quantum Resonator/Cavity Heat Engine

Using a rather different technique—i.e., the correspondence between entropy and statistics of energy transfer that we discuss in the next section—in [15,16], we calculated entropy production for a

resonator/cavity coupled two different environments kept at two different temperatures, see Figure 12. One of the two baths is a probe environment at a temperature of zero for which we calculate the flow of entropy.

Knowing how entropy flows as the result of interactions between the resonator, cavity and other parts of the circuit can help to obtain important information about the possibility of leakage or dephasing in the system and ultimately give rise to modifications of quantum circuits [4]. A good understanding of cavities/resonators is beneficial to search for the nature of non-equilibrium quasiparticles in quantum circuits [42,43]. This can help with detecting light particles like muons whose tunnelling in a quantum circuit can signal a sudden jump in the entropy flow [44–46]. Given that entropy flow can be measured by the full counting statistics of energy transfer, see the next section, it is important to keep track of entropy flow in a resonator.

**Figure 12.** A quantum cavity heat engine.

Again, we use the standard technique that we described above. Let us consider a single harmonic oscillator of frequency *ω*0 and Hamiltonian *H* ˆ = *<sup>ω</sup>*0(*a*ˆ†*a*<sup>ˆ</sup> + 1/2), which is coupled to a number of environments at different temperatures with different coupling strengths. We concentrate on a probe environment that is weakly coupled to the oscillator. In addition, the oscillator is driven by an external force at frequency Ω. We calculate the Renyi flow and consequently the von Neumann entropy flow of the probe environment. The coupling Hamiltonian between the harmonic oscillator and the probe reservoir is *H* ˆ (*t*) = *<sup>X</sup>*<sup>ˆ</sup>(*t*)*a*ˆ†(*t*) + *h*.*c*., with *X*ˆ being the probe reservoir operator. The Fourier transform of the correlator is: *Smn*(*ω*) = exp(−*iω<sup>t</sup>*)*Smn*(*t*)*dω*/2*<sup>π</sup>*. Due to the conservation of energy, the energy exchange occurs either with quantum Ω or with quantum *ω*0.

We note that the time dependence of the average of two operators can be written as *a*ˆ†(*t*)*a*<sup>ˆ</sup>(*t*) = *a*ˆ†*a*<sup>ˆ</sup>*ei<sup>ω</sup>*0(*<sup>t</sup>*−*t*) + *a*<sup>ˆ</sup>(*t*)*a*ˆ†(*t*), where the time dependence of *a*(*t*) is due to the driving force and therefore oscillates at frequency Ω: *a*(*t*) = *a*+ exp(*<sup>i</sup>*Ω*t*) + *a*− exp(−*i*Ω*t*). This corresponds to the fact that the oscillator can oscillate both at its own frequency and at the frequency of external force.

Obtaining the entropy flows from the extended Keldysh correlators is straightforward. The generalized KMS relation in Equation (25) helps to describe the correlators in the thermal bath *B* in terms of their dynamical susceptibility. The result can be summarized as follows:

$$\frac{d S\_M^{(B)}}{dt} = \frac{M \hbar \left(M \omega\_0 / T\_B\right) \tilde{\chi}}{\vec{n} \left( (M - 1) \omega\_0 / T\_B \right) \hbar \left( \omega\_0 / T\_B \right)} \left\{ \langle \langle a^\dagger a \rangle \rangle e^{\frac{\omega\_0}{T\_B}} - \langle \langle a a^\dagger \rangle \rangle \right\},$$

where we defined *T*resonator to be the effective temperature of the harmonic oscillator *aa*† = *n* ¯(*ω*0/*T*resonator) + 1 and *a*†*a* = *<sup>n</sup>*¯(*ω*0/*T*resonator). Taking the derivative with respect to *M* and analytically continuing the result in the limit of *M* → 1 will determine the thermodynamic entropy flow:

$$\frac{d S\_M^{(B)}}{dt} = \frac{1}{T\_B} \left\{ \bar{n} \left( \omega\_0 / T\_{\text{resonant}} \right) - \bar{n} \left( \omega\_0 / T\_B \right) \right\}. \tag{31}$$

The entropy flow changes sign at the onset temperature *Tresonator* = *TB*. Moreover, after the exact evaluation of the incoherent part of the entropy flow, one should notice that it contains some terms proportional to *a* and *a*†. These terms oscillate with the external drive and are nonzero. However, they are all cancelled out by the coherent part of entropy flow such that the overall flow will only depend on the temperatures, and not on the driving force. Therefore, the entropy flow is robust in the sense that it only depends on the temperatures of the probe and harmonic oscillator and is completely insensitive to the external driving force.

The insensitivity of entropy flow to external driving force is interesting and a direct result of including coherent flow of entropy that is absent in semi-classical analysis. The difference can put the coherent entropy flow into an experimental verification.

In the absence of cross-replica correlators, the thermodynamic entropy of a probe environment, coupled to a thermal bath via a resonator, will dramatically depend on the amplitude of the external driving. If no such dependence on the driving amplitude is found, then this is an indication that they are absent; they are in fact eliminated by quantum coherence!

### **4. Linking Information to Physics: A New Correspondence**

As discussed above, the Renyi entropies in quantum physics are considered unphysical, i.e., non-observable quantities, due to their nonlinear dependence on the density matrix. Such quantities cannot be determined from immediate measurements; instead, their quantification seems to be equivalent to determining the density matrix. This requires reinitialization of the density matrix between many successive measurements. Therefore, the Renyi entropy flows between the systems are conserved measures of nonphysical quantities. An interesting and nontrivial question is: Is there any relation between the Renyi entropy flows and the physical flows?

An idea of such a relation was first put forward by Levitov and Klich in [23], where they proposed that entanglement entropy flow in electronic transport can be quantified from the measurement of the full counting statistics (FCS) of charge transfers [22,47–49]. The validity of this relation is restricted to zero temperature and obviously to the systems where interaction occurs by means of charge transfer. Recently, we presented a relation that is similar in spirit [15]. We derived a correspondence for coherent and incoherent second-order diagrams in a general time-dependent situation.

This relation gives an exact correspondence between the informational measure of Renyi entropy flows and physical observables, namely, the full counting statistics of energy transfers [47,50].

We consider reservoir *B* and quantum system *q*. We assume that *B* is infinitely large and is kept in thermal equilibrium at temperature *TB*. System *q* is arbitrary as it may carry several degrees of freedom as well as infinitely many. It does not have to be in thermal equilibrium and is in general subject to time-dependent forces. It is convenient to assume that these forces are periodic with a period of *τ*; however, the period does not explicitly enter the formulation of our result, which is also valid for aperiodic forces. The only requirement is that the flows of physical quantities have stationary limits. The stationary limits are determined after averaging instant flows over a period and—for aperiodic forces—by averaging over a sufficiently long time interval. In the case of energetic interactions, energy transfer is statistical. The statistics can be described by the generating function of the full counting statistics (FCS), namely 'FCS Keldysh actions'.

Recently, in Ref. [15], we proved that the flow of thermodynamic entropy as well as the flow of Renyi entropy between two heat baths via a quantum system is exactly equivalent to the difference between two FCS Keldysh actions of incoherent and coherent energy transfers. In the limit of long *τ* and for a typical reservoir *B* with temperature *TB*, the incoherent and coherent FCS Keldysh actions are *fi* (*ξ*, *TB*) and *fc* (*ξ*, *TB*), with *ξ* being the counting field of energy transfer. These generating functions can be determined using Keldysh diagrams, see [16]. After their evaluation, one finds the statistical *m*-th cumulant function *Cm* by taking the derivative of the generating function in the limit of zero counting function, i.e., *Cm* = lim*ξ*→<sup>0</sup> *∂m f* /*∂ξ<sup>m</sup>*.

In fact, any physical quantity should depend on the cumulants and consequently on a zero counting field. However, informational measures are exceptional. Detailed analysis shows that the flow of Renyi entropy of degree *M* in the reservoir *B* at equilibrium temperature *TB* is exactly, and unexpectedly, the following: *dS M* (*TB*) /*dt* = *M* [ *fi*(*ξ*<sup>∗</sup>, *TB*/*M*) − *fc*(*ξ*<sup>∗</sup>, *T*/*M*)] with *ξ*∗ ≡ *i*(*M* − <sup>1</sup>)/*TB*. Notice that in this correspondence the temperature on the left side is *TB* while it is *TB*/*M*

on the right side. In addition, it is important to notice that the entropy is evaluated by using the generating function of full counting statistics at nonzero counting field *ξ*<sup>∗</sup>. This relation is valid in the weak-coupling limit where the interaction between the systems can be treated perturbatively.
