*3.6. Quantum Entropy Production*

Let us consider that two large heat reservoirs *A* and *B*, each one containing many degrees of freedom and kept at a temperature, are coupled to one another via only a few numbers of shared degrees of freedom. The Hamiltonian can be written as *H* = *HA* + *HB* + *HAB* with *HAB* representing the coupled degrees of freedom.

In order to compute the flow of a quantity between *A* and *B*, that quantity should be conserved in the combined system *A* + *B*. As we discussed in the first section of this paper, Renyi entropy is a conserved quantity in a closed system, therefore *d* ln *S*(*<sup>A</sup>*+*<sup>B</sup>*) *M* /*dt* = 0. However, one should notice that there is a difference between the conservation of physical quantities such as energy and the conservation of entropy. Because physical quantities linearly depend on the density matrix, when it is conserved for a closed system, internally it can flow from a subsystem to another one such that its production in a subsystem is exactly equal to the negative sign of its removal from the other subsystem. However, entropy is not so. In fact, due to nonlinear dependence of entropy on the density matrix, when it is conserved for a bipartite closed system, it is not equally added and subtracted from the subsystem due to the non-equality in Equation (9).

Below, we will present some example systems with rather general Hamiltonians and, using the diagram rules, we evaluated all entropy production diagrams.

### 3.6.1. Example 1: Entropy in a Two-Level Quantum Heat Engine

In Ref. [15], we used the extended Keldysh technique and evaluated entropy flow for the simplest quantum heat engine in which a two-level system couples two heat baths kept at different temperatures, see Figure 7. After taking all physical and informational correlations into account, we found that the exact evaluation in the second order is much different from what physical correlations predict. Here, we reproduce the exact result by giving a pedagogical use of the diagram evaluation described above.

Let us consider two heat baths that are kept at different temperatures weakly interact by exchanging the quantum energy *ωo*. Such a quantum system can be thought of as a two-level system that couples the two heat baths through shared excitations and de-excitations. The Hilbert space of the two-level system contains the states |0 and |1. The free Hamiltonian contains heat bath energy levels *E*(*A*) *α* s and *E*(*B*) *β* s and quantum system energies *En* with *n* = 0, 1, i.e., *H*0 = ∑*α E*(*A*) *α* |*αα*| + ∑*β E*(*B*) *β* |*ββ*| + ∑*<sup>n</sup>*=0,1 *En*|*nn*|.

**Figure 7.** A two-level system quantum heat bath.

We assume the so-called 'transversal' interaction is taken into account between *A*/*B* and the two-level system *q*. This means that they interact via exchanging the quantum of energy *ωo*. Of course, we can generalize the discussion to longitudinal interactions in which no energy is exchanged; however, since such interactions are not of immediate interest for heat transfer in quantum heat devices. we ignore them.

This interaction we assume for the heat bath has the following general form: *Hint* = ∑*<sup>n</sup>*,*m*=0,1 |*n m*| *X*ˆ (*A*) *nm* (*<sup>ω</sup>*0) + *X*ˆ (*B*) *nm* (*<sup>ω</sup>*0) subject to *Em* = *En* and *X*ˆ *nm* representing energy absorption/decay in heat baths. The summation in *Hint* can be generalized to an arbitrary number of heat baths interacting at shared degrees of freedom.

Moreover, the entire system including the two-level system is externally driven. The classical heat baths are naturally not influenced effectively by the driving field; however, the driving can pump in and out energy to the two-level system by the following Hamiltonian *Hdr* = Ω cos(*<sup>ω</sup>dr<sup>t</sup>*)(|01| + |10|).

For simplicity, we take the Hamiltonian into the rotating frame that makes excitation/relaxation with the frequency *ωdr*. In this frame, the excited and ground states are transformed as follows: |1*R* = exp (*<sup>i</sup><sup>ω</sup>dr<sup>t</sup>*)|1 and |0*R* = |0. This will introduce the unitary transformation *UR* = exp (*<sup>i</sup><sup>ω</sup>dr<sup>t</sup>*|11|) on the Hamiltonian, i.e., *HR* = *URHU*† *R* + *<sup>i</sup>*(*∂UR*/*∂t*) *U*† *R*. A few lines of simplification will result in the following Hamiltonian in the rotating frame:

$$\begin{array}{rcl} H\_{R} & \equiv & H\_{0} + V\_{qA} + V\_{qB} + V\_{AB} + V\_{dr} \\ \hline \end{array} \quad \begin{array}{rcl} H\_{0} & = & E\_{0}|0\rangle\langle 0| + (E\_{1} - \omega\_{dr})\,|1\rangle\langle 1| + \sum\_{a} E\_{a}^{(A)}|a\rangle\langle a| + \sum\_{a} E\_{a}^{(B)}|a\rangle\langle a|, \\ \end{array}$$

$$\begin{array}{rcl} V\_{qA} & = & |0\rangle\langle 1| \hat{\mathcal{X}}\_{01}^{(A)}\,(t) \,\epsilon^{j\omega\_{d}t} + |1\rangle\langle 0| \hat{\mathcal{X}}\_{10}^{(A)}\,(t) \,\epsilon^{-i\omega\_{d}t} \equiv & \sum\_{n,m=0,1(n\neq m)} |n\rangle\langle m| \hat{\mathcal{X}}\_{nm}^{(A)}(t) \,\epsilon^{j\omega\_{d}\eta\_{mn}t}, \\ \end{array} \begin{array}{rcl} V\_{q} & = & |0\rangle\langle 1| \hat{\mathcal{X}}\_{01}^{(B)}\,(t) \,\epsilon^{j\omega\_{d}t} + |1\rangle\langle 0| \hat{\mathcal{X}}\_{10}^{(B)}\,(t) \,\epsilon^{-i\omega\_{d}t} \equiv & \sum\_{n,m=0,1(n\neq m)} |n\rangle\langle m| \hat{\mathcal{X}}\_{nm}^{(B)}(t) \,\epsilon^{j\omega\_{d}\eta\_{mn}t}, \\ \end{array} \tag{22}$$

$$\begin{array}{rcl} V\_{AB} & = & 0, \qquad V\_{dr} = \frac{\Omega}{2}\left(|0\rangle\langle 1| + |1\rangle\langle 0|\right), \end{array}$$

with *η*01 = −*η*10 = 1 and *η*00 = *η*11 = 0. Given the fact that there is no direct exchange of energy between *A* and *B*, the density matrix can be represented as *R* = *RqA* ⊗ *RB* + *RA* ⊗ *RqB* in an interaction picture, thus determining entropy flow in the heat bath *B* will depend on the quantum system and the heat bath *B*, although indirectly the heat bath A will influence the quantum system. In general, *d* (*RB*)*<sup>M</sup>* /*dt* = *Trq* #*d RqB<sup>M</sup>* /*dt*\$. Let us recall that this quantity determines the flow of von Neumann entropy and, using Equation (10), it can be simplified to *dS*(*B*)/*dt* = lim*M*→<sup>1</sup> *d* Tr*BTrq* #*dRqB*/*dt RqB<sup>M</sup>*−<sup>1</sup> + ··· + *RqB<sup>M</sup>*−<sup>1</sup> *dRqB*/*dt*\$ /*dM*. Each term in the sum is evaluated in the interaction picture using *dR*/*dt* = (−*<sup>i</sup>*) [*<sup>V</sup>*, *<sup>R</sup>*]. One can show that the external driving will cause the density matrix to evolve as *dRnm*/*dtdr* = (*i*Ω/2) (*Rn*0*δm*<sup>1</sup> + *Rn*1*δm*<sup>0</sup> − *δn*0*R*1*m* − *<sup>δ</sup>n*1*R*0*m*).

The interaction Hamiltonian evolves quantum states and below we evaluate the entropy flow in the *M* = 3 example to the second order perturbation theory. As discussed above, there are in general two types of diagrams in the second order: (1) 'self-interacting' diagrams with second order interaction taking place in one replica, and (2) cross-world-interacting terms in which two different replicas take on each 1st order interaction. The self-interacting diagrams for the two-level system are listed in Figure 8.

**Figure 8.** Self-interacting diagrams for interaction between a quantum system and a heat bath.

These diagrams correspond to the following flows, respectively:

$$\mathbf{T}(a) : \quad \frac{(-1)\int\_{0}^{\infty} d\tau \mathrm{Tr}\_{B}\left\{\sum\_{m,k=0,1} \boldsymbol{\upmu}\_{km}^{(B)}(\mathbf{t'})\boldsymbol{\uplambda}\_{km}^{(B)}(\mathbf{t'}-\tau)\,\boldsymbol{\uplambda}\_{B}\boldsymbol{\upR}\_{mm}e^{-i\boldsymbol{\upmu}\_{d}\boldsymbol{\upmu}\_{km}\tau}e^{i\boldsymbol{\upmu}\_{d}(\boldsymbol{\upmu}\_{mk}+\boldsymbol{\upmu}\_{lm})t'}\boldsymbol{\upR}\_{B}^{2}\right\}}{\mathrm{Tr}\_{B}\left(\hat{\boldsymbol{\mathcal{R}}}\_{B}^{3}\right)},$$

$$\mathbf{T}\_{\mathbf{B}}(b) := \frac{(+1)\int\_{0}^{\infty} d\tau \mathrm{Tr}\_{B}\left\{\sum\_{m,k=0,1(m\neq k)} \boldsymbol{\mathcal{X}}\_{mk}^{(B)}(t'-\tau)\boldsymbol{\mathcal{R}}\_{B}\boldsymbol{\mathcal{R}}\_{kk}\boldsymbol{\mathcal{X}}\_{km}^{(B)}(t')\boldsymbol{e}^{-i\omega\_{dr}\eta\_{mk}\tau}\boldsymbol{e}^{i\omega\_{dr}(\eta\_{mk}+\eta\_{km})t'}\boldsymbol{\mathcal{R}}\_{B}^{2}\right\}}{\mathrm{Tr}\_{B}\left(\boldsymbol{\hat{\mathcal{R}}}\_{B}^{3}\right)},$$

$$\mathbf{T}(c) := \frac{(+1)\int\_{0}^{\infty} d\tau \mathrm{Tr}\_{B}\left\{\sum\_{m,k=0,1(m\neq k)} \boldsymbol{\mathcal{R}}\_{mk}^{(B)}(t')\boldsymbol{\mathcal{R}}\_{B}\boldsymbol{\mathcal{R}}\_{kk}\boldsymbol{\mathcal{R}}\_{km}^{(B)}(t'-\tau)e^{-i\omega\_{dr}\eta\_{km}\tau}e^{i\omega\_{dr}(\eta\_{mk}+\eta\_{km})t'}\boldsymbol{\mathcal{R}}\_{B}^{2}\right\}}{\mathrm{Tr}\_{B}\left(\boldsymbol{\hat{\mathcal{R}}}\_{B}^{3}\right)},$$

$$\mathbf{T}(d) := \frac{(-1)\int\_{0}^{\infty} d\tau \mathrm{Tr}\_{B}\left\{\sum\_{m,k=0,1}^{}\boldsymbol{\upmu}\_{k}\boldsymbol{\upmu}\_{k}\boldsymbol{\upmu}\_{mm}^{\boldsymbol{\upmu}}\boldsymbol{\upmu}\_{mk}^{(\boldsymbol{\upbeta})}(t'-\tau)\boldsymbol{\upbeta}\_{km}^{(\boldsymbol{\upbeta})}(t')\,e^{-i\boldsymbol{\upmu}\_{dr}\boldsymbol{\upeta}\_{mk}\boldsymbol{\upnu}\_{r}\boldsymbol{\upmu}\_{dr}(\boldsymbol{\upeta}\_{mk}+\boldsymbol{\upeta}\_{lm})t'}\boldsymbol{\upbeta}\_{B}^{2}\right\}}{\mathrm{Tr}\_{B}\left(\boldsymbol{\upbeta}\_{B}^{3}\right)}.$$

.

In all these terms, there is a time dependent factor *ei<sup>ω</sup>dr*(*ηmk*+*ηkm*)*t* which is identical to 1 because we always have the following relation valid: *ηmk* = <sup>−</sup>*ηkm*. We assume that heat baths are large and, at equilibrium, therefore the correlation function is the same at all times *t* and only depends on the time difference *τ* between the creation and annihilation of a photon. In the heat bath B, the equilibrium correlation is defined as *S*(*B*) *mn*,*pq* (*τ*) ≡ Tr*B X*ˆ (*B*) *mn* (0) *X*ˆ (*B*) *pq* (*τ*) *RB*. The Fourier transformation of the correlation defines the following frequency-dependent correlation: *S*(*B*) *mn*,*pq* (*ω*) = ∞−∞ *dτ*Tr*B X*ˆ (*B*) *mn* (0) *X*ˆ (*B*) *pq* (*τ*) *RB* exp (*<sup>i</sup>ωτ*). Therefore, in the case of *M* = 1 (i.e., the absence of the last term *<sup>R</sup>*2*B*), the diagrams a–d can be rewritten in terms of *S*(*B*) *mn*,*pq* (*ω*). For example, the diagram (a) for the case of *M* = 1 can be simplified to − ∑*<sup>m</sup>*,*k*=0,1(*<sup>m</sup>* <sup>=</sup>*k*) *R* ˆ *mm* ∞0 *dτ*Tr*B* #*X*ˆ (*B*) *mk* (0)*X*<sup>ˆ</sup> (*B*) *km* (*τ*) *R*ˆ *Be*<sup>−</sup>*i<sup>ω</sup>drηkm<sup>τ</sup>*\$ in which the integral is half of the domain in Fourier transformation and therefore it can be proved to simplify to − ∑*<sup>m</sup>*,*k*=0,1(*<sup>m</sup>* <sup>=</sup>*k*) *R* ˆ *mm* (1/2) *S*(*B*) *mk*,*km* (*<sup>ω</sup>drηmk*) + *<sup>i</sup>*Π*mk*,*km* (*<sup>ω</sup>drηmk*) with <sup>Π</sup>*mn*,*pq* ≡ (*i*/2*π*) *<sup>d</sup>νS*(*B*) *mn*,*pq* (*ν*) / (*ω* − *<sup>ν</sup>*). What is left to be determined is the frequency-dependent correlation function *S*(*B*) *mn*,*pq* (*ω*), which turns out to become completely characterized by the set of reduced frequency-dependent susceptibilities defined as *χ*˜(*B*) *mn*,*pq* (*ω*) ≡ *χ*(*B*) *mn*,*pq* (*ω*) − *χ*(*B*) *pq*,*mn* (−*<sup>ω</sup>*) /*i*, with the dynamical susceptibility in the environment being *χ*(*B*) *mn*,*pq* (*ω*) ≡ (−*<sup>i</sup>*) 0−∞ Tr*B* #*X*<sup>ˆ</sup> (*B*) *mn* (*τ*), *X*ˆ (*B*) *pq* (0) *RB*\$ exp (−*iωτ*). The fluctuation–dissipation theorem provides a link between the equilibrium correlation and the reduced dynamical susceptibility in the classical thermal bath *B* at temperature *TB*. This relation is usually called the Kubo–Martin–Scwinger (KMS) relation: *S*(*B*) *mn*,*pq* (*ω*) = *nB* (*ω*/*TB*) *χ*˜(*B*) *mn*,*pq* (*ω*) with *nB* (*ω*/*TB*) = 1/ (exp (*ωTB*) − 1) being the Bose distribution and *kB* the Boltzmann constant.
