Excess Entropy Production in Quantum System: Quantum Master Equation Approach

Abstract

For open systems described by the quantum master equation (QME), we investigate the excess entropy production under quasistatic operations between nonequilibrium steady states. The average entropy production is composed of the time integral of the instantaneous steady entropy production rate and the excess entropy production. We propose to define average entropy production rate using the average energy and particle currents, which are calculated by using the full counting statistics with QME. The excess entropy production is given by a line integral in the control parameter space and its integrand is called the Berry–Sinitsyn–Nemenman (BSN) vector. In the weakly nonequilibrium regime, we show that BSN vector is described by $\ln \breve{\rho }_0$ and $\rho _0$ where $\rho _0$ is the instantaneous steady state of the QME and $\breve{\rho }_0$ is that of the QME which is given by reversing the sign of the Lamb shift term. If the system Hamiltonian is non-degenerate or the Lamb shift term is negligible, the excess entropy production approximately reduces to the difference between the von Neumann entropies of the system. Additionally, we point out that the expression of the entropy production obtained in the classical Markov jump process is different from our result and show that these are approximately equivalent only in the weakly nonequilibrium regime.

Appendices

Appendix A: Derivative of the von Neumann Entropy

We show that

$$\begin{aligned} \frac{\partial S_\mathrm{{vN}}(\rho _0(\alpha ))}{\partial \alpha ^n}=-\text{ Tr }_S \Big [\ln \rho _0(\alpha ) \frac{\partial \rho _0(\alpha )}{\partial \alpha ^n} \Big ] . \end{aligned}$$

(154)

From the definition of the von Neumann entropy, the LHS of the above equation is given by

$$\begin{aligned} \frac{\partial S_\mathrm{{vN}}(\rho _0(\alpha ))}{\partial \alpha ^n}=-\text{ Tr }_S \Big [\ln \rho _0(\alpha ) \frac{\partial \rho _0(\alpha )}{\partial \alpha ^n} \Big ]-\text{ Tr }_S \Big [ \frac{\partial \ln \rho _0(\alpha )}{\partial \alpha ^n} \rho _0(\alpha ) \Big ]. \end{aligned}$$

(155)

Using (151), the second term of the RHS of the above equation becomes

$$\begin{aligned} -\text{ Tr }_S \Big [ \frac{\partial \ln \rho _0(\alpha )}{\partial \alpha ^n} \rho _0(\alpha ) \Big ]= & {} - \text{ Tr }_S \Big [ \int _0^\infty ds \ \frac{1}{\rho _0(\alpha )+s}\frac{\partial \rho _0(\alpha )}{\partial \alpha ^n}\frac{1}{\rho _0(\alpha )+s} \rho _0(\alpha ) \Big ] \nonumber \\= & {} - \text{ Tr }_S \Big [ \int _0^\infty ds \ \frac{\rho _0(\alpha )}{(\rho _0(\alpha )+s)^2}\frac{\partial \rho _0(\alpha )}{\partial \alpha ^n} \Big ] \nonumber \\= & {} - \text{ Tr }_S \Big [ \frac{\partial \rho _0(\alpha )}{\partial \alpha ^n} \Big ]=0. \end{aligned}$$

(156)

Then, we obtain (154).

Appendix B: Derivation of the Relation Between $k_{i,b}$ and $\rho _{i,b}^{\kappa }$

In this section, we examin the relation of the coefficients of the expansion of $\rho _0^{\kappa }(\alpha )$ and $l_0^\prime (\alpha )$ in (105) of 3.2.

First, we investigate $k_{i,b}$ in (105). (92) can be rewritten as

$$\begin{aligned} \hat{K}^\dagger (\alpha )l_0^\prime (\alpha )+[\mathcal {K}^\prime (\alpha )]^\dagger 1 = J^\sigma _\mathrm{{ss}}(\alpha ). \end{aligned}$$

(157)

Here, $J^\sigma _\mathrm{{ss}}(\alpha ) = \mathcal {O}(\varepsilon ^2)$ holds because $i_\mathrm{{ss}}^{H_b}(\alpha ),i_\mathrm{{ss}}^{N_b}(\alpha )=\mathcal {O}(\varepsilon )$ and

$$\begin{aligned} J_\mathrm{{ss}}^\sigma (\alpha )=\sum _b \left( -i_\mathrm{{ss}}^{H_b}(\alpha )\varepsilon _{1,b}+i_\mathrm{{ss}}^{N_b}(\alpha )\varepsilon _{2,b}\right) \end{aligned}$$

(158)

since

$$\begin{aligned} \sum _b i_\mathrm{{ss}}^{X_b}(\alpha )= -\text{ Tr }_S\Big [X_S \sum _b \mathcal {L}_b(\alpha ) \rho _0(\alpha )\Big ]=0,\ \ \ (X=N,H). \end{aligned}$$

(159)

Then we obtain

$$\begin{aligned} \overline{\partial _{i,b}\mathcal {K}^\prime } ^\dagger 1+\overline{K}^\dagger k_{i,b}+\overline{\partial _{i,b}\mathcal {L}_{b}} ^\dagger \overline{l_0^\prime }= & {} 0, \end{aligned}$$

(160)

in $\mathcal {O}(\varepsilon _{i,b})$. Here, $\partial _{i,b}X {\mathop {=}\limits ^{\mathrm {def}}}\partial X/\partial \alpha _{i,b}$ and $\overline{K}{\mathop {=}\limits ^{\mathrm {def}}}\overline{\hat{K}}$. The first term of the LHS is

$$\begin{aligned} \overline{\partial _{i,b}\mathcal {K}^\prime } ^\dagger 1= & {} \frac{\partial [\mathcal {K}^\prime ]^\dagger 1}{\partial \alpha _{i,b}}\Big \vert _{\alpha _{i,b}=\overline{\alpha _i}} \nonumber \\= & {} \frac{\partial \mathcal {L}_b^\dagger [\alpha _{1,b}H_S-\alpha _{2,b}N_S]}{\partial \alpha _{i,b}}\Big \vert _{\alpha _{i,b}=\overline{\alpha _i}} \nonumber \\= & {} \overline{\partial _{i,b} \mathcal {L}_b} ^\dagger [\overline{\beta }H_S-\overline{\beta \mu }N_S]+\overline{\Pi _b}^\dagger \frac{\partial [\alpha _{1,b}H_S-\alpha _{2,b}N_S]}{\partial \alpha _{i,b}}. \end{aligned}$$

(161)

The third term of the LHS becomes

$$\begin{aligned} \overline{\partial _{i,b}\mathcal {L}_{b}} ^\dagger \overline{l_0^\prime }= & {} \overline{\partial _{i,b}\mathcal {L}_{b}} ^\dagger ( -\overline{\beta }H_S+\overline{\beta \mu }N_S+\overline{c^\prime }^*1) \nonumber \\= & {} -\overline{\partial _{i,b}\mathcal {L}_{b}} ^\dagger (\overline{\beta }H_S-\overline{\beta \mu }N_S). \end{aligned}$$

(162)

Here, we used $\overline{\partial _{i,b}\mathcal {L}_{b}}^\dagger 1=0$ derived from $\hat{K}^\dagger 1=0$. Then, (160) becomes

$$\begin{aligned} \overline{K}^\dagger k_{1,b}+\overline{\Pi _b}^\dagger H_S= & {} 0, \end{aligned}$$

(163)

$$\begin{aligned} \overline{K}^\dagger k_{2,b}-\overline{\Pi _b}^\dagger N_S= & {} 0 . \end{aligned}$$

(164)

Next, we show the relation between $k_{i,b}$ and $\rho _{i,b}^{(-1)}$. (102) leads

$$\begin{aligned} \overline{K_\kappa }\rho _{i,b}^{(\kappa )}+\overline{\partial _{i,b}\mathcal {L}_b}\rho _\mathrm{{gc}} = 0 , \end{aligned}$$

(165)

in $\mathcal {O}(\varepsilon _{i,b})$. Here, $\overline{K_\kappa }{\mathop {=}\limits ^{\mathrm {def}}}\overline{\hat{K}}_\kappa$. By the way,

$$\begin{aligned} \mathcal {L}_b \rho _\mathrm{{gc}}(\alpha _S;\beta _b,\beta _b \mu _b)=0, \end{aligned}$$

(166)

holds. Differentiating this equation by $\alpha _{i,b}$, we obtain

$$\begin{aligned} \overline{\partial _{i,b}\mathcal {L}_b}\rho _\mathrm{{gc}}= & {} -\overline{\mathcal {L}_b}\overline{\frac{\rho _\mathrm{{gc}}(\alpha _S;\beta _b,\beta _b \mu _b)}{\partial \alpha _{i,b}}} = \overline{\mathcal {L}_b}\frac{\partial [\alpha _{1,b} H_S-\alpha _{2,b}N_S]}{\partial \alpha _{i,b}}\rho _\mathrm{{gc}}(\alpha _S;\overline{\beta },\overline{\beta \mu }).\quad \quad \end{aligned}$$

(167)

Substituting these equations into (165), we obtain

$$\begin{aligned} \overline{K}_\kappa \rho _{1,b}^{(\kappa )}+\overline{\Pi _b}(H_S \rho _\mathrm{{gc}})= & {} 0, \end{aligned}$$

(168)

$$\begin{aligned} \overline{K}_\kappa \rho _{2,b}^{(\kappa )}-\overline{\Pi _b}(N_S \rho _\mathrm{{gc}})= & {} 0. \end{aligned}$$

(169)

Now, we use

$$\begin{aligned} \overline{\Pi _b}(\bullet \rho _\mathrm{{gc}})=(\overline{\Pi _b}^\dagger \bullet )\rho _\mathrm{{gc}} , \end{aligned}$$

(170)

which is derived from KMS condition (83). Using this relation, we rewire (168) and (169) as

$$\begin{aligned} \overline{K}_\kappa \rho _{1,b}^{(\kappa )}+(\overline{\Pi _b}^\dagger H_S) \rho _\mathrm{{gc}}= & {} 0, \end{aligned}$$

(171)

$$\begin{aligned} \overline{K}_\kappa \rho _{2,b}^{(\kappa )}-(\overline{\Pi _b}^\dagger N_S) \rho _\mathrm{{gc}}= & {} 0 . \end{aligned}$$

(172)

Multiplying $\rho _\mathrm{{gc}}^{-1}$ from the right, we obtain

$$\begin{aligned} (\overline{K}_\kappa \rho _{1,b}^{(\kappa )})\rho _\mathrm{{gc}}^{-1}+\overline{\Pi _b}^\dagger H_S= & {} 0, \end{aligned}$$

(173)

$$\begin{aligned} (\overline{K}_\kappa \rho _{2,b}^{(\kappa )})\rho _\mathrm{{gc}}^{-1}-\overline{\Pi _b}^\dagger N_S= & {} 0 . \end{aligned}$$

(174)

(170) can be rewritten as

$$\begin{aligned} (\overline{\Pi _b}Y)\rho _\mathrm{{gc}}^{-1}=\overline{\Pi _b}^\dagger (Y\rho _\mathrm{{gc}}^{-1}), \end{aligned}$$

(175)

for any $Y=\bullet \rho _\mathrm{{gc}} \in \mathrm{\varvec{B}}$ by multiplying $\rho _\mathrm{{gc}}^{-1}$ from the right. (175) leads

$$\begin{aligned} (\overline{\Pi }\rho _{i,b}^{(\kappa )})\rho _\mathrm{{gc}}^{-1} =\overline{\Pi }^\dagger (\rho _{i,b}^{(\kappa )}\rho _\mathrm{{gc}}^{-1}) , \end{aligned}$$

(176)

where $\overline{\Pi }{\mathop {=}\limits ^{\mathrm {def}}}\sum _b \overline{\Pi _b}$. By the way, $[H_S(\alpha _S),\rho _0^{(\kappa )}(\alpha )]= 0$ holds similarly to (86). Differentiating this equation by $\alpha _{i,b}$, we obtain

$$\begin{aligned}{}[H_S(\alpha _S),\rho _{i,b}^{(\kappa )}] = 0. \end{aligned}$$

(177)

This relation leads

$$\begin{aligned} (\overline{H_\kappa ^{\times }}\rho _{i,b}^{(\kappa )})\rho _\mathrm{{gc}}^{-1}=\overline{H_\kappa ^{\times }}(\rho _{i,b}^{(\kappa )}\rho _\mathrm{{gc}}^{-1}) = \overline{H_{-\kappa }^{\times }}^\dagger (\rho _{i,b}^{(\kappa )}\rho _\mathrm{{gc}}^{-1}), \end{aligned}$$

(178)

where $H_\kappa ^{\times }\bullet {\mathop {=}\limits ^{\mathrm {def}}}-i[H_S(\alpha _S)+\kappa H_\mathrm{{L}}(\alpha ),\bullet ]$. We used $(H_\kappa ^{\times })^\dagger =-H_\kappa ^{\times }$. In the first equality, we used that $\rho _\mathrm{{gc}}$ commutes with $H_S$ and $H_\mathrm{{L}}$. (176) and (178) lead

$$\begin{aligned} (\overline{K}_\kappa \rho _{i,b}^{(\kappa )})\rho _\mathrm{{gc}}^{-1} = \overline{K}_{-\kappa }^\dagger (\rho _{i,b}^{(\kappa )}\rho _\mathrm{{gc}}^{-1}). \end{aligned}$$

(179)

Substituting this into (173) and (174), we obtain

$$\begin{aligned} \overline{K}_{-\kappa }^\dagger (\rho _{1,b}^{(\kappa )}\rho _\mathrm{{gc}}^{-1})+\overline{\Pi _b}^\dagger H_S= & {} 0 , \end{aligned}$$

(180)

$$\begin{aligned} \overline{K}_{-\kappa }^\dagger (\rho _{2,b}^{(\kappa )}\rho _\mathrm{{gc}}^{-1})-\overline{\Pi _b}^\dagger N_S= & {} 0 . \end{aligned}$$

(181)

Subtracting (180) ((181)) for $\kappa =-1$ from (163) ((164)), we obtain

$$\begin{aligned} \overline{K}^\dagger (k_{i,b}-\rho _{i,b}^{(-1)}\rho _\mathrm{{gc}}^{-1}) = 0. \end{aligned}$$

(182)

This means

$$\begin{aligned} k_{i,b} = \rho _{i,b}^{(-1)}\rho _\mathrm{{gc}}^{-1}+\overline{c_{i,b}}1, \end{aligned}$$

(183)

where $\overline{c_{i,b}}$ is an arbitrary complex number.

Appendix C: Definition of Entropy Production of the Markov Jump Process

Except (192), this section is based on Ref. [21]. We consider the Markov jump process on the states $n=1,2,\ldots ,\mathcal {N}$:

$$\begin{aligned} n(t)= n_k (t_k \le t<t_{k+1}), \ t_0= 0<t_1<t_2 \cdots<t_n<t_{N+1}=\tau . \end{aligned}$$

(184)

where $N=0,1,2,\ldots$ is the total number of jumps. We denote the above path by

$$\begin{aligned} \hat{n}= (N,(n_0,n_1,\ldots ,n_N),(t_1,t_2,\ldots ,t_N)) . \end{aligned}$$

(185)

The probability to find the system in a state n is $p_n(t)$ and it obeys the master equation (119). We suppose the trajectory of the control $\hat{\alpha }=\big ( \alpha (t) \big )_{t=0}^{\tau }$ is smooth. Now we introduce

$$\begin{aligned} \theta _{nm}(\alpha ) {\mathop {=}\limits ^{\mathrm {def}}}\left\{ \begin{array}{ll} - \ln \frac{K_{nm}(\alpha )}{K_{mn}(\alpha )} &{}\quad K_{nm}(\alpha ) \ne 0 \\ 0 &{}\quad K_{nm}(\alpha ) = 0 \end{array} \right. . \end{aligned}$$

(186)

If $n\ne m$, this is entropy production of process $m \rightarrow n$. The entropy production of process (185) is defined by

$$\begin{aligned} \Theta ^{\hat{\alpha }}[\hat{n}] = \sum _{k=1}^N \theta _{n_kn_{k-1}}(\alpha _{t_k}). \end{aligned}$$

(187)

Then the weight (the transition probability density) associated with a path $\hat{n}$ is

$$\begin{aligned} \mathcal {T}^{\hat{\alpha }}[\hat{n}] = \prod _{k=1}^N K_{n_kn_{k-1}}(\alpha _{t_k}) \exp \Big [\sum _{k=0}^N \int _{t_k}^{t_{k+1}}dt \ K_{n_kn_k}(\alpha _t) \Big ] . \end{aligned}$$

(188)

The integral over all the paths is defined by

$$\begin{aligned} \int \mathcal {D}{\hat{n}}\ Y[{\hat{n}}] {\mathop {=}\limits ^{\mathrm {def}}}\sum _{N=0}^\infty \sum _{n_0,n_1,\ldots ,n_N}^{n_{k-1}\ne n_k}\int _0^\tau dt_1 \int _{t_1}^\tau dt_2 \int _{t_3}^\tau dt_3 \cdots \int _{t_{N-1}}^\tau dt_N \ Y[{\hat{n}}], \end{aligned}$$

(189)

and the expectation value of $X[{\hat{n}}]$ is defined by

$$\begin{aligned} \langle X \rangle ^{\hat{\alpha }}{\mathop {=}\limits ^{\mathrm {def}}}\int \mathcal {D}{\hat{n}}\ X[{\hat{n}}]p_{n_0}^\mathrm{{ss}}(\alpha _0) \mathcal {T}^{\hat{\alpha }}[{\hat{n}}] . \end{aligned}$$

(190)

Here, $p_{n}^\mathrm{{ss}}(\alpha )$ is the instantaneous stationary probability distribution characterized by $\sum _m K_{nm}(\alpha )p_m^\mathrm{{ss}}(\alpha ) = 0$. We introduce a matrix $K^\lambda (\alpha )$ by

$$\begin{aligned}{}[K^\lambda (\alpha )]_{nm} {\mathop {=}\limits ^{\mathrm {def}}}K_{nm}(\alpha )e^{i\lambda \theta _{nm}(\alpha )} . \end{aligned}$$

(191)

Then, the k-th order moment of the entropy production is given by

$$\begin{aligned} \langle (\Theta ^{\hat{\alpha }}[{\hat{n}}])^k \rangle ^{\hat{\alpha }}= \frac{\partial ^k}{\partial (i\lambda )^k}\Big \vert _{\lambda =0} \sum _{n,m} \Big [ \mathrm{{T}} \exp \Big [ \int _0^\tau dt \ K^\lambda (\alpha _t) \Big ] \Big ]_{nm} p_m^\mathrm{{ss}}(\alpha _0) . \end{aligned}$$

(192)

In particular, the average is given by

$$\begin{aligned} \sigma ^\mathrm{{C}} {\mathop {=}\limits ^{\mathrm {def}}}\langle \Theta ^{\hat{\alpha }}[{\hat{n}}] \rangle ^{\hat{\alpha }}= \int _0^\tau dt \ \sum _{n,m} \sigma _{nm}^\mathrm{{C}}(\alpha _t)p_m(t), \end{aligned}$$

(193)

where

$$\begin{aligned} \sigma _{nm}^\mathrm{{C}}(\alpha ) {\mathop {=}\limits ^{\mathrm {def}}}K_{nm}(\alpha ) \theta _{nm}(\alpha ) = -K_{nm}(\alpha ) \ln \frac{K_{nm}(\alpha )}{K_{mn}(\alpha )}. \end{aligned}$$

(194)

According to Ref. [21], for a quasi-static operation,

$$\begin{aligned} \sigma _\mathrm{{ex}}^\mathrm{{C}}= S_\mathrm{{Sh}}[p^\mathrm{{ss}}(\alpha _\tau )]-S_\mathrm{{Sh}}[p^\mathrm{{ss}}(\alpha _0)]+\mathcal {O}(\varepsilon ^2\delta ), \end{aligned}$$

(195)

holds where

$$\begin{aligned} \sigma _\mathrm{{ex}}^\mathrm{{C}}{\mathop {=}\limits ^{\mathrm {def}}}\sigma ^\mathrm{{C}}-\int _0^\tau dt \ \sum _{n,m} \sigma _{nm}^\mathrm{{C}}(\alpha _t)p_m^\mathrm{{ss}}(\alpha _t) , \end{aligned}$$

(196)

and $S_\mathrm{{Sh}}[p]{\mathop {=}\limits ^{\mathrm {def}}}-\sum _n p_n\ln p_n$.