Equivalent Definitions of the Quantum Nonadiabatic Entropy Production

Abstract

The nonadiabatic entropy production is a useful tool for the thermodynamic analysis of continuously dissipating, nonequilibrium steady states. For open quantum systems, two seemingly distinct definitions for the nonadiabatic entropy production have appeared in the literature, one based on the quantum relative entropy and the other based on quantum trajectories. We show that these two formulations are equivalent. Furthermore, this equivalence leads us to a proof of the monotonicity of the quantum relative entropy under a special class of completely-positive, trace-preserving quantum maps, which circumvents difficulties associated with the noncommuntative structure of operators.

Appendices

Appendix 1: Time Derivative of Von Neumann Entropy

The derivation of (4) follows by first taking the derivative of the von Neumann entropy $S(t)=-\mathrm{Tr}[\rho _t\ln \rho _t]$,

$$\begin{aligned} \dot{S}(t) = - \mathrm{Tr}[ \dot{\rho }_t \ln \rho _t ] - \mathrm{Tr}\left[ \rho _t \frac{d}{dt}\ln \rho _t \right] . \end{aligned}$$

(45)

Clearly, (4) is true if the second term is zero. To demonstrate this, we decompose $\rho _t$ in its time-dependent eigenbasis $\rho _t = \sum _k P_t^k | k_t \rangle \langle k_t |$, where $\sum _k P_t^k = 1$ and $\{ | k_t \rangle \}$ form an orthonormal basis. We then have

$$\begin{aligned} \mathrm{Tr}\left[ \rho _t \frac{d}{dt}\ln \rho _t \right]&= \mathrm{Tr}\left[ \rho _t \sum _k \frac{\dot{P}_t^k }{P_t^k} | k_t \rangle \langle k_t |\right] + \mathrm{Tr}\left[ \rho _t \sum _k \ln P_t^k \frac{d}{dt}| k_t \rangle \langle k_t |\right] \end{aligned}$$

(46)

$$\begin{aligned}&= \sum _k \dot{P}_t^k + \sum _k P_t^k \ln P_t^k \frac{d}{dt}\mathrm{Tr}[| k_t \rangle \langle k_t |] \end{aligned}$$

(47)

$$\begin{aligned}&= 0. \end{aligned}$$

(48)

Appendix 2: Privileged Representation Identity

To verify (28), we first expand the logarithm in its power series about $I$ and then exploit (19) for the privileged $L_k$ as

$$\begin{aligned} \ln (\pi _\lambda ) L_k(\lambda )&=\sum _{n=1}^\infty \frac{(-1)^{n+1}}{n}(\pi _\lambda -I)^n L_k(\lambda ) \end{aligned}$$

(49)

$$\begin{aligned}&=L_k(\lambda )\sum _{n=1}^\infty \frac{(-1)^{n+1}}{n}(\varpi _k(\lambda )\pi _\lambda -I)^n \end{aligned}$$

(50)

$$\begin{aligned}&=L_k(\lambda )\ln (\varpi _k(\lambda )\pi _\lambda ). \end{aligned}$$

(51)

It is worth noting that this argument will hold for other functions besides the logarithm as long as they have a power series expansion.