Local approach to entropy production in the nonequilibrium dynamics of open quantum systems

Abstract

We discuss fundamental features of the local expression for the entropy production rate of the nonequilibrium quantum dynamics of open systems and its relations to memory effects and the spectrum of the generator of the dynamics. Defining the entropy production rate as negative rate of change of the relative entropy with respect to an instantaneous fixed point, it is shown that positivity of the entropy production rate for all possible initial states implies that the real parts of the eigenvalues of the time-local generator for the quantum master equation are always negative. It is further demonstrated that Markovian dynamics, identified as P-divisibility of the quantum dynamical map, implies positivity of entropy production rate, thus providing a kind of generalized second law in the nonequilibrium regime. We also prove by means of the counterexample of a phase covariant quantum master equation that the converse of this statement is not true, i.e., there are non-Markovian dynamics for which the entropy production rate is always positive. Thus, we conclude that the emergence of negative entropy production rates is a sufficient but not necessary condition for non-Markovianity of the quantum dynamics. Finally, we also consider a recently introduced map-based notion of entropy production and show the equivalence between its positivity and Markovianity for general finite-dimensional systems.

Figures (5)

• Figure 1: Entropy production rate for the phase-covariant model as a function of time for different initial states with Bloch vector $\vec{v}(0) =v(0) (\cos \theta, 0, \sin \theta)^T$. Each subplot considers a different value of $v(0)$, and each differently colored line represents a different value of $\theta$. The parameters of the master equation are $\gamma_+ = 0.2$, $\gamma_- = 0.8$, and a time dependent $\gamma_z(t)$ which can be found in Eq. \ref{['eq:gzt']}. The map violates P-divisibility, but this is not witnessed by the entropy production rate, which remains always positive.
• Figure 2: Projection of the Bloch sphere onto the $xz$ plane, where different colors highlight the different regions used in the proof that entropy production rate remains always positive.
• Figure 3: Functions from Eq. \ref{['eq:conds_CP']} which determine complete positivity of the map. The fact that both $f_1(t)$ and $f_2(t)$ remain smaller or equal to zero confirms that the map studied is CP.
• Figure 4: $x_0$ function (blue solid line) defined in Eq. \ref{['eq:x0']} as a function of time, along with the horizontal black dashed lines marking the boundary values $0$ and $-0.6$.
• Figure 5: $P_0$ function (blue solid line) defined in Eq. \ref{['eq:P0']} as a function of time, along with the horizontal dashed lines marking the values $0.8^2$ and $0.73^2$. The vertical dotted lines mark the times $t = 1$ and $t = 1.5$. Since $v^2(t) \geq P_0(t)$, we can see that for $t \geq 1$, the inequality $v(t) \leq 0.8$ holds, and for $t > 1.5$, so does $v(t) \leq 0.73$.