Stochastic thermodynamics of quantum maps with and without equilibrium

We study stochastic thermodynamics for a quantum system of interest whose dynamics is described by a completely positive trace-preserving (CPTP) map as a result of its interaction with a thermal bath. We define CPTP maps with equilibrium as CPTP maps with an invariant state such that the entropy production due to the action of the map on the invariant state vanishes. Thermal maps are a subgroup of CPTP maps with equilibrium. In general, for CPTP maps, the thermodynamic quantities, such as the entropy production or work performed on the system, depend on the combined state of the system plus its environment. We show that these quantities can be written in terms of system properties for maps with equilibrium. The relations that we obtain are valid for arbitrary coupling strengths between the system and the thermal bath. The fluctuations of thermodynamic quantities are considered in the framework of a two-point measurement scheme. We derive the entropy production fluctuation theorem for general maps and a fluctuation relation for the stochastic work on a system that starts in the Gibbs state. Some simplifications for the probability distributions in the case of maps with equilibrium are presented. We illustrate our results by considering spin $\frac{1}{2}$ systems under thermal maps, nonthermal maps with equilibrium, maps with nonequilibrium steady states, and concatenations of them. Finally, and as an important application, we consider a particular limit in which the concatenation of maps generates a continuous time evolution in Lindblad form for the system of interest, and we show that the concept of maps with and without equilibrium translates into Lindblad equations with and without quantum detailed balance, respectively. The consequences for the thermodynamic quantities in this limit are discussed.

Figure 1

The figure depicts the first two interactions between the system and the copies of the bath.

Figure 2

(a) Hilbert-Schmidt distance between the state of the system at each step of the concatenation and the thermal state. In this concatenation process, we take ${\tau }_1=1,V^1=(J_B+0.3){\sigma }_b^x{\sigma }_S^x+J_B{\sigma }_b^y{\sigma }_S^y,{\tau }_n=4,V^n=J_B{\sigma }_b^x{\sigma }_S^x+J_B{\sigma }_b^y{\sigma }_S^y$ for $n\ge 2$, and the other parameters are $h=1,J_B=3$, and $\beta =1.$ (b) Work and entropy production distributions for the full cycle and the driving alone, $p_1\equiv p_{\text{drive}}\left(\beta w\right)=p_{\text{cycle}}\left(\beta w\right)=p_{\text{cycle}}\left({\mathrm{\Delta }}_{i}s\right)\ne p_2\equiv p_{\text{drive}}\left({\mathrm{\Delta }}_{i}s\right)$. The parameters of the plot are the same as in (a).

Figure 3

(a) For the homogeneous $XX$ spin $\frac{1}{2}$ chain with three sites, cumulated average work (dashed black line), heat (dash-dotted red line), and entropy production (dotted blue line) as a function of the iteration number. The corresponding straight lines are the theoretically computed asymptotic values. (b) The same quantities for the homogeneous $XY$ spin $\frac{1}{2}$ chain with three sites. In both cases, the initial density matrix is the Gibbs state ${\omega }_{\beta }\left(H_S\right)$, which is a nonequilibrium state. The parameters for the plots are $h=2,J_i^x=J_B=3,\beta =1.2$, and $\tau =1$, and $J_i^y=2$ in (b).

Figure 4

(a) Work distribution for the $XX$ spin $\frac{1}{2}$ chain in the equilibrium state. (b) Work fluctuation relation [Eq. (23)] for the $XY$ spin $\frac{1}{2}$ chain, starting from the Gibbs state in the forward and backward processes. In both plots, we have used a chain of two sites and one map. The parameters for the plots are $h=2,J_i^x=J_B=3,\beta =1.2$, and $\tau =1$, and $J_i^y=2$ in (b).