Quantum Martingale Theory and Entropy Production

We employ martingale theory to describe fluctuations of entropy production for open quantum systems in nonequilbrium steady states. Using the formalism of quantum jump trajectories, we identify a decomposition of entropy production into an exponential martingale and a purely quantum term, both obeying integral fluctuation theorems. An important consequence of this approach is the derivation of a set of genuine universal results for stopping-time and infimum statistics of stochastic entropy production. Finally, we complement the general formalism with numerical simulations of a qubit system.

Figure 1

Sample traces of stochastic entropy production in a nonequilibrium stationary process as a function of time. We show the entropy production $\mathrm{\Delta }{S}_{\mathrm{tot}}$ associated with classical (black thin line) and quantum (thick green line) trajectories of duration $t$. When the system state becomes a superposition, $\mathrm{\Delta }{S}_{\mathrm{tot}}$ is not uniquely defined in the quantum case (thick green segments) but only at the final time of the evolution $t$, when a direct measurement on the system is performed. (Inset) Different values that $\mathrm{\Delta }{S}_{\mathrm{tot}}$ would take if system measurements were performed at intermediate times, for a given record of measurements in the environment. How can one then determine stopping times? For example, when does entropy production reach a threshold (red dotted line) for the first time in an open quantum system?

Figure 2

(a) Sample trajectory of $|\psi (t)⟩$ represented in the qubit’s Bloch sphere. The smooth evolution starting from an eigenstate (blue circle) is interrupted by a jump $L_{-}$ (dotted line) which abruptly collapses the qubit state to $|-⟩=[|0⟩-|1⟩]/\sqrt{2}$ along the $x$ axis, after which the evolution continues up to the point marked by the arrow. (b) Convergence of the stopping-time fluctuation theorem for $\mathrm{\Delta }{S}_{\mathrm{mar}}$ as a function of the number of trajectories for three different stopping rules: stopping at a fixed time $t_f\equiv 20{\mathrm{\Gamma }}^{-1}$ (black line); $\min ({\mathcal{T}}_1,t_f)$ with ${\mathcal{T}}_1$ the first-passage time of $\mathrm{\Delta }{S}_{\mathrm{mar}}$ to reach a positive threshold located at 0.3 (brown line); and $\min ({\mathcal{T}}_2,t_f)$, with ${\mathcal{T}}_2$ being the unconditional first-passage time of $\mathrm{\Delta }{S}_{\mathrm{mar}}$ to reach either the thresholds 0.3 or $-0.4$ (red line). The dashed gray line in 1 is a guide for the eye. (Inset) Sample traces of stochastic entropy production (blue dashed line) and its decomposition as the sum of $\mathrm{\Delta }{S}_{\mathrm{mar}}$ (orange solid line) plus $\mathrm{\Delta }{S}_{\mathrm{unc}}$ (green solid line). The horizontal thick lines are the two absorbing boundaries used to compute stopping times. The parameters of the simulation are $\hslash \omega =1$, $\beta =0.2$, $\eta =0.5$, $\mathrm{\Gamma }\equiv {\gamma }_{-}-{\gamma }_{+}={\gamma }_{\downarrow }-{\gamma }_{\uparrow }=0.01$. (c) Empirical cumulative distributions of the finite-time minimum of $\mathrm{\Delta }{S}_{\mathrm{mar}}$ (solid lines) and $\mathrm{\Delta }{S}_{\mathrm{tot}}$ (dashed lines) for different values of the observation time: $\mathrm{\Gamma }t=10$ (red), $\mathrm{\Gamma }t={10}^2$ (blue), $\mathrm{\Gamma }t={10}^3$ (green), and $\mathrm{\Gamma }t={10}^4$ (purple). The gray dashed line is the exponential in the right-hand side of Eq. (11). (Inset) Enlarged view of the distribution for small values of the infima. The parameters for the simulation are $\hslash \omega =1$, $\beta =\eta =0.04$, $\mathrm{\Gamma }=0.01$.