Thermodynamic uncertainty relation under continuous measurement and feedback with quantum-classical-transfer entropy

Abstract

We derive a thermodynamic uncertainty relation (TUR) under quantum continuous measurement and feedback control. By incorporating the quantum-classical-transfer entropy, which quantifies the information gained by continuous measurement, we show that the precision of currents is constrained by information-thermodynamic costs such as the entropy production and information gain. Our result shows that information gain has the potential to enhance the precision of currents beyond the bounds set by the conventional TUR. We illustrate the bound with a driven two-level system under continuous measurement and feedback, demonstrating that feedback achieves higher precision of currents while suppressing the entropy production.

Figures (4)

• Figure 1: Schematic illustration of a quantum system attached to a heat bath under continuous measurement and feedback control.
• Figure 2: Schematic illustration of the dynamics of a quantum system under continuous measurement and feedback control. $Y_{t_n} \coloneqq (y_{t_1}, y_{t_2}, \ldots, y_{t_n})$ denotes accumulated measurement outcomes up to $t_n$. The dynamics during $[t_n, t_{n+1})$ is separated into two processes, interaction with the heat bath and feedback denoted by $\mathcal{E}_{t_{n}}^{Y_{n}}$, and measurement process denoted by $\mathcal{M}$. The states right before and after the process $\mathcal{E}_{t_{n}}^{Y_{n}}$ are denoted by $\rho_{t_{n}}^{Y_{t_n}}$ and $\sigma_{t_n}^{Y_{t_n}}$.
• Figure 3: Numerical example of driven two-level system under continuous measurement and feedback control. (a) Schematic illustration of the feedback protocol. By applying the feedback $\pi$-pulse, we reduce the population of the excited state. We plot the numerical results with feedback (blue circles) and without feedback (red circles) varying parameters randomly. Gray lines connect the points with and without feedback for the same parameters. (b) Scatter plot of the pairs of $\mathrm{Var[\hat{J}]}/\langle \hat{J} \rangle^2$ and $\Sigma + I_{\mathrm{QCT}} - \Delta \chi$. We employ the current $\hat{J}$ as the heat current from the heat bath. The green dotted line represents $\mathrm{Var}[\hat{J}]/\langle \hat{J} \rangle^2 = 2/(\Sigma + I_{\mathrm{QCT}} - \Delta \chi)$. Some points with feedback (blue circles) are below the green dotted line, which shows that TUR without correction terms $\mathcal{Q}, \tilde{\delta} J$ can be violated. (c) Scatter plot of the pairs of $\mathrm{Var}[\hat{J}]/\langle \hat{J} \rangle^2$ and the entropy production $\Sigma$. Most points with feedback control have larger precision of the current and smaller entropy production, compared with those without feedback. We choose $\beta = 1, \tau = 10, \gamma_m = \gamma(2 \bar{N} + 1)$. We randomly sample other parameters from uniform distributions as $\omega \in [0.001, 0.5], \omega_d \in [0, 0.5], \Omega \in [0.0, 0.5], \gamma \in [0.0, 0.1], \eta \in [0.5, 1], \delta \in [0.0, 0.3]$. We calculate $10^4$ trajectories of stochastic Schrödinger equation 10.1093/acprof:oso/9780199213900.001.0001Wiseman_Milburn_2009 for each parameters to obtain the change of von Neumann entropy $\Delta S$, ensemble average and variance of the current $\langle \hat{J} \rangle, \mathrm{Var}[\hat{J}]$. We also calculate $10^3$ trajectories of stochastic master equation \ref{['eq:stochastic_master_eq']} using quantum jump unraveling with the measurement operator $M$ to obtain $\Sigma + I_{\mathrm{QCT}} - \Delta \chi$.
• Figure 4: Thermodynamic quantities of driven two-level system under continuous measurement and feedback control. We plot the numerical results with feedback (blue circles) and without feedback (red circles) varying parameters randomly as mentioned in the main text. Gray lines connect the points with and without feedback for the same parameters. (a) (b) (c) The scatter plots of the pairs of $\Sigma + I_{\mathrm{QCT}} - \Delta \chi$ and (a) the change of von Neumann entropy $\Delta S$, (b) the ensemble average of heat absorbed from the heat bath $Q$, (c) the entropy production $\Sigma = \Delta S - \beta Q$. The plot (a) shows that most points with feedback indicate $\Delta S < 0$, which means that the feedback successfully decreases the entropy and reduces the population of the excited state. The plot (c) shows that most points with feedback indicate $\Sigma < 0$, realizing Maxwell's demon. The plots (a), (b), (c) show that $\Sigma + I_{\mathrm{QCT}} - \Delta \chi$ is non-negative, consistent with Eq. \ref{['eq:generalized_second_law']}.