Thermodynamic Uncertainty Relation with Quantum Feedback

Abstract

Fluctuations are intrinsic to microscopic systems and impose fundamental limits on nonequilibrium precision, as captured by the thermodynamic uncertainty relation (TUR), which links current fluctuations to entropy production. While feedback control is expected to further suppress fluctuations, its role within the TUR framework has remained unclear, particularly in quantum systems where control is inherently information-driven. In this Letter, we consider open quantum systems weakly coupled to a thermal environment, in which quantum jumps are continuously monitored, and Markovian feedback is applied. Using quantum mutual information to quantify the information contribution induced by feedback, we derive a finite-time TUR for arbitrary time-integrated currents in terms of entropy production and mutual information. Our results uncover how feedback control suppresses fluctuations together with thermodynamic cost and establishes a fundamental precision bound imposed by information-based control. As an application, we analyze a quantum clock model and demonstrate that the clock precision can be enhanced by feedback control in the presence of a single thermal reservoir.

Figures (2)

• Figure 1: Schematic illustration of an open quantum system $S$ under Markovian feedback control. Quantum jumps induced by interactions with the environment $E$ are continuously monitored, and feedback control is applied instantaneously based on the detected quantum-jump outcomes.
• Figure 2: (a) Schematic illustration of the quantum clock, composed of a three-level system weakly coupled to a thermal reservoir. (b) Numerical verification of the quantum TUR \ref{['eq:main.result']} with feedback in the stationary state. (b1) Solid and dashed lines represent the relative fluctuation $\mathop{\mathrm{Var}}\nolimits[J]/\ev{J}^2$ and its lower bound $2(1+\delta_{J})^2/\Sigma$, respectively. (b2) Solid and dashed lines depict the average current rate $\dot{\ev{J}}$ and the correction term $\delta_{J}$, respectively. (b3) Solid, dashed, and dash-dotted lines represent $\dot\Sigma$, $\dot S^{\rm tot}$, and $-\dot I$, respectively. The energy level $E_1$ of the state $\ket{1}$ is varied, while the remaining parameters are fixed as $E_0=0$, $E_2=1$, $\beta =1$, $\gamma_{1\rightarrow 0}=10$, $\gamma_{2\rightarrow 1}=0.5$, $\gamma_{2\rightarrow 0}=1$, and $\tau = 1$.