Universal Precision Limits in General Open Quantum Systems

Abstract

The intuition that the precision of observables is constrained by thermodynamic costs has recently been formalized through thermodynamic and kinetic uncertainty relations. While such trade-offs have been extensively studied in Markovian systems, corresponding constraints in the non-Markovian regime remain largely unexplored. In this Letter, we derive universal bounds on the precision of generic observables in open quantum systems that interact with their environments at arbitrary coupling strengths and are subjected to two-point measurements. By introducing an asymmetry term that quantifies the disparity between forward and backward processes, we show that the relative fluctuation of any time-antisymmetric current is constrained not only by entropy production but also by this asymmetry. For general observables, we further prove that their relative fluctuation is always bounded from below by a generalized activity term that characterizes environmental changes. These results establish a comprehensive framework for understanding the fundamental limits of precision in a broad class of open quantum systems, beyond the traditional Markovian setting.

Figures (2)

• Figure 1: Schematic illustration of general open quantum systems interacting with uncorrelated environments and subjected to two-point measurements. (a) The system and environment evolve under a single unitary transformation, with measurements performed at the initial and final times. (b) The system repeatedly interacts with fresh, uncorrelated environments, where each environment is projectively measured before and after the interaction.
• Figure 2: Numerical illustration of the main results \ref{['eq:main.res.1']} and \ref{['eq:main.res.2']} for a qubit interacting with a finite-dimensional environment. (a) Blue circles and orange squares represent the relative fluctuation of current-type observables plotted against $\Sigma+\Sigma_*$ and $\Sigma$, respectively. The solid line depicts the lower-bound function $f(x)$. (b) Blue circles represent the relative fluctuation of generic observables plotted against $\mathscr{P}$. The solid line shows the lower bound $1/(\mathscr{P}^{-1}-1)$. (c) Solid, dashed, dash-dotted lines represent the asymmetry $\Sigma_*$, the entanglement entropy $\overline{S}_{\rm EE}$, and the quality factor $\mathcal{Q}$, respectively, as functions of the coupling strength $\lambda$. Parameters are chosen as follows: the environment dimension $d_E$ is a random integer in the range $[2,5]$; the eigenvalues of the environmental Hamiltonian $H_E$ are sampled from $[0,0.1]$; $V_S$ and $V_E$ are random Hermitian operators with matrix elements in $[-1-i,1+i]$; and fixed parameters are $\omega_z=1$, $\omega_x=0.1$, $\lambda=5$, $\beta=1$, and $T=5$.