Bounds on quantum Fisher information and uncertainty relations for thermodynamically conjugate variables

TL;DR

Addressing thermodynamic uncertainty, the paper extends quantum uncertainty relations to thermodynamically conjugate pairs by linking the estimator variance of a classical intensive parameter $\theta$ to fluctuations of its conjugate quantum operator $\hat{O}$. It introduces a universal integral representation for the quantum Fisher information $\mathcal{F}_{\theta}$ in Gibbs states and derives a chain of bounds $\mathrm{LB} \le \mathcal{F}_{\theta} \le \mathrm{UB}_1 \le \mathrm{UB}_2$ with $\mathrm{UB}_1=\beta\partial_{\theta} \langle \hat{O} \rangle$, $\mathrm{UB}_2=\beta^{2}\langle(\Delta \hat{O})^{2}\rangle$, and $\mathrm{LB}=\frac{(\partial_{\theta}\langle \hat{O}\rangle)^{2}}{\langle(\Delta \hat{O})^{2}\rangle}$. A generalized fluctuation-dissipation relation including a zero-frequency term and a time-domain representation of the symmetric logarithmic derivative $\hat{L}$ are derived, yielding a thermodynamic uncertainty relation $\Delta\theta\,\overline{\Delta O} \ge 1/\beta$ with $\overline{\Delta O} \equiv (\partial_{\theta}\langle\hat{O}\rangle)\,\Delta\theta$. Numerically, the framework is validated in the 1D transverse-field Ising model, showing tight bounds, a $1/T^{2}$ scaling in regimes with ground-state degeneracy, and that the optimal estimator can be well approximated by sums of local operators, indicating experimental relevance.

Abstract

Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which \textit{mechanically} conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for \textit{thermodynamically} conjugate variables -- specifically, a classical intensive parameter $θ$ and its corresponding extensive quantum operator $\hat{O}$ -- in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter $θ$ is quantified by its quantum Fisher information $\mathcal{F}_θ$. The framework is based on an exact integral representation that relates $\mathcal{F}_θ$ to the autocorrelation function of operator $\hat{O}$. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation: $Δθ\,\overline{ΔO} \ge k_\text{B}T$ with $\overline{ΔO}\equiv\partial_θ\langle\hat{O}\rangle\,Δθ$ and $T$ is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.

Figures (3)

• Figure 1: Schematic illustration of the chain inequality $\text{LB}\le\mathcal{F}_{\theta}\le\text{UB}_{1}\le\text{UB}_{2}$. The diagram highlights that these bounds are not independent, the tighter upper bound (UB$_1$) is the geometric mean of the lower bound (LB) and the conventional variance bound (UB$_2$), satisfying $\text{UB}_{1}^{2}=\text{UB}_{2}\times\text{LB}$.
• Figure 2: Temperature dependence of the quantum Fisher information (solid orange) and its bounds --- LB (purple), UB$_1$ (deep blue), and UB$_2$ (light blue). Results are for the exact solution with system size $N=100$. (a) In the ferromagnetic phase ($\gamma < \gamma_c$), all bounds are degenerate. (b) In the paramagnetic phase ($\gamma > \gamma_c$), they are degenerate only at high $T$. In contrast, at low $T$, the quantum Fisher information is saturated by the LB. The dashed line indicates the $1/T^2$ scaling followed by the quantum Fisher information at high $T$ and in the low $T$ ferromagnetic phase, highlighting cooling as a metrological resource.
• Figure 3: Transverse field dependence of the quantum Fisher information (solid orange) and its bounds --- LB (purple), UB$_1$ (deep blue), and UB$_2$ (light blue). Results are for the exact free-fermion solution with system size $N=100$. (a) At high temperature, all bounds are degenerate across the entire range of field strength $\gamma$. (b) At low temperature, the system exhibits dramatically different behavior on either side of the quantum phase transition point at $\gamma_c$. Throughout the entire ferromagnetic phase ($\gamma < \gamma_c$), the quantum Fisher information is significantly enhanced, while in the paramagnetic phase ($\gamma > \gamma_c$) the quantum Fisher information is tightly tracked by the LB.