量子 Fisher 信息的界限及热力学共轭变量的不确定性关系
Bounds on quantum Fisher information and uncertainty relations for thermodynamically conjugate variables
AI总结：
本文基于热力学共轭变量建立不确定性关系，通过量子 Fisher 信息推导出热力学不确定性关系，揭示了热系统中量子传感的精度极限。
摘要：
不确定性关系是量子力学的基本原理，限制了机械共轭变量如位置和动量同时确定的精度。本文建立了热力学共轭变量的类似关系，特别是经典强度参数θ与其对应的量子算符O在平衡态中的关系。我们开发了一个框架，通过精确的积分表示将θ的量子 Fisher 信息与算符O的自相关函数联系起来。由此推导出量子 Fisher 信息的紧上界，得出热力学不确定性关系：Δθ·ΔO≥k_B T，其中ΔO≡∂_θ⟨O⟩·Δθ，T为系统温度。该结果建立了热系统中量子传感和计量的基本精度限制，直接联系到线性响应和波动的热力学性质。
英文摘要：
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.
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