Fluctuation-response inequalities for kinetic and entropic perturbations

Abstract

We derive fluctuation-response inequalities for Markov jump processes that link the fluctuations of general observables to the response to perturbations in the transition rates within a unified framework. These inequalities are derived using the Cramér-Rao bound, enabling broader applicability compared to existing fluctuation-response relations formulated for static responses of current-like observables. The fluctuation-response inequalities are valid for a wider class of observables and are applicable to finite observation times through dynamic responses. Furthermore, we extend these inequalities to open quantum systems governed by the Lindblad quantum master equation and find the quantum fluctuation-response inequality, where dynamical activity plays a central role.

Figures (3)

• Figure 1: Numerical verification of FRIs for general observables. (a) and (b) correspond to \ref{['eq:FRI_B']} and \ref{['eq:FRI_F']}, respectively. For the symmetric and anti-symmetric parameters, $B_{ij}$ and $e^{F_{ij}}$ are randomly sampled from $[-2,2]$ and $[0,10]$, respectively. The observation time is given as $\tau = e^x$ where $x$ is drawn randomly from $[-15,20]$. Weights $g_i$ and $\Lambda_{ij}$ are sampled from $[-2,2]$. The network topology is randomly selected from the four possible configurations shown in the inset of (b). Different point colors represent results from the respective topologies, matching the colors in the inset. The total number of points is $10^5$.
• Figure 2: Numerical verification of FRI \ref{['eq:FRI_B2']} for (a) current-like observables and (b) state-dependent observables. Transition rates, observation times, and weights of observables are sampled within the same ranges as in Fig. \ref{['fig:fig_FRI_classical']}. The network topology is randomly selected from the four configurations in the inset of (a), with data point colors matching the respective topologies. The total number of points is $10^5$.
• Figure 3: Numerical verification of Eq. \ref{['eq:FRI_Q']} for (a) a 2-level system and (b) a 3-level system. The Hamiltonian is constructed as $(A + A^\dagger)/2$, where $A$ is a randomly generated matrix with the real and imaginary parts of each element independently and uniformly sampled from $[-1, 1]$. Jump operators are generated similarly without enforcing the Hermitian condition, with the number of jump operators randomly chosen between one and four. Different numbers of jump operators are represented by distinct colors: blue (1 channel), cyan (2 channels), yellow (3 channels), and dark red (4 channels), respectively. Weights $\Lambda_k$ are randomly sampled from $[-1,1]$, and the observation time $\tau$ is given by $\tau = e^x$, where $x$ is uniformly sampled from $[-15, 20]$.