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 currentlike 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.

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• Fluctuations & noise
• Nonequilibrium statistical mechanics
• Quantum thermodynamics
• Stochastic thermodynamics
• Linear response theory