  The fluctuation-dissipation theorem is a fundamental result in statistical
physics that establishes a connection between the response of a system subject
to a perturbation and the fluctuations associated with observables in
equilibrium. Here we derive its version within a resource-theoretic framework,
where one investigates optimal quantum state transitions under thermodynamic
constraints. More precisely, we first characterise optimal thermodynamic
distillation processes, and then prove a relation between the amount of free
energy dissipated in such processes and the free energy fluctuations of the
initial state of the system. Our results apply to initial states given by
either asymptotically many identical pure systems or arbitrary number of
independent energy-incoherent systems, and allow not only for a state
transformation, but also for the change of Hamiltonian. The
fluctuation-dissipation relations we derive enable us to find the optimal
performance of thermodynamic protocols such as work extraction, information
erasure and thermodynamically-free communication, up to second-order
asymptotics in the number $N$ of processed systems. We thus provide a first
rigorous analysis of these thermodynamic protocols for quantum states with
coherence between different energy eigenstates in the intermediate regime of
large but finite $N$.
