Diagnosing Thermality from Geometric Observables

We address the problem of determining whether a given mixed quantum state corresponds to thermal equilibrium or to a zero-temperature statistical mixture. We show that geometric observables, in particular the quantum Fisher information, provide a direct diagnostic criterion. Thermal states satisfy fluctuation–response relations linking energy variance to parameter sensitivity, while generic mixed states do not. This establishes a geometric test of thermality that does not require prior knowledge of the Hamiltonian and connects equilibrium statistical mechanics with quantum information geometry.