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Fleming's inequality is generalized to the decay function of mixed states. Weshow that for any symmetric hamiltonian $h$ and for any density operator $\rho$on a finite dimensional Hilbert space with the orthogonal projection $\Pi$ ontothe range of $\rho$ there holds the estimate $\Tr(\Pi \rme^{-\rmi ht}\rho\rme^{\rmi ht}) \geq\cos^{2}((\Delta h)_{\rho}t) $ for all real $t$ with$(\Delta h)_{\rho}| t| \leq\pi/2.$ We show that equality either holds for all$t\in\mathbb{R}$ or it does not hold for a single $t$ with $0<(\Deltah)_{\rho}| t| \leq\pi/2.$ All the density operators saturating the bound forall $t\in\mathbb{R},$ i.e. the mixed intelligent states, are determined.

Fleming's bound for the decay of mixed states