Fleming's bound for the decay of mixed states

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

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