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Duality of reduced density matrices and their eigenvalues

For states of quantum systems of $N$ particles with harmonic interactions we prove that each reduced density matrix $ρ$ obeys a duality condition. This condition implies duality relations for the eigenvalues $λ_k$ of $ρ$ and relates a harmonic model with length scales $l_1,l_2, \ldots, l_N $ with another one with inverse lengths $1/l_1, 1/l_2,\ldots, 1/l_N$. Entanglement entropies and correlation functions inherit duality from $ρ$. Self-duality can only occur for noninteracting particles in an isotropic harmonic trap.