I have $N$ pure, but nonorthogonal, states $|\psi_n\rangle$ with density matrix $\rho_n=|\psi_n\rangle\langle\psi_n|$.
Say we call the the total density matrix $\rho=\frac{1}{N}\sum_n \rho_n$.
Are there any formulas to calculate $S_{vn}[\rho]=-\mathrm{Tr}[\rho\log{\rho}]$ solely from the overlaps $\langle\psi_n|\psi_m\rangle$ (and the fact that all individual states are pure)? Gram-Schmidt orthogonalization might be a possibility but perhaps there is an easier way/existing result?
Intuitively it would seem at first that knowledge of the $\mathrm{Tr}[\rho_n\rho_m]=|\langle\psi_n|\psi_m\rangle|^2$ would be sufficient because the mixing between the states is noncoherent, but is this true?