Density matrix off-diagonal elements

The possible values of the diagonal elements of a density matrix are restricted by the condition $\mathrm{Tr}~\rho = 1$. Are there any restrictions on the possible values of off-diagonal elements, apart from the obvious $\mathrm{Re}~\rho_{nm} = \mathrm{Re}~\rho_{mn}$, $\mathrm{Im}~\rho_{nm} = - \mathrm{Im}~\rho_{mn}$? If the off-diagonals are written in the form $\left| \rho_{nm} \right| \exp{i \phi_{nm}}$, do the absolute value and the phase have a simple physical meaning?

The possible values of the diagonal elements of a density matrix are restricted by the condition $\mathrm{Tr}~\rho = 1$. Are there any restrictions on the possible values of off-diagonal elements, apart from the obvious $\mathrm{Re}~\rho_{nm} = \mathrm{Re}~\rho_{mn}$, $\mathrm{Im}~\rho_{nm} = - \mathrm{Im}~\rho_{mn}$? If the off-diagonals are written in the form $\left| \rho_{nm} \right| \exp{i \phi_{nm}}$, do the absolute value and the phase have a simple physical meaning?
$|\rho_{nm}|^2 \le \rho_{nn}\rho_{mm}\le 1$.