Consider an unitary matrix $U$ and an positive definite, invertible and diagonalizable matrix $\rho$ . Then, if the following identity holds (i.e., if there are additional $2\pi I$ factor), \begin{equation} \text{Im}[\text{tr}\ln (\rho U)]=\text{Im}[\text{tr}\ln (\rho)]+\text{Im}[\text{tr}\ln (U)], \end{equation} where the branch cut of $\ln$ function is taken to be $(-\infty, 0]$, and we assume that eigenvalues of $\rho U$ do not locate at the branch cut.
Wiki only mentions that this identity works for product of positive definite matrix, so I wonder if this holds in case one of the matrix is not positive definite.