Will entanglement change if we diagonalize a mixed state?

Question

I know that unitary operations do not alter the eigenvalues of a density matrix, i.e. its purity(mixedness) is conserved. However entanglement is only invariant under local unitary operations of the form $U_A \otimes U_B$, and changeable under nonlocal unitary operations $U_{AB}$.

For a pure entangled state, if we are to diagonalize it we would get a trivial $diag(1,0,...)$, so it is only diagonalizable through nonlocal operation with the loss of entanglement pertaining to the original state. This has prompted me to wonder, For a general undiagonalized bipartite mixed state $\rho = \sum_{ijkl} p_{ijkl} |i_A\rangle \langle j_A| \otimes |k_B\rangle \langle l_B|$, will its entanglement change if we diagonalize it? Since they seem to be only diagonalized through nonlocal unitaries, which always alters entanglement.

Answer 1

Any diagonal density matrix is of the form $$ \rho = \sum p_{ij} \rho_i\otimes \rho_j $$ with $\rho_i=|i\rangle\langle i|$, i.e., it is separable, and thus has zero entanglement.