Any two qubit density matrix can be written as
$$ \rho = \frac{1}{4} \sum_{n,m = 0}^{3} M_{nm} (\sigma_n \otimes \sigma_m), $$
where $\sigma_\mu$'s are the identity and Pauli matrices.
Is it possible to do a local unitary transformation on the system, like $U_1 \otimes U_2$, which will transform $\rho$ into $\rho'$ which looks like this:
$$\rho' = \frac{1}{4} \left( \sigma_0 \otimes \sigma_0 + M^{'}_{11} \sigma_1 \otimes \sigma_1 + M^{'}_{22} \sigma_2 \otimes \sigma_2 + M^{'}_{33} \sigma_3 \otimes \sigma_3 \right)$$
i.e. the matrix whose elements are $\{M_{nm}\}$ is diagonal?