I am considering the density matrix which represents an arbitrary state for a pair of qubits. When written out in terms of the Pauli operators, this is as follows (certain terms vanish for another reason so there are fewer than $16$ terms):
$$\rho = \frac{1}{4} \bigg( I \otimes I + t_{01}I \otimes \sigma_x + t_{10} \sigma_x \otimes I + t_{02} I \otimes \sigma_y + t_{20} \sigma_y \otimes I + t_{11} \sigma_x \otimes \sigma_x + t_{22}\sigma_y \otimes \sigma_y + t_{33} \sigma_z \otimes \sigma_z + t_{12} \sigma_x \otimes \sigma_y + t_{21}\sigma_y \otimes \sigma_x \bigg),$$
where the coefficients take values between $-1$ and $1$.
It is possible to have eigenstates for a density matrix but this is quite a long and complicated expression, so I am just curious to how would one would write down the possible eigenvectors for this matrix? Do the eigenvectors here relate to the usual eigenvectors for the Pauli matrices themselves?
