Can I obtain the pure state corresponding to a density matrix from its main diagonal?

Question

Suppose we have a bipartite pure state as follows:

$$|\psi\rangle=a_1|00\rangle+a_2|01\rangle+a_3|10\rangle+a_4|11\rangle\,.$$

Then, the density matrix is as follows:

$$|\psi\rangle\langle\psi|=\left( \begin{array}{cccc} a_1^2 & a_1 a_2 & a_1 a_3 & a_1 a_4 \\ a_1 a_2 & a_2^2 & a_2 a_3 & a_2 a_4 \\ a_1 a_3 & a_2 a_3 & a_3^2 & a_3 a_4 \\ a_1 a_4 & a_2 a_4 & a_3 a_4 & a_4^2 \\ \end{array} \right)\,.$$

Now, suppose I have the density matrix of the pure state and don't have the state. Can I obtain the pure state using the main diagonal of the density matrix?

Answer 1

No: in writing out your density matrix, you're forgetting that the amplitudes can be complex. So the diagonal elements are things like $|a_1|^2$. You can never determine the complex phase of these values from just the diagonal elements.