Let's say that I have a $3\times3$ complex unitary matrix $\mathbf{M}$, with elements $a_{i,j}$.
If I only know the squared modulus of $\mathbf{M}$ and its elements (that is, I only know $|a_{i,j}|^2$), is there a parameterisation of $\mathbf{M}$ for which I can work out its elements?
So far, I have been using a four-parameter parameterisation equivalent to the CKM matrix (See https://en.wikipedia.org/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix). However, this leaves me with a sign degeneracy in one of the parameters (the phase angle, $\delta$). Is there a better way of doing this?
Alternatively, is this an impossible problem without more information, and will there always be some sort of degeneracy in one or more of the parameters?