It is known that coordinates $C_k\in\mathbb{C}$ of the QM-state vectors $|\psi\rangle$ has an interpretation as probability weights $p_k$ in the whole state through the formula like $|C_k|^2=p_k$. We thus have the two (independent) math operation on the coefficients: making a modulus $C_k\to |C_k|$ and their squaring $|C_k|\to|C_k|^2$. Does anybody know a derivation of that transition, i.e., $C_k\to |C_k|^2=p_k$, from plausible/forceful physical arguments? The key words in the question are 'derivation' and 'forceful'. In other words, can we throw out the word 'interpretation' in the context and derive the rule $|C_k|^2$ is a probability weight? And reversely, the probability weight is exactly $|C_k|^2$ and only this formula?
I've knew right now about the 2005 paper by Aaronson but observed there the following. He begins there with a norm of the space, not with a scalar product of two independent vectors of $H$. The problem here, in my view, is that I prefere the object $(\psi,\phi)$ to be more fundamental (in the QM context) than $(\psi,\psi)$. This is because the derivation of the former requires some extra/exotic arguments on properties of the norm (polarization procedure). Consideration in the reverse direction does not. So, one could rephrase the question about 'interpretation/derivation' of $|(\psi,\phi)|^2$, not about $|(\psi,\psi)|^2$.
