Let $U$ be an $n \times n$ unitary matrix.
And let $|\cdot |^2: \mathbb C^n \rightarrow \mathbb R_+^n$ be the function such that $|\mathbb w|^2$ is the vector which has its $i$-th entry equal to $|w_i|^2$, where $w_i$ is the $i$-th entry of $\mathbb w$.
When is there a function $f : \mathbb R_+^n \rightarrow \mathbb R_+^n$ such that for each unit vector $\mathbb z \in \mathbb C^n$, $f(\mathbb |\mathbb z|^2) = |U\mathbb z|^2$?
Note: I am asking this question, because it is related to quantum mechanics; if there exists a linear function $f$ such that the above is true, then quantum mechanics could be interpreted classically. But this cannot be so, since $|U\mathbb z|^2$ is not linear. But what if $f$ is nonlinear?
Of course, your conjecture is false. For instance, take $U=Rot(\theta),z=[\cos(\alpha),\sin(\alpha)]^T,y=[\cos(\alpha),-\sin(\alpha)]^T$. Then $|z|^2=|y|^2$ and $|Uz|^2=[\cos^2(\theta+\alpha),\sin^2(\theta+\alpha)^T$ and $|Uy|^2=[\cos^2(\theta-\alpha),\sin^2(\theta-\alpha)]^T$.
Keep your bounty for further opportunity.