Is the bra-ket scalar product the only function invariant under unitary transformations? In Sabine's recent paper on a proof of Born's rule, she asserts that transition probabilities must be functions of the scalar product of the initial and final states.

Section 2 Proof
Since $P_N$ is invariant under unitary operations, transition probabilities
can only be functions of scalar products

How does one prove this statement?

[1] S. Hossenfelder, "A derivation of Born’s rule from symmetry", Annals Phys. 425 (2021) 168394, arXiv:2006.14175.
 A: Let $|\psi_1  \rangle, |\phi_1 \rangle,|\psi_2 \rangle,| \phi_2 \rangle$ be unit vectors such that :
$$ \langle \psi_1 |\phi_1 \rangle = \langle \psi_2 |\phi_2 \rangle$$
Then, there is a unitary map $U$ such that :
$$U |\psi_1 \rangle = |\psi_2 \rangle $$
and
$$U |\phi_1 \rangle = |\phi_2 \rangle $$
Since, $P_N$ is invariant under unitary transformations, we have :
$$P_N( |\psi_1  \rangle \to |\phi_1 \rangle) = P_N(|\psi_2 \rangle \to | \phi_2 \rangle)$$
This means that $P_N( |\psi_1  \rangle \to |\phi_1 \rangle)$ only depends on $\langle \psi_1 |\phi_1 \rangle$ : QED
A: One can show that it must consist of such scalar products with the aid of the Haar measure. (It is also sometimes called "twirling")
Consider a quantity
$$ Q = \phi_a^* A_{a,b} \phi_b , $$
that is invariant under unitary transformations. It implies that
$$ Q = \phi_a^* U_{a,x}^{\dagger} A_{x,y} U_{y,b} \phi_b , $$
for any unitary transformations $U$.
In that case one can apply an integration over all such unitary transformation to the quantity. It produces
$$ Q = \int \phi_a^* U_{a,x}^{\dagger} A_{x,y} U_{y,b} \phi_b\ \text{d}U , $$
where $\text{d}U$ is the Haar measure.  The result of such an integration produces
$$ \int U_{a,x}^{\dagger} U_{y,b}\ \text{d}U = \delta_{a,b} \delta_{x,y} . $$
Hence
$$ Q = \text{tr}\{A\}\ (\phi^* \cdot \phi) . $$
There can also be higher order quantities with two or more pairs, such as
$$ Q = \phi_a^* \phi_c^* A_{a,b,c,d} \phi_b \phi_d . $$
One can also evaluate the integration over the unitary transformations for such cases. In all such cases, the integration would produce contractions between $\phi^*$ and $\phi$ together with contraction between indices of $A$. Therefore, the invariance implies the scalar products.
