transformation $U \in U(n)$ such that $U {\bf a} ={\bf a'}$ and $U {\bf b} = {\bf b'}$. Indeed, one can split $U$ into real and imaginary parts: $U = X - iY$, and it is easy to check that $R=R(X,Y)$ is such that $R{\bf x} ={\bf x'}$ and $R{\bf y} = {\bf y'}$.

Let us introduce the following notations: $A \equiv ||{\bf a}||^2=||{\bf a'}||^2, B \equiv ||{\bf b}||^2=||{\bf b'}||^2$ and $C \equiv \mathrm{Re}\langle {\bf a}| {\bf b} \rangle = \mathrm{Re}\langle {\bf a'}| {\bf b'} \rangle$.

Consider first the case where ${\bf a}$ and ${\bf b}$ are colinear. This means that ${\bf b} = C/A {\bf a}$ and $C = \pm \sqrt{AB}$. Using the Cauchy-Schwarz inequality, $|C| = |{\bf a'} \cdot {\bf b'}| \leq ||{\bf a'}|| \cdot||{\bf b'}|| =  \sqrt{AB}$ with equality if and only if ${\bf a'}$ and ${\bf b'}$ are colinear. This means that ${\bf a'}$ and ${\bf b'}$ are colinear and that ${\bf b'} = (C/A) \, {\bf a'}$. Because $||{\bf a}|| = ||{\bf a'}||$, the reflexion $U$ across the mediator hyperplane of ${\bf a}$ and ${\bf a'}$ is a unitary transformation that maps ${\bf a}$ to ${\bf a'}$. This reflexion also maps ${\bf b}$ to ${\bf 