Zak Transform and non-uniqueness in an extension of Pauli's phase retrieval problem

The aim of this paper is to pursue the investigation of the phase retrieval problem for the fractional Fourier transform $\text{ff}\mathrm{\_}\alpha$ started by the second author. We here extend a method of A.E.J.M Janssen to show that there is a countable set $\text{qq}$ such that for every finite subset $\mathring{a}\subset \text{qq}$, there exist two functions $f,g$ not multiple of one an other such that $|\ff_\alpha f|=|\ff_\alpha g|$forevery$\alpha\in \aa$. Equivalently, in quantum mechanics, this result reformulates as follows: if $Q\mathrm{\_}\alpha =Q\cos \alpha +P\sin \alpha$ ($Q,P$ be the position and momentum observables), then $\{Q\mathrm{\_}\alpha ,\alpha \in \mathring{a}\}$ is not informationally complete with respect to pure states. This is done by constructing two functions $\text{ffi},\psi$ such that $\text{ff}\mathrm{\_}\alpha \text{ffi}$ and $\text{ff}\mathrm{\_}\alpha \psi$ have disjoint support for each $\alpha \in \mathring{a}$. To do so, we establish a link between $\text{ff}\mathrm{\_}\alpha [f]$, $\alpha \in \text{qq}$ and the Zak transform $Z[f]$ generalizing the well known marginal properties of $Z$.