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

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