Paper detail

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 $\ff\_α$ 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 $å\subset \qq$, there exist two functions $f,g$ not multiple of one an other such that $|\ff\_αf|=|\ff\_αg|$ for every $α\in å$. Equivalently, in quantum mechanics, this result reformulates as follows: if $Q\_α=Q\cosα+P\sinα$ ($Q,P$ be the position and momentum observables), then $\{Q\_α,α\inå\}$ is not informationally complete with respect to pure states. This is done by constructing two functions $\ffi,ψ$ such that $\ff\_α\ffi$ and $\ff\_αψ$ have disjoint support for each $α\in å$. To do so, we establish a link between $\ff\_α[f]$, $α\in \qq$ and the Zak transform $Z[f]$ generalizing the well known marginal properties of $Z$.