Zak Transform and non-uniqueness in an extension of Pauli's phase retrieval problem
Résumé
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 $Z$.
Origine : Fichiers produits par l'(les) auteur(s)