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

TL;DR

This paper investigates non-uniqueness in phase retrieval for the fractional Fourier transform, showing that certain sets of transforms do not uniquely determine functions, with implications for quantum measurement completeness.

Contribution

It extends Janssen's method to demonstrate non-uniqueness in phase retrieval for fractional Fourier transforms using Zak transform techniques.

Findings

Existence of non-unique functions with identical fractional Fourier transforms on specific sets.

Construction of functions with disjoint support in their transforms for given sets.

Linking fractional Fourier transforms to Zak transform to analyze properties.

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 $Z$.