Nonuniqueness of phase retrieval for three fractional Fourier transforms

Abstract

We prove that, regardless of the choice of the angles $θ_1,θ_2,θ_3$, three fractional Fourier transforms $F_{θ_1}$, $F_{θ_2}$ and $F_{θ_3}$ do not solve the phase retrieval problem. That is, there do not exist three angles $θ_1$, $θ_2$, $θ_3$ such that any signal $ψ\in L^2(R)$ could be determined up to a constant phase by knowing only the three intensities $|F_{θ_1}ψ|^2$, $|F_{θ_2}ψ|^2$ and $|F_{θ_3}ψ|^2$. This provides a negative argument against a recent speculation by P. Jaming, who stated that three suitably chosen fractional Fourier transforms are good candidates for phase retrieval in infinite dimension. We recast the question in the language of quantum mechanics, where our result shows that any fixed triple of rotated quadrature observables $Q_{θ_1}$, $Q_{θ_2}$ and $Q_{θ_3}$ is not enough to determine all unknown pure quantum states. The sufficiency of four rotated quadrature observables, or equivalently fractional Fourier transforms, remains an open question.