Nonuniqueness of phase retrieval for three fractional Fourier transforms

Abstract

We prove that, regardless of the choice of the angles $\theta_1,\theta_2,\theta_3$, three fractional Fourier transforms $F_{\theta_1}$, $F_{\theta_2}$ and $F_{\theta_3}$ do not solve the phase retrieval problem. That is, there do not exist three angles $\theta_1$, $\theta_2$, $\theta_3$ such that any signal $\psi\in L^2(R)$ could be determined up to a constant phase by knowing only the three intensities $|F_{\theta_1}\psi|^2$, $|F_{\theta_2}\psi|^2$ and $|F_{\theta_3}\psi|^2$. 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 $Q_{\theta_1}$, $Q_{\theta_2}$ and $Q_{\theta_3}$ 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.