Nonuniqueness of phase retrieval for three fractional Fourier transforms

We prove that, regardless of the choice of the angles θ1θ1, θ2θ2, θ3θ3, three fractional Fourier transforms Fθ1Fθ1, Fθ2Fθ2 and Fθ3Fθ3 do not solve the phase retrieval problem. That is, there do not exist three angles θ1θ1, θ2θ2, θ3θ3 such that any signal ψ∈L2(R)ψ∈L2(R) could be determined up to a constant phase by knowing only the three intensities |Fθ1ψ|2|Fθ1ψ|2, |Fθ2ψ|2|Fθ2ψ|2 and |Fθ3ψ|2|Fθ3ψ|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 Qθ1Qθ1, Qθ2Qθ2 and Qθ3Qθ3 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.