We prove that, regardless of the choice of the angles defining them, three fractional Fourier transforms do not solve the phase retrieval problem. That is, there do not exist three angles such that any square integrable signal could be determined up to a constant phase by knowing only the three intensities of the corresponding fractional Fourier transforms. 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 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.
Nonuniqueness of phase retrieval for three fractional Fourier transforms
CARMELI, CLAUDIO;
2014-01-01
Abstract
We prove that, regardless of the choice of the angles defining them, three fractional Fourier transforms do not solve the phase retrieval problem. That is, there do not exist three angles such that any square integrable signal could be determined up to a constant phase by knowing only the three intensities of the corresponding fractional Fourier transforms. 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 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.