Uniqueness results in an extension of Pauli's phase retrieval problem

Abstract

International audienceIn this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if $u$ and $v$ are such that fractional Fourier transforms of order $\alpha$ have same modulus $|F_\alpha u|=|F_\alpha v|$ for some set $\tau$ of $\alpha$'s, then $v$ is equal to $u$ up to a constant phase factor. The set $\tau$ depends on some extra assumptions either on $u$ or on both $u$ and $v$. Cases considered here are $u$, $v$ of compact support, pulse trains, Hermite functions or linear combinations of translates and dilates of Gaussians. In this last case, the set $\tau$ may even be reduced to a single point ({\it i.e.} one fractional Fourier transform may suffice for uniqueness in the problem)

Wavelet transform modulus: phase retrieval and scattering

“…When d ≥ 2, almost all functions become uniquely determined, but not all [5]. A similar analysis can be done for the fractional Fourier transform [7].…”

“…This property allows us to apply to our problem classical harmonic analysis techniques, as has also been done in [1,7].…”

Pure state ‘really’ informationally complete with rank-1 POVM

“…Let 7], the true value of m 0 (d) is given in [8]. We have m 0 (2, 3, 4, 5, 6, 7) = (4,8,10,16,18,23). We compare this with the value of 3d − 2: (4,7,10,13,16,19).…”

Stable pure state quantum tomography from five orthonormal bases

“…This poor scaling has led to the search for more efficient methods which allow for a reduction of resources in specific cases.Recent focus has been on the identification of unknown pure (or more generally low rank) states [1][2][3][4][5][6][7]. Any two pure states can be distinguished with a measurement having just ∼ 4d outcomes [1] or, when restricting to projective measurements, with only four orthonormal bases [3,7,8]. The drawback of these approaches is that the measurements they provide cannot distinguish pure states from all states, implying that one needs to know that the state is pure prior to the measurement in order not to confuse it with mixed states having the same measurement outcome distributions.…”

“…where k n is the leading coefficient of p n (see [3,8] for more details). This formula evaluated at n = d − 1 and x = y = x j also yields the normalization factor v 1…”