Nineteen vectors and a certificate of their phase-retrieval injectivity in ${\mathbb{C}}^6$

Abstract

We exhibit nineteen explicit vectors in ${\mathbb{C}}^6$, with Gaussian-integer entries, that do phase retrieval: the intensities $|⟨a_j,x⟩{|}^2$, $j=1,\dots ,19$, determine every $x\in {\mathbb{C}}^6$ up to a unimodular scalar. This gives $m\mathbb{C}(6)\le 19=4d-5$, one below the generic count $4d-4$; combined with the lower bound $m\mathbb{C}(6)\ge 18$ of Wang and Xu, the minimal measurement number in dimension six is 18 or 19. The claim is verified by an exact rational computation, archived and rerunnable: after an elementary reduction, phase retrieval is equivalent to the emptiness of the real zero set of a system of five quadratic equations in five unknowns, and the archived eliminant of that system is a squarefree integer polynomial of degree fourteen with no real roots.