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 $|\langle a_j,x\rangle|^2$, $j=1,\dots,19$, 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.