In quantum information, symmetric informationally complete measurements (SIC-POVMs) serve as elegant and efficient tools for numerous tasks like state tomography, QKD, and more. These objects have appeared in other contexts such as frame theory and design theory; for example, they are minimal spherical 2-designs. There are even intriguing connections to open problems in number theory. It has been an open problem for 25 years to prove that SIC-POVMs exist in infinitely many dimensions. Here we give a putatively complete construction of all SIC-POVMs having Weyl-Heisenberg symmetry in dimensions d > 3. The construction is conditionally correct assuming a conjecture from number theory known as Stark's Conjecture and a special value identity of a certain special function. The construction is unusual: it proceeds by first constructing an object we call a “ghost” and then showing that this object is (conditionally) a Galois conjugate of a SIC-POVM. This talk will be an elementary overview of these results, and no number theory is assumed. This is joint work with Marcus Appleby and Gene Kopp based on arXiv:2501.03970.
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