Codes and Expansions (CodEx) Seminar

Gene Kopp (University of Bristol)
SIC-POVM existence and the Stark conjectures

The existence of a configuration of equiangular lines in \(d\)-dimensional complex Hilbert space of cardinality achieving the theoretical upper bound of \(d^2\) is known only for finitely many dimensions d. Such configurations have been studied extensively in the context of quantum information theory, in which they are known as symmetric informationally complete positive operator-valued measures (SIC-POVMs or SICs), and in frame theory and design theory, where they are maximal equiangular tight frames (ETFs) and minimal 2-designs.
We give an explicit conjectural construction of SIC-POVMs in an infinite family of dimensions. Our construction uses values of derivatives of zeta functions at \(s=0\) and is closely connected to the Stark conjectures over real quadratic fields. Moreover, in the same family, we prove a conditional result stating that SIC-POVMs exist under a strong algebraic hypothesis about units in a certain number field. The talk will include a worked example in dimension \(d=5\) and an overview of some number-theoretic background necessary to understand the main results.