SIC-POVMs and algebraic number theory
John Yard, Institute for Quantum Computing
SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) are certain extremal rank-1 projective measurements corresponding to maximal sets of complex equiangular lines as well as to minimal complex projective 2-designs. They are conjectured to exist in every finite-dimensional complex Hilbert space as orbits of generalized Pauli groups. To date, they are however only mathematically proven to exist for finitely many dimensions. The proofs in these cases are mainly computational and require computer-assisted calculations in number fields of degree growing roughly quadratically with the dimension. In this talk, I will illustrate how the structure of these number fields can be explained using class field theory. The resulting uniformity of these fields constitutes overwhelming evidence in favor of their existence for every dimension and uncovers a surprising connection to Hilbert's still-unsolved 12th problem of giving explicit generators for class fields of real quadratic fields. This talk is partially based on joint work with Appleby, Flammia and McConnell.