Zauner's conjecture predicts that in any dimension d, you can find d² complex vectors in ℂᵈ that are equiangular — every pair makes the same angle. These configurations, called SIC-POVMs, are optimal quantum measurements. The conjecture has been verified numerically in hundreds of dimensions but proved in only a handful.
The paper constructs SIC-POVMs from number theory. The method starts not with the SICs themselves but with “ghost SICs” — algebraic objects derived from special values of the Shintani-Faddeev modular cocycle over real quadratic fields. The ghost SICs are not measurement configurations but number-theoretic shadows that encode the structure a SIC would need.
The construction converts ghosts to SICs using the abelian Stark conjecture — a deep, largely unproven conjecture in algebraic number theory about the values of L-functions at s = 0. If Stark is true, the ghost SICs produce actual equiangular lines in every dimension. The paper provides a complete putative list for all d > 3 and validates it computationally, including four previously unknown SICs in dimension 100.
The structural point: a conjecture in quantum measurement theory is equivalent to a conjecture in algebraic number theory. Zauner (quantum) holds if and only if Stark (arithmetic) holds. The bridge is not metaphorical — the SIC vectors are literally constructed from the special values that Stark's conjecture predicts. The ghosts are the L-function values; the SICs are what they build. Two seemingly unrelated open problems in different branches of mathematics turn out to be the same problem viewed from different sides.