Zauner's Conjecture (SIC-POVMs)Open Problem
In a 1999 Vienna doctoral thesis, Gerhard Zauner conjectured that in every finite dimension d one can find d² unit vectors in complex d-space that are mutually as spread out as possible — any two sharing the same squared overlap 1/(d+1). Such a configuration, a symmetric informationally complete positive operator-valued measure (SIC-POVM), is the optimal minimal measurement for reconstructing an unknown quantum state, which is why the idea was rediscovered and named by Renes, Blume-Kohout, Scott, and Caves in 2004 and became central to quantum tomography, quantum cryptography, and the QBist reading of quantum mechanics. Geometrically these are maximal sets of complex equiangular lines; physically they are the most efficient quantum measurements; and, remarkably, they appear to be governed by deep number theory — recent work by Appleby, Flammia, Kopp, and others ties exact SICs to Stark units and Hilbert's twelfth problem on explicit class field theory. Exact solutions have been hand-built in scores of dimensions and numerical ones found in every dimension checked, yet a general existence proof remains out of reach. Formalizing Zauner's conjecture gives this problem — straddling quantum information, geometry, and algebraic number theory — a precise shared target.