Algebraic combinatorics seminar - Huangjun Zhu

Super-symmetric informationally complete measurements

Abstract:

Symmetric informationally complete measurements (SICs) are highly symmetric structure in the Hilbert space. They possess many nice properties which render them an ideal candidate of fiducial measurements. The symmetry of SICs is intimately connected with the geometry of the quantum state space and also has profound implications for foundational studies. Here we study those SICs that are most symmetric according to a natural criterion and show that all of them are covariant with respect to the Heisenberg-Weyl groups, which are characterized by the discrete analog of the canonical commutation relation. Moreover, their symmetry groups are subgroups of the Clifford groups. In particular, we prove that the SIC in dimension~2, the Hesse SIC in dimension~3, and the set of Hoggar lines in dimension~8 are the only three SICs up to unitary equivalence whose symmetry groups act transitively on pairs of SIC projectors. Our work is of intrinsic interest to studying the geometry of quantum state space and foundational issues entangled with the geometry.