Seeking Symmetries of SIC-POVMs

          @misc{ scivideos_PIRSA:08100069,
            doi = {10.48660/08100069},
            url = {https://pirsa.org/08100069},
            author = {Grassl, Markus},
            keywords = {Quantum Information, Quantum Foundations},
            language = {en},
            title = {Seeking Symmetries of SIC-POVMs},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2008},
            month = {oct},
            note = {PIRSA:08100069 see, \url{https://scivideos.org/pirsa/08100069}}
          }
          

Abstract

By definition, SIC-POVMs are symmetric in the sense that the magnitude of the inner product between any pair of vectors is constant. All known constructions are based on additional symmetries, mainly with respect to the Weyl-Heisenberg group. Analyzing solutions for small dimensions, Zauner has identified an additional symmetry of order three and conjectured that these symmetries can be used to construct SIC-POVMs for all dimensions. Appleby has confirmed that all numerical solutions of Renes et al. indeed have that additional symmetry. This leads to the main questions addressed in the talk: Do all SIC-POVMs necessarily possess these symmetries, or can we construct SIC-POVMs without or with other symmetries?