SICs, Convex Cones, and Algebraic Sets

          @misc{ pirsa_PIRSA:08100074,
            doi = {10.48660/08100074},
            url = {https://pirsa.org/08100074},
            author = {Barnum, Howard},
            keywords = {Quantum Information, Quantum Foundations},
            language = {en},
            title = {SICs, Convex Cones, and Algebraic Sets},
            publisher = {Perimeter Institute},
            year = {2008},
            month = {oct},
            note = {PIRSA:08100074 see, \url{https://pirsa.org}}
          }
          

Abstract

The question whether SICs exist can be viewed as a question about the structure of the convex set of quantum measurements, or turned into one about quantum states, asserting that they must have a high degree of symmetry. I\'ll address Chris Fuchs\' contrast of a \'probability first\' view of the issue with a \'generalized probabilistic theories\' view of it. I\'ll review some of what\'s known about the structure of convex state and measurement spaces with symmetries of a similar flavor, including the quantum one, and speculate on connections to recent SIC triple product results. And I\'ll present some old calculations, which will look familiar to old hands but may be worth contemplating yet again, reducing the Heisenberg-symmetric-SIC existence problem to the existence of solutions to a set of simultaneous polynomials in unit-modulus complex variables.