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QI: Fachverband Quanteninformation

QI 1: Quantum Foundations

QI 1.9: Vortrag

Montag, 18. März 2024, 12:00–12:15, HFT-FT 101

Attempting to Simplify the Search for SIC-POVMs — •Ghislaine Coulter-de Wit^{1, 2}, David Llamas¹, Matt Weiss¹, and Christopher Fuchs¹ — ¹University of Massachusettes Boston, Boston, USA — ²Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Deutschland

Symmetric informationally complete quantum measurements or SIC-POVMs are interesting from a number of perspectives. For example in QBism, SIC-POVMs give a way of representing the Born Rule such that the form minimizes the distinction between it and the classical law of total probability [DeBrota, Fuchs, and Stacey, Phys. Rev. Res. 2, 013074 (2020)]. However, it is unclear whether they exist in all dimensions. In the search for SIC-POVMs, we are most interested in the group covariant case, where the problem boils down to finding a single fiducial vector for generating the whole structure. For finite dimensions d, this amounts to finding a solution to d² simultaneous fourth-order polynomial equations generated by the discrete Weyl-Heisenberg group. However, it has been conjectured that it is already enough to satisfy only 3d/2 of the defining equations to find a solution [Appleby, Dang, and Fuchs, Entropy 16, 1484 (2014)]. Using techniques of gradient descent we find strong correlations in the solutions between the conjecture and full d² equations. These numerical results imply that the conjecture is true, dropping the complexity of numerically searching for SICs from a quadratic to a linear number of equations in d.

Keywords: QBism; SIC-POVM; Computational results; Weyl-Heisenberg group