A Linear Dependency Structure Arising from Weyl-Heisenberg Symmetry

The Weyl-Heisenberg (WH) group was used by Hermann Weyl to construct finite-dimensional quantum mechanics in the earliest days of the theory and, through its ubiquitous use in quantum information theory, is even more important today. While investigating properties of symmetric informationally-complete (SIC) measurements, we found a linear dependency structure in a class of Weyl-Heisenberg covariant sets when certain conditions on the dimensionality of the Hilbert space are met. This result reveals more structure in WH symmetry than previously noted and helps us gain a better understanding of quantum state space. For example in the Quantum Bayesian framework of Fuchs and collaborators, the number of zeros of a quantum state in a SIC representation is directly related to this linear dependency.