Geometrical Mapping of Algebraic Representations of Qudits

While quantum computation architectures typically rely on qubits, more generalized frameworks involve quantum states with d possible measurement outcomes, known as a qudits. This talk focuses on the geometrical visualization of qudit states, and more specifically builds a bridge between their algebraic representations and geometrical representations of the states as points in high-dimensional Euclidean spaces. Starting from the case of a d=3 qudit, a "qutrit," we will explore how components of the density matrix representations of quantum states are directly related to geometrical lengths in the Euclidean map. These relations can be extended to the d=4 case, which also reveals some subtleties in the geometrical visualizations and mapping of lengths to matrix elements. Finally, we will present tools, ideas, and techniques behind general geometrical relations that hold for qudits of any dimension d.