Generalizing the Bloch Sphere:  Geometric Visualizations of Qudit States for Any Dimension

At both the pedagogical and research level, the Bloch Sphere has proved a useful tool as an intuitive map of single qubit states. However, limited use has been made of analogous Euclidean maps of general qudit state spaces for any d-dimensional Hilbert space, in part because the high dimensionality of the mapping spaces is viewed as prohibitive to our intuition. In this talk we demonstrate how careful selection of cross-sections of generalized qudit mapping spaces can make these high-dimensional spaces more intuitive. In particular, we show how the entries in column vector and density matrix representations of states can be understood as geometrical coordinates locating the points corresponding to states in the Euclidean mapping space. Our work leads to a revised metric for the basis-invariant "distance" between states that can be directly related to representation entries and measurement probabilities.