Geometric Visualizations of Single and Entangled Qubits

The one-to-one correspondence of single qubit states to points on a Bloch sphere provides a useful visualization for understanding some fundamental concepts of quantum information processing. However, the exponentially increasing dimensionality of state representations as the number of qubits in a system increases precludes similar one-to-one mappings of two-qubit and higher systems, hindering visualizations of even the simplest quantum algorithms. Taking a cue from visualizations of special relativity, where Minkowski diagrams in reduced-dimensional subspaces of 2 or 3 dimensions provide useful pedagogical and conceptual tools for understanding basic concepts, we discuss how taking a subspace of the full two-qubit space allows for the construction of one-to-one maps that visualize this subspace in 2 and 3 dimensions. These maps have physically meaningful lengths and angles, a non-trivial topology reflecting the full state space, and allow visualizations of such concepts as entanglement and separability not possible with the single-qubit Bloch sphere. Interactive, online versions of these maps have been developed to aid both students and researchers learning the fundamentals of quantum algorithms.