Pauli graph and finite projective lines/geometries

The commutation relations between the generalized Pauli operators of N -qudits (i. e., N p-level quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical pattern. One may identify vertices/points with the operators so that edges/lines join commuting pairs of them to form the so-called Pauli graph PpN . As per two-qubits (p = 2, N = 2) all basic properties and partitionings of this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W (2). The structure of the two-qutrit (p = 3, N = 2) graph is more involved; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the geometry of generalized quadrangle Q(4, 3), the dual of W (3). Finally, the generalized adjacency graph for multiple (N > 3) qubits/qutrits is shown to follow from symplectic polar spaces of order two/three. The relevance of these mathematical concepts to mutually unbiased bases and to quantum entanglement is also highlighted in some detail.