On the Pauli graphs of N-qudits

Abstract

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; 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 generalized quadrangle Q(4, 3), the dual ofW(3).

17 pages. Expanded section on two-qutrits, Quantum Information and Computation (2007) accepté

References in corpus (4)

• Mutually Unbiased Bases and Trinary Operator Sets for N Qutrits
• Projective Ring Line Encompassing Two-Qubits
• Quantum Kaleidoscopes and Bell's theorem
• Tomography of one and two qubit states and factorisation of the Wigner distribution in prime power dimensions

Cited by in corpus (8)

• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• Mermin's Pentagram as an Ovoid of PG(3,2)
• The Veldkamp Space of the Smallest Slim Dense Near Hexagon
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• On the Veldkamp Space of GQ(4, 2)
• A Classification of the Veldkamp Lines of the Near Hexagon L_3 times GQ(2, 2)
• Moebius Pairs of Simplices and Commuting Pauli Operators