dorsal/arxiv
View SchemaOn the Pauli graphs of N-qudits
| Authors | Michel Planat, Metod Saniga |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0701211 |
| URL | https://arxiv.org/abs/quant-ph/0701211 |
| Journal | Quantum Information and Computation 8, 1-2 (2008) 127-146 |
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).
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"abstract": "A comprehensive graph theoretical and finite geometrical study of the\ncommutation relations between the generalized Pauli operators of N-qudits is\nperformed in which vertices/points correspond to the operators and edges/lines\njoin commuting pairs of them. As per two-qubits, all basic properties and\npartitionings of the corresponding Pauli graph are embodied in the geometry of\nthe generalized quadrangle of order two. Here, one identifies the operators\nwith the points of the quadrangle and groups of maximally commuting subsets of\nthe operators with the lines of the quadrangle. The three basic partitionings\nare (a) a pencil of lines and a cube, (b) a Mermin\u0027s array and a bipartite-part\nand (c) a maximum independent set and the Petersen graph. These factorizations\nstem naturally from the existence of three distinct geometric hyperplanes of\nthe quadrangle, namely a set of points collinear with a given point, a grid and\nan ovoid, which answer to three distinguished subsets of the Pauli graph,\nnamely a set of six operators commuting with a given one, a Mermin\u0027s square,\nand set of five mutually non-commuting operators, respectively. The generalized\nPauli graph for multiple qubits is found to follow from symplectic polar spaces\nof order two, where maximal totally isotropic subspaces stand for maximal\nsubsets of mutually commuting operators. The substructure of the (strongly\nregular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic\nto a graph of a partial geometry. Finally, the (not strongly regular) Pauli\ngraph of a two-qutrit system is introduced; here it turns out more convenient\nto deal with its dual in order to see all the parallels with the two-qubit case\nand its surmised relation with the generalized quadrangle Q(4, 3), the dual\nofW(3).",
"arxiv_id": "quant-ph/0701211",
"authors": [
"Michel Planat",
"Metod Saniga"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"journal_ref": "Quantum Information and Computation 8, 1-2 (2008) 127-146",
"title": "On the Pauli graphs of N-qudits",
"url": "https://arxiv.org/abs/quant-ph/0701211"
},
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