dorsal/arxiv
View SchemaPauli graph and finite projective lines/geometries
| Authors | Michel R. P. Planat, Metod Saniga |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0703154 |
| URL | https://arxiv.org/abs/quant-ph/0703154 |
| DOI | 10.1117/12.721687 |
| Journal | Dans Photon Counting Applications, Quantum Optics and Quantum Cryptography - Optics and Optoelectronics, Prague : Tch\`eque, R\'epublique (2007) |
Abstract
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 P_{p^N} . 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.
{
"annotation_id": "8fcf5c6d-03e2-412e-8834-f934b5e3aa67",
"date_created": "2026-03-02T18:02:37.252000Z",
"date_modified": "2026-03-02T18:02:37.252000Z",
"file_hash": "c2cccc52f3d4883fb73f04e7653061dd8afdf158c70d431a0de594de0b26cffc",
"private": false,
"record": {
"abstract": "The commutation relations between the generalized Pauli operators of N-qudits\n(i. e., N p-level quantum systems), and the structure of their maximal sets of\ncommuting bases, follow a nice graph theoretical/geometrical pattern. One may\nidentify vertices/points with the operators so that edges/lines join commuting\npairs of them to form the so-called Pauli graph P_{p^N} . As per two-qubits (p\n= 2, N = 2) all basic properties and partitionings of this graph are embodied\nin the geometry of the symplectic generalized quadrangle of order two, W(2).\nThe structure of the two-qutrit (p = 3, N = 2) graph is more involved; here it\nturns out more convenient to deal with its dual in order to see all the\nparallels with the two-qubit case and its surmised relation with the geometry\nof generalized quadrangle Q(4, 3), the dual of W(3). Finally, the generalized\nadjacency graph for multiple (N \u003e 3) qubits/qutrits is shown to follow from\nsymplectic polar spaces of order two/three. The relevance of these mathematical\nconcepts to mutually unbiased bases and to quantum entanglement is also\nhighlighted in some detail.",
"arxiv_id": "quant-ph/0703154",
"authors": [
"Michel R. P. Planat",
"Metod Saniga"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1117/12.721687",
"journal_ref": "Dans Photon Counting Applications, Quantum Optics and Quantum\n Cryptography - Optics and Optoelectronics, Prague : Tch\\`eque, R\\\u0027epublique\n (2007)",
"title": "Pauli graph and finite projective lines/geometries",
"url": "https://arxiv.org/abs/quant-ph/0703154"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "9807af0f-5812-4dae-893f-aa708b90530f",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}