dorsal/arxiv
View SchemaQuantum Entanglement and Projective Ring Geometry
| Authors | Michel R. P. Planat, Metod Saniga, Maurice R. Kibler |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0605239 |
| URL | https://arxiv.org/abs/quant-ph/0605239 |
| Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2 (2006) Paper 066, 14 pages |
Abstract
The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15$\times$15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of $n$ copies of the Galois field GF(2), with $n$ = 2, 3 and 4.
{
"annotation_id": "63face91-5884-4a46-baec-bfebad477e70",
"date_created": "2026-03-02T18:02:27.500000Z",
"date_modified": "2026-03-02T18:02:27.500000Z",
"file_hash": "bfd81e343a072596c0a64b2093cd0a33f4de65a45c2ed62d8b466da604854722",
"private": false,
"record": {
"abstract": "The paper explores the basic geometrical properties of the observables\ncharacterizing two-qubit systems by employing a novel projective ring geometric\napproach. After introducing the basic facts about quantum complementarity and\nmaximal quantum entanglement in such systems, we demonstrate that the\n15$\\times$15 multiplication table of the associated four-dimensional matrices\nexhibits a so-far-unnoticed geometrical structure that can be regarded as three\npencils of lines in the projective plane of order two. In one of the pencils,\nwhich we call the kernel, the observables on two lines share a base of Bell\nstates. In the complement of the kernel, the eight vertices/observables are\njoined by twelve lines which form the edges of a cube. A substantial part of\nthe paper is devoted to showing that the nature of this geometry has much to do\nwith the structure of the projective lines defined over the rings that are the\ndirect product of $n$ copies of the Galois field GF(2), with $n$ = 2, 3 and 4.",
"arxiv_id": "quant-ph/0605239",
"authors": [
"Michel R. P. Planat",
"Metod Saniga",
"Maurice R. Kibler"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"journal_ref": "Symmetry, Integrability and Geometry: Methods and Applications\n (SIGMA) 2 (2006) Paper 066, 14 pages",
"title": "Quantum Entanglement and Projective Ring Geometry",
"url": "https://arxiv.org/abs/quant-ph/0605239"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "2a18570d-fd34-4f7d-a4fb-df114c8653a7",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}