dorsal/arxiv
View SchemaMubs and Hadamards of Order Six
| Authors | Ingemar Bengtsson, Wojciech Bruzda, Asa Ericsson, Jan-Åke Larsson, Wojciech Tadej, Karol Zyczkowski |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0610161 |
| URL | https://arxiv.org/abs/quant-ph/0610161 |
| DOI | 10.1063/1.2716990 |
| Journal | J. Math. Phys. 48, 052106 (2007) |
Abstract
We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organise our results. Finally we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.
{
"annotation_id": "c667c781-f93d-455f-a741-0b27ce9a1f92",
"date_created": "2026-03-02T18:02:31.043000Z",
"date_modified": "2026-03-02T18:02:31.043000Z",
"file_hash": "207b9453ba3d3a4597d13180fbd60585d591558318b0e600bf5e1b0a18491bf9",
"private": false,
"record": {
"abstract": "We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We\nfind only triplets of MUBs, and thus do not come close to the theoretical upper\nbound 7. However, we point out that the natural habitat for sets of MUBs is the\nset of all complex Hadamard matrices of the given order, and we introduce a\nnatural notion of distance between bases in Hilbert space. This allows us to\ndraw a detailed map of where in the landscape the MUB triplets are situated. We\nuse available tools, such as the theory of the discrete Fourier transform, to\norganise our results. Finally we present some evidence for the conjecture that\nthere exists a four dimensional family of complex Hadamard matrices of order 6.\nIf this conjecture is true the landscape in which one may search for MUBs is\nmuch larger than previously thought.",
"arxiv_id": "quant-ph/0610161",
"authors": [
"Ingemar Bengtsson",
"Wojciech Bruzda",
"Asa Ericsson",
"Jan-\u00c5ke Larsson",
"Wojciech Tadej",
"Karol Zyczkowski"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.2716990",
"journal_ref": "J. Math. Phys. 48, 052106 (2007)",
"title": "Mubs and Hadamards of Order Six",
"url": "https://arxiv.org/abs/quant-ph/0610161"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "bf679543-b1ef-43d4-be27-b59f33f4e2b9",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}