dorsal/arxiv
View SchemaMutually Unbiased Bases and The Complementarity Polytope
| Authors | Ingemar Bengtsson, Asa Ericsson |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0410120 |
| URL | https://arxiv.org/abs/quant-ph/0410120 |
| Journal | Open Sys. & Information Dyn. (2005) 12: 107-120 |
Abstract
A complete set of N+1 mutually unbiased bases (MUBs) forms a convex polytope in the N^2-1 dimensional space of NxN Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown whether it can be made to lie within the body of density matrices unless N=p^k, where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N^2 facets in such a way that their centers form a regular simplex if and only if there exists an affine plane of order N. Affine planes of order N are known to exist if N=p^k; perhaps they do not exist otherwise. However, the link to the existence of MUBs--if any--remains to be found.
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"abstract": "A complete set of N+1 mutually unbiased bases (MUBs) forms a convex polytope\nin the N^2-1 dimensional space of NxN Hermitian matrices of unit trace. As a\ngeometrical object such a polytope exists for all values of N, while it is\nunknown whether it can be made to lie within the body of density matrices\nunless N=p^k, where p is prime. We investigate the polytope in order to see if\nsome values of N are geometrically singled out. One such feature is found: It\nis possible to select N^2 facets in such a way that their centers form a\nregular simplex if and only if there exists an affine plane of order N. Affine\nplanes of order N are known to exist if N=p^k; perhaps they do not exist\notherwise. However, the link to the existence of MUBs--if any--remains to be\nfound.",
"arxiv_id": "quant-ph/0410120",
"authors": [
"Ingemar Bengtsson",
"Asa Ericsson"
],
"categories": [
"quant-ph"
],
"journal_ref": "Open Sys. \u0026 Information Dyn. (2005) 12: 107-120",
"title": "Mutually Unbiased Bases and The Complementarity Polytope",
"url": "https://arxiv.org/abs/quant-ph/0410120"
},
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