dorsal/arxiv
View SchemaKochen-Specker Vectors
| Authors | Mladen Pavicic, Jean-Pierre Merlet, Brendan McKay, Norman D. Megill |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0409014 |
| URL | https://arxiv.org/abs/quant-ph/0409014 |
| DOI | 10.1088/0305-4470/38/7/013 |
| Journal | Journal of Physics A: Mathematical and General, 38, (7), 1577-1592 (2005) |
Abstract
We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in n-dim Hilbert space, H^n, n>3 to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in R^n, on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all 4-dim KS vector systems containing up to 24 vectors were generated and described, all 3-dim vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.
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"abstract": "We give a constructive and exhaustive definition of Kochen-Specker (KS)\nvectors in a Hilbert space of any dimension as well as of all the remaining\nvectors of the space. KS vectors are elements of any set of orthonormal states,\ni.e., vectors in n-dim Hilbert space, H^n, n\u003e3 to which it is impossible to\nassign 1s and 0s in such a way that no two mutually orthogonal vectors from the\nset are both assigned 1 and that not all mutually orthogonal vectors are\nassigned 0. Our constructive definition of such KS vectors is based on\nalgorithms that generate MMP diagrams corresponding to blocks of orthogonal\nvectors in R^n, on algorithms that single out those diagrams on which algebraic\n0-1 states cannot be defined, and on algorithms that solve nonlinear equations\ndescribing the orthogonalities of the vectors by means of statistically\npolynomially complex interval analysis and self-teaching programs. The\nalgorithms are limited neither by the number of dimensions nor by the number of\nvectors. To demonstrate the power of the algorithms, all 4-dim KS vector\nsystems containing up to 24 vectors were generated and described, all 3-dim\nvector systems containing up to 30 vectors were scanned, and several general\nproperties of KS vectors were found.",
"arxiv_id": "quant-ph/0409014",
"authors": [
"Mladen Pavicic",
"Jean-Pierre Merlet",
"Brendan McKay",
"Norman D. Megill"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/38/7/013",
"journal_ref": "Journal of Physics A: Mathematical and General, 38, (7), 1577-1592\n (2005)",
"title": "Kochen-Specker Vectors",
"url": "https://arxiv.org/abs/quant-ph/0409014"
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