dorsal/arxiv
View SchemaThe Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$< x^{3} - x>$ and Quantum Entanglement II. The Mermin "Magic" Square/Pentagram
| Authors | Metod Saniga, Michel Planat, Milan Minarovjech |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0603206 |
| URL | https://arxiv.org/abs/quant-ph/0603206 |
| DOI | 10.1007/s11232-007-0049-5 |
| Journal | Theoretical and Mathematical Physics 151 (2007) 625-631 |
Abstract
In 1993, Mermin (Rev. Mod. Phys. 65, 803--815) gave lucid and strikingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight by making use of what has since been referred to as the Mermin(-Peres) "magic square" and the Mermin pentagram, respectively. The former is a $3 \times 3$ array of nine observables commuting pairwise in each row and column and arranged so that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by similar contradiction. An interesting one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring ${\rm GF}(2) \otimes \rm{GF}(2)$ is established. Under this mapping, the concept "mutually commuting" translates into "mutually distant" and the distinguishing character of the third column's observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are both either zero-divisors, or units. The ten operators of the Mermin pentagram answer to a specific subset of points of the line over GF(2)[$x$]/$<x^{3} - x>$. The situation here is, however, more intricate as there are two different configurations that seem to serve equally well our purpose. The first one comprises the three distinguished points of the (sub)line over GF(2), their three "Jacobson" counterparts and the four points whose both coordinates are zero-divisors; the other features the neighbourhood of the point ($1, 0$) (or, equivalently, that of ($0, 1$)). Some other ring lines that might be relevant for BKS proofs in higher dimensions are also mentioned.
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"abstract": "In 1993, Mermin (Rev. Mod. Phys. 65, 803--815) gave lucid and strikingly\nsimple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of\ndimensions four and eight by making use of what has since been referred to as\nthe Mermin(-Peres) \"magic square\" and the Mermin pentagram, respectively. The\nformer is a $3 \\times 3$ array of nine observables commuting pairwise in each\nrow and column and arranged so that their product properties contradict those\nof the assigned eigenvalues. The latter is a set of ten observables arranged in\nfive groups of four lying along five edges of the pentagram and characterized\nby similar contradiction. An interesting one-to-one correspondence between the\noperators of the Mermin-Peres square and the points of the projective line over\nthe product ring ${\\rm GF}(2) \\otimes \\rm{GF}(2)$ is established. Under this\nmapping, the concept \"mutually commuting\" translates into \"mutually distant\"\nand the distinguishing character of the third column\u0027s observables has its\ncounterpart in the distinguished properties of the coordinates of the\ncorresponding points, whose entries are both either zero-divisors, or units.\nThe ten operators of the Mermin pentagram answer to a specific subset of points\nof the line over GF(2)[$x$]/$\u003cx^{3} - x\u003e$. The situation here is, however, more\nintricate as there are two different configurations that seem to serve equally\nwell our purpose. The first one comprises the three distinguished points of the\n(sub)line over GF(2), their three \"Jacobson\" counterparts and the four points\nwhose both coordinates are zero-divisors; the other features the neighbourhood\nof the point ($1, 0$) (or, equivalently, that of ($0, 1$)). Some other ring\nlines that might be relevant for BKS proofs in higher dimensions are also\nmentioned.",
"arxiv_id": "quant-ph/0603206",
"authors": [
"Metod Saniga",
"Michel Planat",
"Milan Minarovjech"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1007/s11232-007-0049-5",
"journal_ref": "Theoretical and Mathematical Physics 151 (2007) 625-631",
"title": "The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$\u003c x^{3} - x\u003e$ and Quantum Entanglement II. The Mermin \"Magic\" Square/Pentagram",
"url": "https://arxiv.org/abs/quant-ph/0603206"
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