Mermin's pentagram as an ovoid of PG(3, 2)

M. Saniga, P. Lévay

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Mermin's pentagram, a specific set of ten three-qubit observables arranged in quadruples of pairwise commuting ones into five edges of a pentagram and used to provide a very simple proof of the Kochen-Specker theorem, is shown to be isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective space of order two, PG(3, 2). This demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W(5, 2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5, 2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3, 2).

Original languageEnglish
Article number50006
JournalEPL
Volume97
Issue number5
DOIs
Publication statusPublished - Mar 2012

Fingerprint

theorems
operators
geometry

ASJC Scopus subject areas

  • Physics and Astronomy(all)

Cite this

Mermin's pentagram as an ovoid of PG(3, 2). / Saniga, M.; Lévay, P.

In: EPL, Vol. 97, No. 5, 50006, 03.2012.

Research output: Contribution to journalArticle

Saniga, M. ; Lévay, P. / Mermin's pentagram as an ovoid of PG(3, 2). In: EPL. 2012 ; Vol. 97, No. 5.
@article{d0b232080e41409181cba6d200f48294,
title = "Mermin's pentagram as an ovoid of PG(3, 2)",
abstract = "Mermin's pentagram, a specific set of ten three-qubit observables arranged in quadruples of pairwise commuting ones into five edges of a pentagram and used to provide a very simple proof of the Kochen-Specker theorem, is shown to be isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective space of order two, PG(3, 2). This demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W(5, 2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5, 2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3, 2).",
author = "M. Saniga and P. L{\'e}vay",
year = "2012",
month = "3",
doi = "10.1209/0295-5075/97/50006",
language = "English",
volume = "97",
journal = "Europhysics Letters",
issn = "0295-5075",
publisher = "IOP Publishing Ltd.",
number = "5",

}

TY - JOUR

T1 - Mermin's pentagram as an ovoid of PG(3, 2)

AU - Saniga, M.

AU - Lévay, P.

PY - 2012/3

Y1 - 2012/3

N2 - Mermin's pentagram, a specific set of ten three-qubit observables arranged in quadruples of pairwise commuting ones into five edges of a pentagram and used to provide a very simple proof of the Kochen-Specker theorem, is shown to be isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective space of order two, PG(3, 2). This demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W(5, 2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5, 2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3, 2).

AB - Mermin's pentagram, a specific set of ten three-qubit observables arranged in quadruples of pairwise commuting ones into five edges of a pentagram and used to provide a very simple proof of the Kochen-Specker theorem, is shown to be isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective space of order two, PG(3, 2). This demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W(5, 2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5, 2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3, 2).

UR - http://www.scopus.com/inward/record.url?scp=84857969818&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857969818&partnerID=8YFLogxK

U2 - 10.1209/0295-5075/97/50006

DO - 10.1209/0295-5075/97/50006

M3 - Article

AN - SCOPUS:84857969818

VL - 97

JO - Europhysics Letters

JF - Europhysics Letters

SN - 0295-5075

IS - 5

M1 - 50006

ER -