### 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 language | English |
---|---|

Article number | 50006 |

Journal | EPL |

Volume | 97 |

Issue number | 5 |

DOIs | |

Publication status | Published - Mar 2012 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*EPL*,

*97*(5), [50006]. https://doi.org/10.1209/0295-5075/97/50006

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

Research output: Contribution to journal › Article

*EPL*, vol. 97, no. 5, 50006. https://doi.org/10.1209/0295-5075/97/50006

}

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 -