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

¹ Astronomical Institute, Slovak Academy of Sciences - SK-05960 Tatranská Lomnica, Slovak Republic, EU
² Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics H-1521 Budapest, Hungary, EU

^a msaniga@astro.sk

Received: 2 December 2011
Accepted: 26 January 2012

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).