Mermin's pentagram as an ovoid of PG(3, 2)
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
Accepted: 26 January 2012
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).
PACS: 02.40.Dr – Euclidean and projective geometries / 03.65.Aa – Quantum systems with finite Hilbert space / 03.67.-a – Quantum information
© EPLA, 2012