Mermin's Pentagram as an Ovoid of PG(3,2)

Metod Saniga (1) , Peter Levay (2)

1 Astronomical Institute of the Slovak Academy of Sciences
2 BME - Budapest University of Technology and Economics [Budapest]

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).

Mots clés: Three-Qubit Pauli Group; Finite Geometry; Mermin's Pentagram

Domaines: Physique Quantique [quant-ph]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Metod Saniga, Peter Levay. Mermin's Pentagram as an Ovoid of PG(3,2). EPL - Europhysics Letters, 2012, pp.50006. ⟨10.1209/0295-5075/97/50006⟩. ⟨hal-00644786⟩