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

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

2011-11-25

arxiv.org/abs/1111.5923v1

EPL / Europhysics Letters 97 (2012) 50006

Physics

Quantum Physics

5 pages, 4 figures

Scientific paper

10.1209/0295-5075/97/50006

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).

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