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

TL;DR

This paper reveals that Mermin's pentagram, a key structure in quantum contextuality proofs, is mathematically equivalent to an ovoid in PG(3,2), linking quantum observables to projective geometry.

Contribution

It establishes a novel geometric interpretation of Mermin's pentagram as an ovoid in PG(3,2), connecting quantum contextuality to projective geometry.

Findings

Mermin's pentagram is isomorphic to an ovoid in PG(3,2)

The pentagram's observables correspond to points on a hyperbolic quadric

The geometric structure provides new insights into quantum contextuality

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).