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

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).

5 pages, 4 figures

References in corpus (6)

• Hidden Variables and the Two Theorems of John Bell
• Black Hole Entropy and Finite Geometry
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Projective Ring Line Encompassing Two-Qubits
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• On the Pauli graphs of N-qudits

Cited by in corpus (14)

• The black-hole/qubit correspondence: an up-to-date review
• On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits
• Grassmannian Connection Between Three- and Four-Qubit Observables, Mermin's Contextuality and Black Holes
• Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon
• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
• Mermin pentagrams arising from Veldkamp lines for three qubits
• Singularity of type $D_4$ arising from four qubit systems
• Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank
• $X$-States From a Finite Geometric Perspective
• 'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon
• New and improved bounds on the contextuality degree of multi-qubit configurations
• A Finite-Geometric Classification of Three-Qubit Mermin Pentagrams
• A Class of Three-Qubit Contextual Configurations Located in Fano Pentads
• Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic Quadric of PG(7,2)