Finite Geometry Behind the Harvey-Chryssanthacopoulos Four-Qubit Magic Rectangle

Metod Saniga (1) , Michel Planat (2)

1 Astronomical Institute of the Slovak Academy of Sciences
2 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)

A ''magic rectangle" of eleven observables of four qubits, employed by Harvey and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a 16-dimensional Hilbert space, is given a neat finite-geometrical reinterpretation in terms of the structure of the symplectic polar space $W(7, 2)$ of the real four-qubit Pauli group. Each of the four sets of observables of cardinality five represents an elliptic quadric in the three-dimensional projective space of order two (PG$(3, 2)$) it spans, whereas the remaining set of cardinality four corresponds to an affine plane of order two. The four ambient PG$(3, 2)$s of the quadrics intersect pairwise in a line, the resulting six lines meeting in a point. Projecting the whole configuration from this distinguished point (observable) one gets another, complementary ''magic rectangle" of the same qualitative structure.

Keywords: Magic Rectangle" of Observables; Bell-Kochen-Specker Theorem; ''Magic Rectangle" of Observables; Four-Qubit Pauli Group; Finite Geometry

Domains: Quantum Physics [quant-ph]; Mathematical Physics [math-ph]; Mathematical Physics [math-ph]; Combinatorics [math.CO]

Metod Saniga, Michel Planat. Finite Geometry Behind the Harvey-Chryssanthacopoulos Four-Qubit Magic Rectangle. Quantum Information & Computation, 2012, 11, pp.1011-1016. ⟨hal-00692040⟩