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

TL;DR

This paper provides a finite-geometrical reinterpretation of the four-qubit magic rectangle used in quantum contextuality proofs, revealing its structure within symplectic polar spaces and projective geometries.

Contribution

It introduces a novel finite-geometrical framework for understanding the four-qubit magic rectangle, connecting it to elliptic quadrics and affine planes in projective spaces.

Findings

The magic rectangle corresponds to elliptic quadrics and affine planes in PG(3,2).

The four quadrics intersect pairwise in lines meeting at a point.

Projecting from a key observable yields a complementary magic rectangle.

Abstract

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.