Hexagons govern three-qubit contextuality

TL;DR

This work leverages finite geometry, notably the three-qubit symplectic space $\mathcal{W}(5,2)$ and the split Cayley hexagon $\mathcal{H}$, to provide a unifying geometric account of three-qubit contextuality. It shows that classically-embedded copies of $\mathcal{H}$ encode the contextuality content of major three-qubit configurations, with a precise degree of contextuality $d=63$ for the full line context set, and that the complements of skew-embedded hexagons contain grids that underpin contextuality. A concrete layering and transformation procedure relates skew and classical embeddings, and the results extend to elliptic/hyperbolic quadrics and doilies, tying disparate contextual configurations under a single geometric mechanism. The authors verify aspects of the theory experimentally via Cabello inequalities on NISQ hardware, demonstrating contextuality beyond noncontextual bounds, and outline a clear route toward higher-qubit generalizations, including analogies to multipartite entanglement structures.

Abstract

Split Cayley hexagons of order two are distinguished finite geometries living in the three-qubit symplectic polar space in two different forms, called classical and skew. Although neither of the two yields observable-based contextual configurations of their own, {\it classically}-embedded copies are found to fully encode contextuality properties of the most prominent three-qubit contextual configurations in the following sense: for each set of unsatisfiable contexts of such a contextual configuration there exists some classically-embedded hexagon sharing with the configuration exactly this set of contexts and nothing else. We demonstrate this fascinating property first on the configuration comprising all 315 contexts of the space and then on doilies, both types of quadrics as well as on complements of skew-embedded hexagons. In connection with the last-mentioned case and elliptic quadrics we also conducted some experimental tests on a Noisy Intermediate Scale Quantum (NISQ) computer to substantiate our theoretical findings.

Figures (15)

• Figure 1: The Fano plane depicted in its standard rendering. This self-dual geometry comprises seven points (bullets) and seven lines (six straight segments and a circle), with three points per line and, dually, three lines through a point, being equivalent to the projective plane over the two-element field $\mathbb{F}_2=\{0,1\}$.
• Figure 2: The Fano plane labelled by three-qubit observables encapsulates the fact that these seven observables form a set of mutually commuting operators. In fact, it is an example of a maximal set of mutually commuting observables in the three-qubit Pauli group.
• Figure 3: Mermin pentagram: This configuration of ten three-qubit observables provides an operator-based proof of the Kochen-Specker Theorem. There is no NCHV model that can reproduce the outcomes predicted by QM for this configuration.
• Figure 4: A Mermin-Peres magic square, aka$\mathcal{Q}^{+}(3,2)$, and the doily, aka$\mathcal{W}(3,2)$, are two 'extremal' two-qubit observable-based contextual configurations. The minimal number of constraints that cannot be satisfied by an NCHV model, i. e. the degree of contextuality, is one for the grid and three for the doily. For these 'peculiar' configurations, with a single exception, the unsatisfiable constraints can be identified solely with the corresponding negative contexts (represented here by the doubled lines).
• Figure 5: A copy of the Heawood graph accommodating, as sets of observables, a pair of three-qubit Fano planes, one represented by seven black bullets and the other by seven big circles; also shown are the remaining 21 points (gray) on the lines represented by the 21 edges of the graph.