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Eisenstein 33-Vector Kochen-Specker Set (3D)

Basic Metadata

Property Value
Name Eisenstein 33-Vector KS Set
Dimension 3 (spin-1 Hilbert space)
Number of Rays 33
Ring \(\mathbb{Z}[\omega]\), where \(\omega = e^{2\pi i/3}\)
Type State-independent KS set
Author Michael Kernaghan
Year 2026
Cancellation Identity \(1 + \omega + \omega^2 = 0\)

Overview

The Eisenstein 33-vector KS set is constructed over the ring of Eisenstein integers \(\mathbb{Z}[\omega]\), where \(\omega\) is a primitive cube root of unity. The key algebraic identity \(1 + \omega + \omega^2 = 0\) provides a cancellation mechanism that enables exact orthogonalities among triples of vectors in \(\mathbb{C}^3\).

This construction is significant for several reasons:

  • Rigid: With 78 orthogonal pairs (6 more than the Peres-33 set), the Eisenstein set is infinitesimally rigid — the additional constraints kill the flex mode present in the Peres graph
  • Smallest BPQS: Achieves the exact bipartite perfect quantum strategy (BPQS) count of \(5 \times 9 = 45\), the smallest among all known 3D KS sets
  • Connected to Cabello 2025: Independently discovered via the Wigner-Heisenberg construction by Cabello (arXiv:2508.07335), who identified it as the "simplest Kochen-Specker set"

Key features:

  • 57-ray pool: The full Eisenstein alphabet with \(|c| \leq 2\) generates 57 rays
  • Norm-2 cancellation: The identity \(1 + \omega + \omega^2 = 0\) has norm 1, satisfying the norm-\(\leq 2\) threshold for KS-uncolorability
  • Three distinct 33-sets: The Eisenstein-33, Peres-33, and Conway-Kochen-33 (CK-33) are three structurally distinct KS sets, all with 33 vectors but different graphs

Structure

The 33 rays live in \(\mathbb{C}^3\), using coordinates from the Eisenstein integer alphabet \(\{0, \pm 1, \pm \omega, \pm \omega^2\}\). They are arranged into orthogonal triads whose orthogonality relies on the vanishing sum \(1 + \omega + \omega^2 = 0\).

Orthogonality Structure

Property Value
Orthogonal pairs 78
Orthogonal triads —
Degree distribution Includes degree-5 vertices (12 of them)

The 78 orthogonal pairs provide 12 more constraints than the Peres-33 set's 72 pairs. These additional constraints — arising from 12 degree-5 vertices — are what make the Eisenstein set rigid while Peres is flexible.

Rigidity

The Eisenstein-33 set is infinitesimally rigid: the Jacobian null space of the orthogonality constraints has dimension equal to the symmetry dimension (41), leaving zero deformation degrees of freedom.

Property Eisenstein-33 Peres-33
Orthogonal pairs 78 72
Null space dim 41 42
Symmetry dim 41 41
Deformation modes 0 (rigid) 1 (flex)

The rigidity is a consequence of the graph structure, not the algebraic field: the 6 additional orthogonal pairs provide enough constraints to eliminate the infinitesimal flex present in the Peres graph.

BPQS (Bipartite Perfect Quantum Strategy) — Exact

The Eisenstein-33 set defines a bipartite perfect quantum strategy with:

\[|S_A| \times |S_B| = 5 \times 9 = 45\]

This is the exact minimum BPQS for this construction, verified computationally. It is the smallest BPQS among all known 3D KS sets, consistent with the high constraint density of the Eisenstein graph.

Connection to Cabello 2025

Cabello's 2025 paper "Simplest Kochen-Specker Set" (arXiv:2508.07335, PRL) constructs a 33-vector, 14-basis KS set via the Wigner-Heisenberg (WH) group acting on \(\mathbb{C}^3\). This construction naturally produces Eisenstein-integer coordinates because the WH group for dimension 3 involves cube roots of unity.

Our algebraic islands programme discovered the same set independently through systematic alphabet enumeration over \(\mathbb{Z}[\omega]\), providing a complementary algebraic perspective on why this construction exists.

Relation to Other KS Sets

Construction Dimension Vectors Pairs Rigid? Ring
Conway-Kochen (CK-31) 3 31 71 Yes \(\mathbb{Z}\)
Eisenstein 3 33 78 Yes \(\mathbb{Z}[\omega]\)
Peres 3 33 72 No (flex) \(\mathbb{Z}[\sqrt{2}]\)
Z[\(\sqrt{-2}\)] 3 33 72 No (flex) \(\mathbb{Z}[\sqrt{-2}]\)
Conway-Kochen (CK-33) 3 33 76 Yes \(\mathbb{Z}\)

References

  • M. Kernaghan, "The Algebraic Landscape of Kochen-Specker Sets in Dimension Three" (2026)
  • A. Cabello, "Simplest Kochen-Specker Set," Phys. Rev. Lett. (2025), arXiv:2508.07335
  • A. Cabello, "Simplest Bipartite Perfect Quantum Strategies," Phys. Rev. Lett. 134, 010201 (2024), arXiv:2311.17735