The GHZ-Fano Game is a Complete Self-Test: Linear Robustness from PSL(2,7) Symmetry

This paper proves that the GHZ-Fano game — a nonlocal game on seven constraints defined by the Fano plane PG(2,2) acting as the constraint graph of the GHZ stabiliser group — is a complete self-test: any quantum strategy achieving the maximum winning score must be locally unitarily equivalent to the GHZ state (|000⟩ + |111⟩)/√2 with Pauli measurements {σ_z, σ_x} for each party.

The proof proceeds in four exact steps:

1. A sum-of-squares (SOS) certificate {K_i = √(2/7) P⁻_i} whose common kernel is span{|GHZ⟩} — the state self-test.
2. Fano incidence forces anti-commutation {M^Z_P, M^X_P} = 0 on each party's local observables.
3. Jordan's lemma decomposes each party's Hilbert space into 2-dimensional Pauli blocks.
4. Irreducibility of the M_2(ℂ) representation identifies each block with (σ_z, σ_x).

All four steps hold for arbitrarily large (possibly infinite-dimensional) Hilbert spaces.

The state self-test achieves linear robustness 1 − F ≤ 0.875ε rather than the generic √ε scaling. This follows from the uniform spectrum of the Fano constraint operator on the complement of |GHZ⟩ — itself a consequence of the transitive action of PSL(2,7) on the seven Fano lines.

As a corollary, the game provides a device-independent protocol for certifying the Steane [[7,1,3]] stabiliser code without trusting the measurement apparatus.

Keywords

Self-Test, Nonlocal Games, GHZ State, Fano Plane, PG(2,2), PSL(2,7), Sum-of-Squares Certificate, Linear Robustness, Device-Independent Certification, Steane Code, [[7,1,3]] Code, GHZ Stabiliser, Anti-Commutation, Jordan Lemma, Quantum Entanglement, Bell Inequality, Quantum Verification, Origami ISA, W(5,2), Symplectic Polar Space, Quantum Error Correction