The Fano Monogamy Paradox: Irreducible Three-Body Entanglement and the Tripartite Hardy Impossible Event

Monogamy of entanglement states that quantum correlations cannot be freely shared: if two parties are maximally entangled, neither can be entangled with a third. The Fano plane $\mathrm{PG}(2,2)$ appears to violate this principle — its Hamiltonian ground state simultaneously correlates all seven directed triples. Hardy's paradox is a complementary phenomenon: four conditions on a multi-qubit state force the classically impossible conclusion $P(Z_+,\ldots,Z_+) = 0$, yet quantum mechanics permits $P(Z_+,\ldots,Z_+) > 0$. Both are well-understood within standard quantum mechanics; neither was ever mysterious to quantum theory itself.

This paper identifies a single necessary condition that unifies both phenomena: irreducible three-body entanglement, as measured by the Coffman–Kundu–Wootters three-tangle $\tau_3$. The Fano monogamy apparent paradox dissolves once one observes that $C(i,j) < 1$ for all pairs and $\tau_3 > 0$ for Fano-line triples — the correlations are triangular, not pairwise. For the tripartite Hardy impossible event ($N=3$), we show numerically that $\tau_3 > 0$ is a necessary condition: no W-class or product state ($\tau_3 = 0$) satisfies all four Hardy conditions simultaneously, confirmed across 54 independent free-state optimisations (JAX/Adam, A100 GPU). The open question — whether this is provable algebraically — is stated explicitly.

The novel contribution is the connection between the Dür–Vidal–Cirac entanglement classification and Hardy-type nonlocality: GHZ-class states ($\tau_3 > 0$) can exhibit the tripartite impossible event; W-class states cannot. Within the Adelic Simplicial Architecture (ASA), both phenomena share a further geometric explanation: the octonion associator $|\mathcal{A}(e_i,e_j,e_k)| = 0$ on Fano-line triples and $|\mathcal{A}| = 2$ elsewhere is the algebraic signature of the same triangular irreducibility, connecting the monogamy resolution and the Hardy condition to the associator firewall of the companion papers on the Spacelike Associator Paradox and Hardy's Paradox and the Fano Associator.