Z-boson pair entanglement against all separable states: a decay-angle witness for H → ZZ*

ATLAS has reported evidence of Z-boson pair entanglement in H → ZZ* → 4ℓ at 4.7σ (4.9σ expected) with 304 fb⁻¹. The separable hypothesis it tests is a single state, the pure product state |00⟩, which ATLAS describes as the only separable state consistent with its assumptions. That is true only for pure states. By ATLAS's own criterion, every state whose two measured coefficients vanish is separable, and that set contains the CP-invariant mixtures of the helicity states |+−⟩, |00⟩ and |−+⟩, not just |00⟩. Restricting the null to |00⟩ therefore presupposes that the spin state is pure, which an entanglement test should not assume. We give a witness built from one event score, Q = (n̂·m̂)², where n̂ and m̂ are the lepton directions in the two Z rest frames. Its mean c lies between 0.32 and 0.36 for every separable two-qutrit state, does not depend on the leptons' analysing power, equals a fixed linear combination of three standard angular coefficients, and gives a tight lower bound on the negativity. In a generator-level model calibrated to the ATLAS result, an ATLAS-style test keeps less than a quarter of its significance when the null is the dephased Standard Model state, which has the Standard Model Z polarizations and no helicity coherence: 1.15σ expected. In this model no test on the same events has an expected significance above 2.4–3.1σ against that state, and the published coefficients give 1.4σ observed evidence of coherence. The witness, valid against every separable state, has an expected significance of 1.8σ (1.6–2.0σ) at 304 fb⁻¹ and 5.6σ (4.9–6.4σ) at 3000 fb⁻¹, from statistics alone. This record includes the paper, its LaTeX source and figures, and the Python scripts, with their outputs, that reproduce every number in it.