Exceeding the biseparable Bell bound with the trigemini Hamiltonian

Determine whether alternative initial states or parameter choices for the three-mode trigemini Hamiltonian can produce a Mermin–Klyshko parameter exceeding the quantum biseparable bound of 2√2, thereby certifying genuine multipartite entanglement.

Background

The paper uses displaced-parity measurements to evaluate the tripartite Mermin–Klyshko inequality for states generated by the three-mode trigemini Hamiltonian. In the parameter regimes numerically explored, the largest violation is approximately 2.21, which exceeds the local-realism bound of 2 but remains below the biseparable bound 2√2 ≈ 2.82. Consequently, the reported test certifies nonlocality and entanglement but not genuine multipartite entanglement.

The authors explicitly state that the explored regime does not establish a fundamental limitation of the Hamiltonian and leave unresolved whether other initial states or parameter choices can yield a violation above the biseparable threshold. Such a violation would provide a direct certification of genuine multipartite entanglement using the displaced-parity protocol.

References

We stress that this is a statement about the parameter regime explored here, and not a fundamental limitation of the trigemini Hamiltonian: we cannot exclude that other initial states or parameter choices may push $B_3$ beyond the biseparable bound, a possibility that we leave for future investigation.

— Quantum computational resources and validation protocols for a three-mode non-Gaussian trilinear Hamiltonian  (2609.10043 - Laurora et al., 9 Sep 2026) in Section 3.1, subsection “Displaced-parity test for the trigemini Hamiltonian”; reiterated in Section 7, Conclusions