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arXiv · 2605.13832

Combining moment matrices, symmetric extension, and Lov\'asz theta: $\Phi_{\text{E8}}$ is entangled

Jȩdrzej Stempin·Gerard Anglès Munné·Santiago Llorens·Felix Huber
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Abstract

We solve an open problem in entanglement theory posed by Yu et al., {\it Nature Communications 12, 1012 (2021)}. The problem is to show, via an entanglement witness, that the $14$-qubit state $\Phi_{\text{E8}}$ is entangled. Inspired by a method from quantum codes, we combine symmetric extension with moment matrices to prove that $\Phi_{\text{E8}}$ is entangled. The proof has the form of a rational infeasibility certificate for a semidefinite program, yielding an explicit entanglement witness. Our approach unifies and extends several earlier methods that involve the Lov\'asz theta number of the Pauli anti-commutativity graph, promising scalability and flexibility in further applications.

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Jȩdrzej Stempin, Gerard Anglès Munné, Santiago Llorens, Felix Huber. 2026-05-13. Combining moment matrices, symmetric extension, and Lov\'asz theta: $\Phi_{\text{E8}}$ is entangled. https://arxiv.org/abs/2605.13832

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