Combining moment matrices, symmetric extension, and Lovász theta: $Φ_{\text{E8}}$ is entangled
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917492220231680 |
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| author | Stempin, Jȩdrzej Munné, Gerard Anglès Llorens, Santiago Huber, Felix |
| author_facet | Stempin, Jȩdrzej Munné, Gerard Anglès Llorens, Santiago Huber, Felix |
| contents | 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 $Φ_{\text{E8}}$ is entangled. Inspired by a method from quantum codes, we combine symmetric extension with moment matrices to prove that $Φ_{\text{E8}}$ 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 Lovász theta number of the Pauli anti-commutativity graph, promising scalability and flexibility in further applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Combining moment matrices, symmetric extension, and Lovász theta: $Φ_{\text{E8}}$ is entangled Stempin, Jȩdrzej Munné, Gerard Anglès Llorens, Santiago Huber, Felix Quantum Physics 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 $Φ_{\text{E8}}$ is entangled. Inspired by a method from quantum codes, we combine symmetric extension with moment matrices to prove that $Φ_{\text{E8}}$ 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 Lovász theta number of the Pauli anti-commutativity graph, promising scalability and flexibility in further applications. |
| title | Combining moment matrices, symmetric extension, and Lovász theta: $Φ_{\text{E8}}$ is entangled |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2605.13832 |