Combining moment matrices, symmetric extension, and Lovász theta: $Φ_{\text{E8}}$ is entangled

• URL: http://arxiv.org/abs/2605.13832v1
• Date: Wed, 13 May 2026 17:54:51 GMT
• Title: Combining moment matrices, symmetric extension, and Lovász theta: $Φ_{\text{E8}}$ is entangled
• Abstract summary: We show, via an entanglement witness, that the $14$-qubit state $_textE8$ is entangled.<n>Inspired by a method from quantum codes, we combine symmetric extension with moment matrices to prove that $_textE8$ is entangled.
• Score: 1.3999481573773072
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $Φ_{\text{E8}}$ 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.

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