The Triangle Criterion detects mixed-state magic, proves multi-qubit distillation is strictly stronger than single-qubit schemes, and identifies a purity bound plus undetectable unfaithful magic states.
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The 14-qubit state Φ_E8 is entangled, certified by an explicit entanglement witness obtained as a rational infeasibility certificate for a semidefinite program that combines moment matrices with symmetric extension.
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Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection
The Triangle Criterion detects mixed-state magic, proves multi-qubit distillation is strictly stronger than single-qubit schemes, and identifies a purity bound plus undetectable unfaithful magic states.
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Combining moment matrices, symmetric extension, and Lov\'asz theta: $\Phi_{\text{E8}}$ is entangled
The 14-qubit state Φ_E8 is entangled, certified by an explicit entanglement witness obtained as a rational infeasibility certificate for a semidefinite program that combines moment matrices with symmetric extension.