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arXiv · 2512.16777

Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection

Zhenhuan Liu·Tobias Haug·Qi Ye·Zi-Wen Liu·Ingo Roth
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Abstract

We introduce a mixed-state magic criterion, the Triangle Criterion, which plays a role for magic analogous to the Positive Partial Transposition (PPT) Criterion for entanglement: it combines strong detection capability, a clear geometric interpretation, and an operational link to magic distillation. Using this criterion, we uncover several new features of multi-qubit magic distillation and detection. We prove that genuinely multi-qubit magic distillation protocols are strictly more powerful than all single-qubit schemes by showing that the Triangle Criterion is not stable under tensor products. Moreover, we show that, with overwhelming probability, multi-qubit magic states with relatively low rank cannot be distilled by any single-qubit distillation protocol. We derive an upper bound on the minimal purity of magic states, which is conjectured to be tight with both numerical and constructive evidences. Using this minimal-purity result, we predict the existence of unfaithful magic states, namely states that cannot be detected by any fidelity-based magic witness, and reveal fundamental limitations of mixed-state magic detection in any single-copy scheme.

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Zhenhuan Liu, Tobias Haug, Qi Ye, Zi-Wen Liu, Ingo Roth. 2026-04-20. Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection. https://arxiv.org/abs/2512.16777

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