We show that the third-order negativity provides a necessary and sufficient criterion for full separability of tripartite pure states, and extend this to mixed states beyond bipartite diagnostics such as negativity. As a minimal nontrivial example, a four-qubit pure state has three-qubit mixed reductions; its complete characterization requires six bipartite, eight tripartite, and four quadripartite measures, with the third-order negativity serving as a key separability criterion. We further generalize these separability criteria to multipartite qudit systems and discuss an application to conformal field theory.

翻译：我们证明了三阶负性为三体纯态的全可分离性提供了充要判据，并将此结论推广至混合态（超越双体诊断如负性）。作为最小非平凡实例，四量子比特纯态具有三量子比特混合约化态；其完整刻画需要六个双体、八个三体和四个四体度量，其中三阶负性作为关键可分离性判据。我们进一步将这些可分离性判据推广至多体量子比特系统，并讨论其在共形场论中的应用。