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arXiv:2605.02097 · Physical review. D/Physical review. D.

Separability from Multipartite Measures

Chen-Te Ma, Ma-Ke Yuan

quant-phcond-mat.stat-mechcs.IThep-thmath-phSubmitted May 3, 2026• Announced May 5, 2026• 0 citations• Revised Oct 16, 2026 (v3)

v1 · May 3, 2026 v2 · May 11, 2026 v3 · Oct 16, 2026

Abstract

We show that multi-invariants provide necessary and sufficient criteria for full separability in finite-dimensional multipartite systems. For tripartite three-qubit states, the third-order negativity gives an exact criterion for full separability for both pure and mixed states. We then extend the framework to four-qubit pure states, whose full characterization requires eighteen independent real parameters. We identify eighteen independent entanglement measures, consisting of six bipartite, eight tripartite, and four quadripartite measures, with the third-order negativity serving as a key separability criterion. Finally, we generalize the construction to multipartite qudit systems and discuss an application to conformal field theory.

Comments: 52 pages, 4 figures, 2 tables, minor changes, reference changed

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A new criterion necessary and sufficient for the separability of pure bipartite systems for arbitrary finite dimensions is demonstrated; and the corresponding finer quantitative measures or characterizations of entanglement (beyond mere separability or non-separability determination) are discussed. Based on this criterion, we proved that the well-known Peres-Horodecki positivity-of-partial-transform criterion is also necessary and sufficient for separability in the case of pure bipartite systems. The maximum value of entanglement, and the corresponding maximally-entangled states are also worked out in detail.

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We extend the classification of mixed states of quantum systems composed of arbitrary numbers of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives

1+18+1

partial separability classes in the tripartite case contrary to a former

1+8+1

. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.

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We propose a method of constructing the separability criteria for multipartite quantum states on the basis of entanglement witnesses. The entanglement witnesses are obtained by finding the maximal expectation values of Hermitian operators and then optimizing over all possible Hermitian operators. We derive a set of tripartite separability criteria for the four-qubit Greenberger–Horne–Zeilinger (GHZ) diagonal states. The derived criterion set contains four criteria that are necessary and sufficient for the tripartite separability of the highly symmetric four-qubit GHZ diagonal states; the criteria completely account for the numerically obtained boundaries of the tripartite separable state set. One of the criteria is just the tripartite separability criterion of the four-qubit generalized Werner states.