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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 May 11, 2026 (v2)

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

Abstract

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.

Comments: 45 pages, 1 figure, 2 tables, minor changes, reference changed

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In this paper, we present a method to construct full separability criterion for tripartite systems of qubits. The spirit of our approach is that a tripartite pure state can be regarded as a three-order tensor that provides an intuitionistic mathematical formulation for the full separability of pure states. We extend the definition to mixed states and give out the corresponding full separability criterion. As applications, we discuss the separability of several bound entangled states, which shows that our criterion is feasible.

We derive a family of necessary separability criteria for finite-dimensional systems based on inequalities for variances of observables. We show that every pure bipartite entangled state violates some of these inequalities. Furthermore, a family of bound entangled states and true multipartite entangled states can be detected. The inequalities also allow us to distinguish between different classes of true tripartite entanglement for qubits. We formulate an equivalent criterion in terms of covariance matrices. This allows us to apply criteria known from the regime of continuous variables to finite-dimensional systems.

Let H 1 , H 2 , …, H n , K ( n ⩾ 2) be complex Hilbert spaces of finite or infinite dimension. A necessary and sufficient criterion of full separability for multipartite states is established, which says that a state ρ in H 1 ⊗ H 2 ⊗⋅⋅⋅⊗ H n ⊗ K is fully separable if and only if (Λ⊗Id)ρ ⩾ 0 for all (finite-rank) LPP elementary operators Λ from into . Furthermore, a characterization of LPP elementary operators and a method of constructing LPP elementary operators by positive elementary operators are given. An example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states.

In this Letter we find the new criteria of separability of multipartite qubit density matrixes. Especially, we discuss in detail the criteria of separability for tripartite qubit density matrixes. We find the sufficient and necessary conditions of separability for tripartite qubit density matrixes, and give two corollaries. The second corollary can be taken as the criterion of existence of entanglement for tripartite qubit density matrixes. In concrete application, its steps are quite simple and easy to operate. Some examples, discussions and the generalization to more high dimensional multipartite qubit density matrixes are given.

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Abstract Based on the generalized Bloch representation, we study the separability and entanglement of arbitrary dimensional multipartite quantum states. Some sufficient and some necessary criteria are presented. For certain states, these criteria together are both sufficient and necessary. Detailed examples show that our criteria are better than some existing ones in identifying separability. Based on these criteria, the largest separable ball around the maximally mixed state for arbitrary multi-qubit systems is found, and it is proved that its radius defined via the l 1-norm of the Bloch vector is the constant 1. Furthermore, the criteria in this paper can be implemented experimentally.