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arXiv:2604.00875 · Fluctuation and Noise Letters

Conditions for 3-partite and 4-partite genuine entanglement

Mark Hillery

quant-phSubmitted Apr 1, 2026• Announced Apr 2, 2026• 0 citations
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Abstract

A system of three or four particle can be entangled in a number of different ways. It may be the case that only subsets of the particles are entangled, and these subsets are not entangled with each other. It may also be the case that the state is the sum of states in which entanglement only exists within subsets. If this is not the case, the state is said to be genuinely entangled. GHZ states, for example, are genuinely entangled. Deciding whether a state is genuinely entangled is not simple, but conditions do exist to detect it. Here we would like to propose additional sufficient conditions based on one for bipartite entanglement.

Journal reference
Fluctuation and Noise Letters, 25, 2540029 (2026)
DOI
10.1142/S0219477525400292

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A system of three or four particle can be entangled in a number of different ways. It may be the case that only subsets of the particles are entangled, and these subsets are not entangled with each other. It may also be the case that the state is the sum of states in which entanglement only exists within subsets. If this is not the case, the state is said to be genuinely entangled. GHZ states, for example, are genuinely entangled. Deciding whether a state is genuinely entangled is not simple, but conditions do exist to detect it. Here we would like to propose additional sufficient conditions based on one for bipartite entanglement.
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Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.
We investigate the detection of entanglement in nnn-partite quantum states. We obtain practical separability criteria to identify genuinely entangled and nonseparable mixed quantum states. No numerical optimization or eigenvalue evaluation is needed, and our criteria can be evaluated by simple computations involving components of the density matrix. We provide examples in which our criteria perform better than all known separability criteria. Specifically, we are able to detect genuine nnn-partite entanglement which has previously not been identified. In addition, our criteria can be used in today's experiments.
Positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement. In the multipartite case, however, the direct application of a positive but not completely positive map cannot distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition. We thus generalise this bipartite concept to the multipartite setting by introducing non-positive maps that are positive on the subset of biseparable states but can map to a non-positive element if applied to a genuine multipartite entangled state. We explicitly construct examples of multipartite non-positive maps, obtained from positive maps via a lifting procedure, that in this fashion can reveal genuine multipartite entanglement in a robust way.