# Maximum $N$-body correlations do not in general imply genuine multipartite entanglement

1Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
2Departamento de Química Física, Universidad del País Vasco UPV/EHU, E-48080 Bilbao, Spain
3IKERBASQUE Basque Foundation for Science, E-48013 Bilbao, Spain

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### Abstract

The existence of correlations between the parts of a quantum system on the one hand, and entanglement between them on the other, are different properties. Yet, one intuitively would identify strong $N$-party correlations with $N$-party entanglement in an $N$-partite quantum state. If the local systems are qubits, this intuition is confirmed: The state with the strongest $N$-party correlations is the Greenberger-Horne-Zeilinger (GHZ) state, which does have genuine multipartite entanglement. However, for high-dimensional local systems the state with strongest $N$-party correlations may be a tensor product of Bell states, that is, partially separable. We show this by introducing several novel tools for handling the Bloch representation.

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### Cited by

[1] Andreas Ketterer, Nikolai Wyderka, and Otfried Gühne, "Entanglement characterization using quantum designs", Quantum 4, 325 (2020).

[2] Haiqing Huang, Irfan Ahmed, Ahmed Ali, Xin-wei Zha, Raymond Hon-Fu Chan, and Yanpeng Zhang, "Relations between the average bipartite entanglement and N-partite correlation functions", Laser Physics 32 7, 075201 (2022).

[3] Andreas Ketterer, Satoya Imai, Nikolai Wyderka, and Otfried Gühne, "Statistically significant tests of multiparticle quantum correlations based on randomized measurements", Physical Review A 106 1, L010402 (2022).

[4] Matthias Miller and Daniel Miller, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 378 (2021) ISBN:978-1-6654-1691-7.

[5] Cornelia Spee, "Certifying the purity of quantum states with temporal correlations", Physical Review A 102 1, 012420 (2020).

[6] Jens Siewert, "On orthogonal bases in the Hilbert-Schmidt space of matrices", Journal of Physics Communications 6 5, 055014 (2022).

[7] Satoya Imai, Nikolai Wyderka, Andreas Ketterer, and Otfried Gühne, "Bound Entanglement from Randomized Measurements", Physical Review Letters 126 15, 150501 (2021).

[8] Daniel Miller, "Small quantum networks in the qudit stabilizer formalism", arXiv:1910.09551.

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