Genuine multipartite entanglement of quantum states in the multiple-copy scenario

Carlos Palazuelos1,2 and Julio I. de Vicente3

1Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, E-28040 Madrid, Spain
2Instituto de Ciencias Matemáticas, E-28049 Madrid, Spain
3Departamento de Matemáticas, Universidad Carlos III de Madrid, E-28911, Leganés (Madrid), Spain

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Genuine multipartite entanglement (GME) is considered a powerful form of entanglement since it corresponds to those states that are not biseparable, i.e. a mixture of partially separable states across different bipartitions of the parties. In this work we study this phenomenon in the multiple-copy regime, where many perfect copies of a given state can be produced and controlled. In this scenario the above definition leads to subtle intricacies as biseparable states can be GME-activatable, i.e. several copies of a biseparable state can display GME. We show that the set of GME-activatable states admits a simple characterization: a state is GME-activatable if and only if it is not partially separable across one bipartition of the parties. This leads to the second question of whether there is a general upper bound in the number of copies that needs to be considered in order to observe the activation of GME, which we answer in the negative. In particular, by providing an explicit construction, we prove that for any number of parties and any number $k\in\mathbb{N}$ there exist GME-activatable multipartite states of fixed (i.e. independent of $k$) local dimensions such that $k$ copies of them remain biseparable.

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[1] Marcel Seelbach Benkner, Jens Siewert, Otfried Gühne, and Gael Sentís, "Characterizing generalized axisymmetric quantum states in d×d systems", Physical Review A 106 2, 022415 (2022).

[2] Jonathan Schluck, Gláucia Murta, Hermann Kampermann, Dagmar Bruß, and Nikolai Wyderka, "Continuity of robustness measures in quantum resource theories", Journal of Physics A: Mathematical and Theoretical 56 25, 255303 (2023).

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