Nonlocality without entanglement and its subsequent generalizations offer deep information-theoretic insights and subsequently find several useful applications. The concept of a genuinely nonlocal set of product states emerges as a natural multipartite generalization of this phenomenon. The existence of such sets eventually raises the problem concerning their entanglement-assisted discrimination. Here, we construct examples of genuinely nonlocal product states for an arbitrary number of parties. The strength of genuine nonlocality of these sets can be considered minimal as their perfect discrimination is possible with entangled resources residing in Hilbert spaces having the smallest possible dimensions. Our constructions lead to fully separable measurements that are impossible to implement even if all but one party come together. Furthermore, they also provide the opportunity to compare different multipartite states that otherwise are incomparable under single copy local manipulation.

中文翻译：

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具有最小真正非定域性的多方正交产品状态

没有纠缠的非定域性及其随后的概括提供了深刻的信息理论见解，并随后找到了几个有用的应用。真正的非局部产品状态集的概念是作为这种现象的自然多方概括而出现的。此类集合的存在最终会引发有关其纠缠辅助区分的问题。在这里，我们为任意数量的参与方构建了真正非本地产品状态的示例。这些集合的真正非定域性的强度可以被认为是最小的，因为它们的完美区分是可能的，因为纠缠资源位于具有最小可能维度的希尔伯特空间中。我们的结构导致完全可分离的测量，即使只有一方聚集在一起，也无法实施。此外，