A set of quantum states is said to be absolutely entangled when at least one state in the set remains entangled for any definition of subsystems, i.e., for any choice of the global reference frame. In this work we investigate the properties of absolutely entangled sets (AESs) of pure quantum states. For the case of a two-qubit system, we present a sufficient condition to detect an AES, and use it to construct families of $N$ states such that $N-3$ (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition $d=d_1{d}_2$, we prove that sets of $N>⌊(d_1+1)(d_2+1)/2⌋$ states are AESs with Haar measure 1. Then, we define AESs for multipartitions. We derive a general lower bound on the number of states in an AES for a given multipartition, and also construct explicit examples. In particular, we exhibit an AES with respect to any possible multipartitioning of the total system.

中文翻译：

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双分区和多分区的绝对纠缠纯状态集

当一组量子态中的至少一个状态对于子系统的任何定义（即，对于全局参考系的任何选择）保持纠缠状态时，就称该组量子态是绝对纠缠的。在这项工作中，我们研究了纯量子态的绝对纠缠集 (AES) 的性质。对于双量子位系统的情况，我们提出了检测 AES 的充分条件，并用它来构建$N$ 陈述这样的 $N-3$（最大可能数量）对子系统的任何定义保持纠缠。对于一般的二分法$d=d_1{d}_2$，我们证明集合 $N>⌊(d_1+1)(d_2+1)/2⌋$状态是具有 Haar 测度 1 的 AES。然后，我们为多分区定义 AES。对于给定的多分区，我们推导出 AES 中状态数的一般下限，并构建显式示例。特别是，我们针对整个系统的任何可能的多分区展示了 AES。