Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.

中文翻译：

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多个多部分状态的局部变换

了解多方纠缠至关重要，因为它支持物理学中的广泛现象。通过经典通信 (LOCC) 辅助的局部操作对状态转换的研究允许人们定量分析纠缠，因为它在希尔伯特空间中引入了偏序。然而，已经表明，对于具有固定局部维度的系统，这个顺序通常是微不足道的，这阻止了关于任何纠缠度量的多方状态相互关联。为了获得非平凡的偏序，我们研究了 LOCC 的物理动机扩展：多状态 LOCC。在这里，我们考虑同时作用于有限数量的纠缠纯态的 LOCC 变换。我们研究多部分和二部分多状态转换。在多方情况下，我们证明了可以通过仅应用局部幺正 (LU) 来更改单个初始状态的随机 LOCC (SLOCC) 类。我们表明，通过将纠缠从一种状态转移到另一种状态，可以执行在单副本情况下不可能的状态转换；提供多粒子纠缠催化的例子；并展示改进的概率协议。在二分情况下，我们确定了许多非平凡的 LU 转换，并表明源纠缠不是可加的。这些结果表明，多态 LOCC 比单态 LOCC 具有更丰富的景观。可以执行在单个副本情况下不可能的状态转换；提供多粒子纠缠催化的例子；并展示改进的概率协议。在二分情况下，我们确定了许多非平凡的 LU 转换，并表明源纠缠不是可加的。这些结果表明，多态 LOCC 比单态 LOCC 具有更丰富的景观。可以执行在单个副本情况下不可能的状态转换；提供多粒子纠缠催化的例子；并展示改进的概率协议。在二分情况下，我们确定了许多非平凡的 LU 转换，并表明源纠缠不是可加的。这些结果表明，多态 LOCC 比单态 LOCC 具有更丰富的景观。