AI 中文总结
该研究针对量子技术中多粒子纠缠维度与系统维度关系不明确的问题，提出基于纯态分解的划分秩概念，构造了纯态和混合态划分秩的表征方法，可用于识别最大关联态及量子态分类。
AI 中文摘要
在量子技术中，纠缠作为资源的实用性随系统规模增大而提升，即当考虑更多粒子或更高维量子系统时，纠缠的效用会增强。然而，维度与多粒子纠缠之间的相互作用尚未得到充分理解。目前仅对于两粒子系统，基于施密特分解存在明确且一致的纠缠维度概念。我们引入一种概念，用于表征多粒子态的纠缠维度，该概念基于将纯态分解为无真正多粒子纠缠态的叠加。我们提供了构造性方法，以表征纯态和混合态的划分秩，这使得能够识别新型最大关联态，以及在随机局域操作和经典通信（SLOCC）下对量子态进行离散分类。从数学角度来看，我们的方法可通过张量的切片秩和划分秩来表述，且我们的结果可通过将这些概念与广义内射张量范数关联起来进行表征。
英文摘要
The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.
Comments19 pages, 2 figures
