量子纠缠、割线簇和球性之间的联系
A link between Quantum Entanglement, Secant varieties and Sphericity
摘要：
本文阐述了表示论、代数几何和量子信息理论中三个概念之间的关系。射影空间上约化群的第一球作用。齐次射影簇的第二割线簇，以及与之相关的秩和边秩的概念.第三，量子纠缠。我们的主要结果涉及到由其约化单粒子密度矩阵重构态的问题与其分解中可分和项的最小数目之间的关系。更确切地说，我们表明，球形意味着一个给定的秩的状态不能近似的状态的一个较低的秩。我们把这种近似是可能的状态称为例外状态。对于三个，重要的是从量子纠缠的角度来看的情况下，可区分的，费米子和玻色子粒子，我们还表明，非球形意味着存在例外的状态。值得注意的是，例外状态属于非二体纠缠类。特别是，我们表明，$W$型国家和他们适当的修改是例外的状态，源于上述三种情况下的第二割线品种。指出例外态的存在是用单粒子约化密度矩阵确定态的局域幺正等价性的一个物理障碍。最后，对于一些已知轨道结构的可区分粒子系统，我们列出了所有例外的状态，并讨论了它们在纠缠理论中可能的重要性。
英文摘要：
In this paper, we shed light on relations between three concepts studied in representations theory, algebraic geometry and quantum information theory. First - spherical actions of reductive groups on projective spaces. Second - secant varieties of homogeneous projective varieties, and the related notions of rank and border rank. Third - quantum entanglement. Our main result concerns the relation between the problem of the state reconstruction from its reduced one-particle density matrices and the minimal number of separable summands in its decomposition. More precisely, we show that sphericity implies that states of a given rank cannot be approximated by states of a lower rank. We call states for which such approximation is possible exceptional states. For three, important from quantum entanglement perspective cases of distinguishable, fermionic and bosonic particles, we also show that non-sphericity implies the existence of exceptional states. Remarkably, the exceptional states belong to non-bipartite entanglement classes. In particular, we show that the $W$-type states and their appropriate modifications are exceptional states stemming from the second secant variety for three cases above. We point out that the existence of the exceptional states is a physical obstruction for deciding the local unitary equivalence of states by means of the one-particle reduced density matrices. Finally, for a number of systems of distinguishable particles with known orbit structure we list all exceptional states and discuss their possible importance in entanglement theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
