Let R be an invariant polynomial ring of a reductive group acting on a vector space, and let d be the minimum integer such that R is generated by those polynomials in R of degree no more than d. To upper bound such d is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary groups - the direct product of unitary groups acting on the tensor product of Hilbert spaces, and local general linear groups - the direct product of general linear groups acting on the tensor product of Hilbert spaces. For these two group actions, we prove explicit upper bounds on the degrees needed to generate the corresponding invariant polynomial rings. On the other hand, systematic methods are provided to construct all homogeneous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if r-copies of them are local unitary equivalent for some r.

中文翻译：

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多部分纠缠的局部等价


令 R 为作用于向量空间的约简群的不变多项式环，令 d 为使得 R 由 R 中次数不大于 d 的多项式生成的最小整数。自 19 世纪对不变理论的最初研究以来，这种 d 的上限一直是一个长期悬而未决的问题。受其在描述多部分纠缠方面的重要作用的启发，我们研究了局部酉群的不变多项式环（酉群作用于希尔伯特空间张量积的直积）和局部广义线性群（一般线性群作用于希尔伯特空间张量积的直积）。关于希尔伯特空间的张量积。对于这两个群动作，我们证明了生成相应的不变多项式环所需的次数的明确上限。另一方面，提供了系统的方法来构造所有齐次多项式，这些多项式在这两组下对于任何固定次数都是不变的。因此，我们的结果可以被视为不变多项式环的完整表征。作为一个有趣的应用，我们证明了多部分纠缠是可加的，因为两个多部分状态是局部酉等价的当且仅当它们的 r 副本对于某个 r 是局部酉等价的。