Walgate and Scott have determined the maximum number of generic pure quantum states that can be unambiguously discriminated by an LOCC measurement [Journal of Physics A: Mathematical and Theoretical, 41:375305, 08 2008]. In this work, we determine this number in a more general setting in which the local parties have access to pre-shared entanglement in the form of a resource state. We find that, for an arbitrary pure resource state, this number is equal to the Krull dimension of (the closure of) the set of pure states obtainable from the resource state by SLOCC. Surprisingly, a generic resource state maximizes this number.
Local state discrimination is closely related to the topic of entangled subspaces, which we study in its own right. We introduce $r$-entangled subspaces, which naturally generalize previously studied spaces to higher multipartite entanglement. We use algebraic-geometric methods to determine the maximum dimension of an $r$-entangled subspace, and present novel explicit constructions of such spaces. We obtain similar results for symmetric and antisymmetric $r$-entangled subspaces, which correspond to entangled subspaces of bosonic and fermionic systems, respectively.

中文翻译：

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具有预共享纠缠的纠缠子空间和通用本地状态判别

Walgate 和 Scott 已经确定了可以通过 LOCC 测量明确区分的通用纯量子态的最大数量 [Journal of Physics A: Mathematical and Theoretical, 41:375305, 08 2008]。在这项工作中，我们在更一般的环境中确定这个数字，其中本地各方可以访问资源状态形式的预共享纠缠。我们发现，对于任意的纯资源状态，这个数等于 SLOCC 从资源状态中获得的纯状态集（闭包）的 Krull 维数。令人惊讶的是，通用资源状态使这个数字最大化。
地方状态歧视与纠缠子空间的主题密切相关，我们自己研究了这个主题。我们引入了$r$-纠缠子空间，它自然地将先前研究的空间推广到更高的多方纠缠。我们使用代数几何方法来确定$r$-纠缠子空间的最大维数，并提出这种空间的新颖显式构造。我们对对称和反对称$r$-纠缠子空间获得了类似的结果，它们分别对应于玻色子和费米子系统的纠缠子空间。