RUI: Space of Multiparticle Entanglement Types

量子系统的纠缠类型的分类和描述共享给定纠缠类型的状态集是量子信息论领域中的一个重要的公开问题。它的重要性来源于量子计算算法和量子通信协议，这些应用依赖于纠缠作为一种重要的资源。大多数已知结果利用多项式不变量(状态向量系数的多项式函数)来分析纠缠。这种方法的复杂性随着量子比特数呈指数增长，到目前为止已被证明不适用于量子比特数较少的系统。我们将探索一种使用基于局部酉变换群的李代数的几何不变量的替代方法。与多项式不变量相比，这适用于具有任意数量的量子比特的系统。PI已经取得了区分单态乘积在纠缠类型中具有唯一极值性质的一般结果，PI建议进一步发展这一机制来提炼现有的不变量，构造新的不变量，并证明一些猜想，包括关于乘积态和n-CAT态的纠缠分类陈述。

Classification of entanglement types and describing the set of states whichshare a given entanglement type for quantum systems whose subsystems arequantum bits (qubits) is an important open problem in the field of quantuminformation theory. Its importance derives from applications including quantumcomputation algorithms and quantum communications protocols whichrely on entanglement as an essential resource.Most known results utilize polynomial invariants (polynomial functions ofstate vector coefficients) to analyze entanglement. The complexity of thisapproach grows exponentially with the number of qubits and thus far hasproved impractical to apply to systems with more than a small number ofqubits.An alternative approach using geometric invariants based on the Lie algebra of the group of local unitary transformations will be explored. In contrast to polynomial invariants, this is applicable to systems with an arbitrary number of qubits. The PIs have achieved general results that distinguish products of singletstates as having a unique extremal property among entanglement types.The PIs propose to further develop this machinery to refine existing invariants,to construct new invariants, and to prove a number of conjectures includingentanglement classification statements about product states and n-cat states.