四量子比特系统的纠缠：具有多项式罗盘的几何图谱I（有限世界）
Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
摘要：
我们通过代数几何和不变量理论研究了这四个量子比特系统的几何结构，这使得我们能够将某些纠缠态解释为代数簇。更准确地说，我们描述的零锥，即，由所有不变多项式湮灭的状态的集合，以及所谓的第三割线簇，其可以被解释为多于三个量子比特的GHZ状态的推广。我们所有的几何描述都伴随着算法，这些算法使我们能够将零锥或第三割线簇中的任何给定状态识别为本文中描述的47个簇之一的点。这47个变种对应于47种非等价纠缠模式，如果我们允许量子比特的排列，则可以减少到15种不同的类别。
英文摘要：
We investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory, which allows us to interpret certain entangled states as algebraic varieties. More precisely we describe the nullcone, i.e., the set of states annihilated by all invariant polynomials, and also the so called third secant variety, which can be interpreted as the generalization of GHZ-states for more than three qubits. All our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper. These 47 varieties correspond to 47 non-equivalent entanglement patterns, which reduce to 15 different classes if we allow permutations of the qubits.
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。
