纠缠态集合的拓扑结构
Topology of the Set of Entangled State
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中文总结 AI 辅助
研究作用于\(\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\)上的两体纠缠密度算符集合\(\mathsf E\)的拓扑结构，通过多种方法计算其同调群等，发现除两比特情况外的连通性及各维度下相关群的消失情况，还证明其在各域上有非平凡约化同调。
中文摘要 AI 辅助
我们研究了作用于\(\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\)上的两体纠缠密度算符集合\(\mathsf E\)的拓扑结构。首先表明\(\mathsf E\)是路径连通的，除两比特情况外是单连通的。在两比特情况下，\(\mathsf E\)与最大纠缠态集合同伦等价，而最大纠缠态集合与\(\mathbb{RP}^3\)同胚。还计算了\(\mathsf E\)的闭包和内部的完整同调群。在所有更大维度中，表明\(\mathsf E\)的同调群和同伦群在\(1\leq k\leq 2(n_1 - 1)(n_2 - 1) - 2\)度时消失，\(k\geq (n_1n_2)^2 - 3\)度的所有同调群也消失。通过计算欧拉特征，利用环面作用不动点论证和亚历山大对偶性，表明对于所有\(n_1,n_2\geq 2\)，\(\mathsf E\)在每个域上都有非平凡的约化同调。
英文摘要
We investigate the topology of the set $\mathsf E$ of entangled bipartite density operators acting on $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}$. We start by showing that $\mathsf E$ is path-connected, and even simply connected except in the two-qubit case. In this exceptional case $\mathsf E$ turns out to be homotopy equivalent to the set of maximally entangled states, which itself is homeomorphic to $\mathbb{RP}^3$. Here we also compute the complete homology of the closure and interior of $\mathsf E$. In all larger dimensions, we show that the homology and homotopy groups of $\mathsf E$ vanish in degrees $1\leq k\leq 2(n_1-1)(n_2-1)-2$, and all homology groups of degree $k\geq (n_1n_2)^2-3$ also vanish. This range is controlled by the space $\mathsf W$ of entanglement witnesses, which we show is highly connected beyond two qubits and homotopy equivalent to $\mathsf E$. By computing the Euler characteristic, using a torus-action fixed point argument together with Alexander duality, we show that $\mathsf E$ nevertheless has non-trivial reduced homology over every field for all $n_1, n_2 \geq 2$.
