Multi-boundary entanglement in Chern-Simons theory and link invariants
Multi-boundary entanglement in Chern-Simons theory and link invariants
复制标题
DOI：
10.1007/jhep04(2017)061
复制
发表时间：
2016-11
影响因子：
5.4
通讯作者：
V. Balasubramanian;J. Fliss;R. Leigh;Onkar Parrikar
中科院分区：
文献类型：
--
作者：
V. Balasubramanian;J. Fliss;R. Leigh;Onkar Parrikar
我们考虑 3 流形 M n 上 k 级规范组 G 的 Chern-Simons 理论，其边界由 n 个拓扑连接的环面组成。 M n 上的欧几里得路径积分定义了边界上的量子态，位于环面希尔伯特空间的 n 倍张量积中。我们重点关注 M n 是三球体 S 3 内某个 n 分量链路的链路补集的情况。结果状态的纠缠熵定义了与帧无关的链路不变量，这些不变量对所选链路的拓扑敏感。对于 k 级的阿贝尔理论 (G= U (1) k)，我们给出了与将通用 n 分量链接任意 (m| n− m) 划分为子链接相关的纠缠熵的通用公式。该公式涉及某些丢番图方程的解数，其系数与两个子链接之间的高斯链接数 (mod k) 相关。该公式连接了量子信息论、结论和数论中的简单概念，并表明当且仅当子链接具有零高斯链接（mod k）时，子链接之间的纠缠熵才会消失。对于 G= SU (2) k，我们研究各种二元和三元链接。我们证明了 2 分量 Hopf 链路是最大纠缠的，因此类似于贝尔对，并且具有零高斯链路的怀特海链路具有纠缠熵。最后，我们证明博罗梅安环具有“W 型”纠缠结构（即，追踪出一个环面不会导致可分离状态），并给出了其他具有“GHZ 型”纠缠的三元环的示例（即，追踪出一个环面确实导致了可分离状态）。
We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where M n is the link-complement of some n-component link inside the three-sphere S 3. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G= U (1) k) we give a general formula for the entanglement entropy associated to an arbitrary (m| n− m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G= SU (2) k, we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (ie, tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (ie, tracing out one torus does lead to a separable state).