A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases—of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the ${\mathbb{Z}}_2$ Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by $G\times H$ symmetry gives rise to anomalous LRE upon measuring $G$-charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and we uncover the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way.

中文翻译：

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测量对称保护拓扑相的长程纠缠


多体量子态之间的根本区别在于短程和长程纠缠（SRE 和 LRE）。后者不能由有限深度电路创建，这强调了薛定谔猫态、拓扑序和量子临界性的非局域性质。值得注意的是，通过对 SRE 执行单点测量来获得 LRE 的例子是已知的，例如通过测量 2D 簇状态的子晶格获得环面码。然而，仍然缺乏对 SRE 测量何时以及如何产生 LRE 的系统理解。在这里，我们确定 LRE 是在对对称保护拓扑 (SPT) 相执行测量时出现的，簇状态就是其中一个例子。例如，我们展示了如何通过将集群 SPT 添加到输入状态然后进行测量来实现 Kramers-Wannier 变换。这种转换自然地将状态与 SRE 和 LRE 联系起来。一个应用是当输入状态为 ${\mathbb{Z}}_2$ Levin-Gu SPT时双倍数序的实现。类似地，费米子 SPT 和测量的添加导致了一般状态的 Jordan-Wigner 变换的实现。更一般地说，我们认为一大类受 $G\times H$ 对称性保护的 SPT 相在测量 $G$ 电荷时会产生异常 LRE，并且我们证明这对于特定条件下的SPT阶段。我们的工作引入了一种新的实用工具，使用 SPT 相作为创建 LRE 的资源，并且我们发现了分类结果，一旦我们增加有限-具有单点测量的深度电路。 特别是，任何具有可解有限规范群的拓扑或分形阶都可以通过这种方式从乘积状态获得。