Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases

IF 11.6 1区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Physical Review X Pub Date : 2024-06-07 DOI:10.1103/physrevx.14.021040
Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath, Ruben Verresen
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Abstract

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 Z2 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×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.

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通过测量受对称保护的拓扑相位实现远距离纠缠
多体量子态的一个基本区别是具有短程和长程纠缠(SRE 和 LRE)的量子态。后者无法通过有限深度电路产生,这突出了薛定谔猫态、拓扑秩序和量子临界的非局域性。值得注意的是,我们已经知道通过对 SRE 进行单点测量获得 LRE 的例子,例如通过测量二维簇态的子晶格获得环形代码。然而,我们仍然缺乏对 SRE 测量何时以及如何产生 LRE 的系统了解。在这里,我们确定了 LRE 出现在对对称保护拓扑(SPT)相进行测量时--簇态就是一个例子。例如,我们展示了如何通过在输入状态中添加一个簇态 SPT,然后进行测量来实现克拉默-万尼尔变换。这种变换自然地将状态与 SRE 和 LRE 联系起来。当输入态为 Z2 Levin-Gu SPT 时,双海米阶的实现就是一个应用。同样,加入费米子 SPT 和测量,就能实现一般状态的乔丹-维格纳变换。更广义地说,我们认为一大类受 G×H 对称性保护的 SPT 相在测量 G 电荷时会产生反常的 LRE,而且我们证明,在某些条件下,SPT 相中的一般点也会出现这种情况。我们的工作引入了一种新的实用工具,将 SPT 相作为创建 LRE 的资源,并揭示了这样一个分类结果:一旦我们用单点测量增强有限深度电路,所有通过顺序测量阿贝尔群或约旦-维格纳变换相关的状态都属于同一等价类。特别是,任何具有可解有限规整群的拓扑阶或分形阶,都可以通过这种方法从乘积态中获得。
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来源期刊
Physical Review X
Physical Review X PHYSICS, MULTIDISCIPLINARY-
CiteScore
24.60
自引率
1.60%
发文量
197
审稿时长
3 months
期刊介绍: Physical Review X (PRX) stands as an exclusively online, fully open-access journal, emphasizing innovation, quality, and enduring impact in the scientific content it disseminates. Devoted to showcasing a curated selection of papers from pure, applied, and interdisciplinary physics, PRX aims to feature work with the potential to shape current and future research while leaving a lasting and profound impact in their respective fields. Encompassing the entire spectrum of physics subject areas, PRX places a special focus on groundbreaking interdisciplinary research with broad-reaching influence.
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