[Paper Review] Long-range entanglement from measuring symmetry-protected topological phases

Nathanan Tantivasadakarn, Ryan Thorngren+2 more|arXiv (Cornell University)|Dec 2, 2021

Advanced Electron Microscopy Techniques and Applications158 references42 citations

TL;DR

The paper shows that long-range entanglement can be generated by measuring symmetry-protected topological (SPT) phases, via cluster-state based entanglers, implementing Kramers-Wannier and Jordan-Wigner transformations.

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 $\mathbb Z_2$ 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 $G imes H$ 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 uncovers 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.

Motivation & Objective

• Motivate the distinction between short-range and long-range entanglement in many-body states and identify when LRE can be created by measurements on SRE states.
• Demonstrate that measuring SPT phases induces LRE, and connect this to anomalies and boundary physics of SPTs.
• Show how to realize KW and JW transformations by measurement-assisted SPT entangling operations.
• Provide examples of topological orders (e.g., double semion) and fermionic SPTs arising from this framework.
• Propose a general classification: states related by sequential gauging or Jordan-Wigner transformations lie in the same equivalence class when augmented with measurements.

Proposed method

• Define the cluster state entangler U_CZ as a product of CZ gates on nearest neighbors.
• Show that measuring a sublattice after entangling with U_CZ implements the Kramers-Wannier (KW) duality.
• Demonstrate that KW maps a product state to a long-range entangled state (GHZ in 1D, toric code in 2D).
• Extend the framework to include SPTs with G×H symmetry and show LRE arises upon measuring G-charges.
• Apply similar procedures to fermionic SPTs to realize JW transformations.
• Use symmetry fractionalization arguments to argue LRE is robust across the SPT phase.

Experimental results

Research questions

• RQ1 Under what conditions does measuring an SPT phase produce long-range entanglement?
• RQ2 How can the KW and JW transformations be implemented via measurement-augmented cluster-state protocols?
• RQ3 What topological orders arise from measuring SPTs, and how are they related to known models (e.g., double semion, toric code)?
• RQ4 How general is the mechanism when extending from bosonic to fermionic SPTs and higher G×H symmetries?
• RQ5 What is the relation between sequential gauging, JW transformations, and the resulting equivalence classes of states under finite-depth circuits plus measurements?

Key findings

• LRE appears when measuring SPT phases, not just fixed-point SPT wavefunctions.
• The cluster-state entangler combined with single-site measurements implements KW duality on the input state.
• Measuring the Levin-Gu Z2 SPT on the A sublattice yields double semion topological order.
• In the fermionic case, the procedure implements the Jordan-Wigner transformation for a general state.
• A broad class of SPT phases protected by G×H symmetry yields anomalous LRE upon measuring G-charges, linking SPT data to emergent LRE.
• States related by sequential gauging of Abelian groups or by JW transformation belong to the same equivalence class when augmented with finite-depth circuits and measurements.