Long-range entanglement from measuring symmetry-protected topological phases

Abstract

A fundamental distinction between many-body quantum states are those with short-and longrange 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. This 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 singlesite measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way.

Efficiently preparing Schrödinger's cat, fractons and non-Abelian topological order in quantum devices

“…Usually, gauging is a nonlocal transformation which would require a linear-depth circuit. However, in our companion work [52], we show that an arbitrary initial state can be gauged by a finite-depth circuit (made of controlled-Z gates) and single-site projections or measurements. Here we use this general fact to produce D4 topological order.…”

“…At a conceptual level, this construction can be interpreted as effectively gauging the Z 2 charge-conjugation symmetry of the Z 3 toric code, in line with S 3 = Z 3 Z 2 . In a companion work, we explore more generally how gauging can be implemented using finite-depth timeevolution and measurements [52].…”

“…Moreover, the authors thank Ryan Thorngren for collaboration on the companion work Ref. [52]. DMRG simulations were performed using the TeNPy Library [57].…”

“…Non-Abelian D 4 or S 3 topological order can be obtained by using a four-step protocol where one timeevolves and measures twice, creating the color code [51] or Z 3 toric code [31] as an intermediate state. This can be conceptually interpreted as sequentially gauging certain symmetries, thereby expanding the possible states of matter which can be efficiently prepared by finite-time evolution and measurement, the theoretical ramifications of which we explore further in a companion work [52].…”

Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

“…These so-called categorical symmetries have received widespread attention in recent years, e.g. [23][24][25][26][27][28][29][30][31][32]. In the case that these models are critical, their low-energy physics is described by a conformal field theory, in which these non-local symmetries are identified with the topological defects [33][34][35][36][37][38].…”