Long-range entanglement from measuring symmetry-protected topological phases

TL;DR

This work provides a unified framework showing that long-range entanglement can be generated from short-range entangled (SPT) phases by single-site measurements, effectively realizing nonlocal dualities such as Kramers-Wannier and Jordan-Wigner as measurement-based processes. By adding SPT entanglers and performing measurements on sublattices, the authors demonstrate how to obtain twisted gauge theories and non-Abelian topological orders from Abelian starting points, and extend the construction to gapless states via KW on the XY chain. They also develop an MPS-based formalism proving that LRE post-measurement is robust across a broad class of SPTs and provide concrete analytic and numerical examples (deformed cluster, AKLT, and spin-1 chains) showcasing cat-state generation. The work further generalizes to higher-form and subsystem symmetries, suggesting wide applicability in realizing exotic topological phases and linking measurement-based protocols to a broader landscape of topological quantum matter.

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 $G \times H$ 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 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.

Figures (6)

• Figure 1: From the cluster state entangler to the Kramers-Wannier transformation. (a) Relation between the cluster state entangler and the Kramers-Wannier duality in arbitrary dimensions, with $A$ legs drawn in red and $B$ legs drawn in blue. Here the entangler is simply a product of controlled-$Z$ on nearest-neighbor sites. (b) Proof of this equality at the level of operators where $X$ on the red sites is interchanged with $ZZ$ on the blue sites.
• Figure 2: The Kramers-Wannier transformation from finite-depth circuit and measurements. The cluster state entangler can be used to implement Kramers-Wannier duality by measurement. The final state depends on $s_n = 0,1$ corresponding to measurement outcomes $X_n =1,-1$, respectively, which we can express as a product $\prod_n Z^{s_n}$ applied to $\mathinner{|{\psi}\rangle}$ before KW transformation. These operators can be pushed through the KW transformation to obtain a product of $X$ operators on the $B$ sublattice (blue). Hence, by acting with this product on the postmeasurement state, one can obtain the KW transformation of $\mathinner{|{\psi}\rangle}$ without postselection.
• Figure 3:
• Figure 4: Cat state from measuring the Haldane SPT phase. We consider the ground state of the spin-$1$ Heisenberg chain, which is in a nontrivial SPT phase for the $\mathbb Z_2\times \mathbb Z_2$ symmetry of $\pi$-rotations. In accordance with its short-range entanglement, we find that the Fisher information scales linearly with system size (blue dots). In contrast, if we measure the $R^z_n=e^{i \pi S^z_n}$-charge on every site, the remaining state has Fisher information $F \sim L^2$ (red dots), signaling long-range entanglement in the post-meaurement state (here we have chosen different random measurement outcomes for each $L$). This finding confirms that measuring one $\mathbb Z_2$ symmetry of the Haldane SPT phase creates a cat state for the remaining $\mathbb Z_2$ symmetry, even if one is not at a fine-tuned fixed-point limit.
• Figure 5: Anomalous symmetry from measuring an SPT phase. In an SPT phase, applying the symmetry in a region is equivalent to applying a unitary operator just near the boundary of that region; equivalently, the membrane operator has long-range order if we include the appropriate unitary operator along its boundary. In the $G \times H$ SPT fixed point models of the linear form $A_p F(B_q)$, $G$ acts only on the $A$ sublattice and the boundary operator acts only on the $B$ sublattice. If we then measure the spins of the $A$ sublattice, this boundary operator remains as a symmetry, now locally defined along the boundary. Because the boundary is codimension $p$, this defines a $G$$p$-form symmetry, which acts as the entangler for a nontrivial $H$ SPT phase. This implies that the $G$$p$-form symmetry in the post-measured state has a mixed anomaly with $H$, implying that the state cannot be short-range entangled.