Long-range entanglement from measuring symmetry-protected topological phases

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 Schr\"odinger 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.

Recently, there has been growing interest in explicitly incorporating measurements into the study of many-body quantum states.For instance, a multitude of works have studied entanglement reduction from measurements, giving rise to surprising new structures [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50].However, there are also examples where measurements increase the entanglement.For example, it is known that performing single-site measurements on a subset of sites of a cluster state (with SRE) can produce a Greenberger-Horne-Zeilinger (GHZ) cat state [51], the toric code [52][53][54], and certain fracton codes via a layered construction [55,56].In fact, it has been remarked that all states realized by CSS stabilizer codes [57,58] (i.e., stabilizers that are of the form i∈S Z i or i∈S X i ) can be obtained by measuring an appropriate cluster state [59].
The existence of these examples begs the following question: What is the general framework for when, how, and why one can create LRE from SRE states and singlesite measurements?In this work, we argue that the essential fact in the above examples is that the cluster state is an SPT.This deeper understanding confers at least four advantages.First, in contrast to earlier studies, we argue that LRE states are obtained on measuring not just the fixed-point wave function of the SPT but any state within the same phase.Second, the origin of LRE under measurement is tied to a specific anomaly involving the symmetries-related to the anomaly living at the boundary of the original SPT phase-thereby constraining the nature of the resulting LRE.Third, it allows for the preparation of states that are not realized by stabilizer codes, such as topological order described by twisted gauge theories or non-Abelian fracton orders [60][61][62][63][64][65][66][67][68][69].Fourth, we achieve a new perspective on Kramers-Wannier (KW) [18,[70][71][72][73][74][75][76][77][78] and Jordan-Wigner (JW) [79][80][81][82][83][84][85][86][87] transformations.Indeed, we show how these nonlocal transformations can be efficiently implemented in a finite time by adding SPT entanglers to arbitrary initial states 3 and subsequently performing single-site measurements.In a companion work [88], we explain how this general understanding can be utilized to prepare, e.g., Z 3 , S 3 , and D 4 topological order in quantum devices such as Rydberg atom arrays.
This work is structured as follows.In Sec.II, we set the stage by reviewing some known examples, explaining how the 1D GHZ and 2D toric code states can be obtained by measuring particular cluster states.In Sec.III, we generalize these cases by reinterpreting the act of measuring cluster states as effectively implementing a KW transformation.To give illustrative examples, we explain how this allows one to transform the nontrivial Z 2 SPT in 2D to the double-semion topological order, and to transform the 1D XY chain into two decoupled critical Ising models by using finite-depth circuits and single-site measurements.Moreover, we discuss how certain types of non-Abelian topological order can be obtained by sequential applications of this scheme.Sec.IV generalizes this to the fermionic case, where a similar procedure implements the JW transformation, illustrated by creating the Kitaev chain from a trivial spin chain.Sec.V broadens our scope further: First, we argue that this procedure is a robust property of the SPT phase (which we exemplify by obtaining cat states via measuring the spin-1 Heisenberg chain), and second we argue that anomalous symmetries and LRE are generically obtained by measuring a broad class of SPT states (which we discuss in detail for the Z 3  2 SPT in 2D).We conclude with directions for future research in Sec.VI.

II. MOTIVATING EXAMPLES
We begin by reviewing how measuring cluster states in 1D and 2D can produce GHZ states [51] and the toric code [52], respectively.Consider a 1D chain with 2N qubits.The cluster state |ψ⟩ on this chain is the unique state that satisfies Z n−1 X n Z n+1 |ψ⟩ = |ψ⟩ for all n, where X, Y, Z denote the Pauli matrices.It can be prepared from the product state in the X basis by applying controlled-Z gates on all nearest neighboring qubits: We call the above unitary U CZ the cluster state entangler.Now suppose we measure X on all odd sites, with outcomes X 2n+1 = (−1) s2n+1 .Since Z 2n−2 X 2n−1 Z 2n commutes with the measurement, the state after the measurement |ψ out ⟩ satisfies Z 2n−2 Z 2n |ψ out ⟩ = (−1) s2n−1 |ψ out ⟩. On the other hand, the even stabilizers do not commute with the measurement; only their prod- . If all the s m = 0, then |ψ out ⟩ is the GHZ state on the even qubits: Otherwise, it is the GHZ state up to single-site spin flips conditioned on the measurement outcomes: |GHZ⟩ = |ψ out ⟩. Thus, regardless of the outcome, |ψ out ⟩ has long-range entanglement, as can, for example, be quantified by quantum Fisher information [89,90] (see also Sec.V A 5).
In 2D, we can consider a cluster state on the vertices and edges of the square lattice [52].The stabilizers of the cluster state for each vertex and edge are X v e⊃v Z e and X e v⊂e Z v respectively, where e ⊃ v and v ⊂ e denote edges e that contain the vertex v, and vertices v that are contained in e, respectively.Measuring X on all the edges will give a GHZ state on the vertices (up to spin flips that depend on measurement outcomes).On the other hand, measuring X on all the vertices gives a state of the toric code: We have the vertex term of the toric code, e⊃v Z e = ±1 depending on the measurement outcome, and we have the plaquette operator e⊂p X e = 1 coming from a product of four edge stabilizers around a plaquette, which commutes with the measurement.Note that while the topological order of this state is independent of the sign of the aforementioned stabilizers, one can always bring this to a state with e⊃v Z e = +1 by applying string operators that pair up the vertices with e⊃v Z e = −1.

III. KRAMERS-WANNIER TRANSFORMATION FROM MEASURING CLUSTER STATE SPT PHASES
We have seen that long-range entangled states can be obtained by performing single-site measurements on the cluster state.To explore a deeper reason for this finding, we will show how the cluster state secretly encodes the KW transformation.For simplicity, we will first discuss the 1D case, where the KW transformation is defined as the map X n → Z n Z n+1 and Z n−1 Z n → X n ; although this map preserves the locality of Z 2 -symmetric operators, it is a nonlocal mapping, relating SRE to LRE.
A first hint of the connection between the cluster state and the KW transformation is the fact that Z n Z n+2 and X n+1 act the same way on the cluster state.Moreover, , where U CZ is the cluster entangler, Eq. ( 1).Let us divide the sites into the odd and even sublattices, denoted A and B, respectively, and define the states |+⟩ A,B on these subspaces.We find that the operator σ := ⟨+| A U CZ |+⟩ B : H A → H B gives the KW transformation.For example, we show that X A is correctly mapped to Z B Z B , i.e., σX A = Z B Z B σ: and vice versa.This example is depicted graphically in Fig. 1.Note that this method works on any bipartite graph using a suitably generalized cluster state in any dimension, in which case, the Z B 's that appear act on the B vertices adjacent to where X A acts and vice versa.Eq. ( 3) suggests a method to apply KW by measurement.We begin with a state in H A and then introduce the ancillas |+⟩ B .We then apply U CZ to the combined system and measure the X spins on A. If the measurement outcomes are all + spins, then we have exactly implemented the KW duality.Otherwise, we have instead implemented the closely related operator where s a ∈ {0, 1} are the measurement outcomes of site a.By pushing through the excess operators from the A sites to the B sites using σ, we can rewrite this formula as where the s b are functions of the s a that depend on the graph.For example, in 1D, where A and B are the odd and even sublattices of the chain respectively, we have s b = 1<a<b s a .Thus, we see that further applying b∈B X s b restores the exact KW mapping σ.See Fig. 2.
This finding explains why the measured 1D cluster state has long-range order-it produces the KW dual of the trivial state |+⟩ A , which is a GHZ state.Likewise in FIG. 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 sn = 0, 1 corresponding to measurement outcomes Xn = 1, −1, respectively, which we can express as a product n Z sn applied to |ψ⟩ 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 |ψ⟩ without postselection.
2D we obtain the KW dual of the trivial state which is a toric code state 4 .We later argue that the long-range order holds for any state in the same SPT phase as the cluster state.Indeed, this fact can be seen by symmetry fractionalization for the two Z 2 symmetries a∈A X a and b∈B X b (acting on the odd and even sublattices, respectively) protecting the SPT phase.If we act on any state |ψ⟩ in the same SPT phase by the Z A 2 symmetry in a region R, it will reduce to some Z B 2 charged operators at the boundary of the region: where O is some operator with finite support situated at the left and right boundaries of R, which anticommutes with Z B 2 .Intuitively, this means that |ψ⟩ has the KW property, exchanging order operators and disorder operators, at long distances.See Sec.V A 1.
In higher dimensions, the cluster state is an SPT for higher form or subsystem symmetries that depend on the lattice.For example, if A and B are sites at the vertices and edges of the square lattice, then we have symmetries a∈A X a and b∈γ⊂B X b , where we have a symmetry for each closed curve γ drawn along the edges of the direct lattice.The KW so constructed is the duality between the Ising model and Ising gauge theory in 2+1D.
A summary of examples that arise from the KW transformation of various symmetries is given in Table I.

A. Twisted gauge theory from measuring cluster + SPT phases
As a first application, we discuss what happens when we apply this procedure to other states on the A sublattice, such as an SPT.As in Fig. 2, we add |+⟩ B ancillas, couple A and B with the cluster state entangler, and then perform measurements on the A sublattice.The result of this procedure is equivalent to gauging the SPT phase 5 .
To illustrate this procedure, we discuss how beginning with the A sublattice in the pure Z 2 or "Levin-Gu" SPT state |ψ⟩ [31] we obtain the double semion topological order [12] after entangling and measuring.The Levin-Gu SPT is defined on the vertices of the triangular lattice (A) and is an eigenstate of the following (non-Pauli) stabilizers: where ⟨vuu ′ ⟩ are the six triangles around v, and the wavy lines denote e πi 4 ZuZ u ′ between vertices u and u ′ .Note that this stabilizer is not simply a product of Pauli operators.Let us also stress that since this is an SPT phase, it is possible to prepare this state by a finite-depth circuit 6 .Following our procedure, we add the B sublattice consisting of edges of the triangular lattice, supporting a product with the trivial stabilizer Next, we couple the two sublattices with the cluster state 5 Alternatively, by viewing the SPT and the cluster state as a single state, performing the measurement on this combined SPT can be thought of as a different way of performing the KW duality on the product state.This choice of adding an extra SPT before gauging is also known discrete torsion [95], or defectification classes [96] in the literature. 6The unitary that creates the Levin entangler, resulting in the stabilizers Before we perform the measurements on all A sites (the vertices of the triangular lattice), we note that the vertex stabilizer does not commute with the measurement.Thus, it would not directly give us a useful condition on the postmeasurement state.However, using the fact that Z u Z u ′ |ψ⟩ = X (uu ′ ) |ψ⟩, where (uu ′ ) is the edge with u and u ′ as end points, the following is an equally valid set of stabilizers of |ψ⟩: where R e = e πi 4 Xe .The vertex stabilizers now commute with the measurement.However, the stabilizers in Eq. (10) do not commute for adjacent vertices.However, this problem is cured by restricting to the subspace: We can therefore circumvent having non-commuting stabilizers by attaching which is a projector into this subspace on each triangle.Finally, |ψ⟩ is identified as the unique state that has eigenvalue +1 under the following operators: Performing the measurement with outcomes X v = (−1) sv , the postmeasurement state is the unique state that has eigenvalue +1 under the operators: which is the ground state of the double semion model [31] up to single site X-rotations on edges that pair up the vertices where s v = 1 to remove the signs, and swapping X e with Z e to match the choice in Ref. [31].
Our implementation of gauging via combining measurements with a cluster state entangler (including Z n generalizations) implies that we can produce all twisted quantum double models of a finite Abelian gauge group via stacking general SPTs prior to measuring-which can be prepared by finite-depth circuits [27].Note that these models already contain certain non-Abelian phases, e.g., D 4 topological order arises upon gauging the Z 3 2 symmetry of an SPT phase with a type-III cocycle [97,98].(For obtaining non-Abelian topological order associated with any solvable group, see Sec.III C.) Similarly, our procedure allows for the creation of twisted fracton phases by gauging 3D subsystem SPT phases [65,84,85,99,100].Thus a much wider class of states can be obtained from local unitary circuits and local operations and classical communications (LOCC) [54] than previously established.

B. Physically applying the Kramers-Wannier transformation to a gapless state
Here, we discuss an example where the input state |ψ⟩ (in Fig. 2) itself has long-range entanglement.In particular, we focus on a well-known example of how the XY chain-an example of a gapless state-can be transformed into two decoupled critical Ising chains by gauging particle-hole symmetry 7 .Here, we achieve this gauging by using a finite-depth circuit and single-site measurements.
We place the XY chain on the odd sites (A) and initialize with |+⟩ states on the even sites (B).The aforementioned state can be considered the ground state of the following Hamiltonian Next, we gauge the Z 2 subgroup n X 2n−1 of the full U (1) symmetry of the XY chain.To do so, we couple the even and odd sites with the cluster state entangler U = n CZ n,n+1 , resulting in Note that since Z 2n−1 X 2n Z 2n+1 is an integral of motion, the following Hamiltonian also has the same wave function as its ground state: Now, we perform a measurement on the odd sites with measurement outcomes X = (−1) s ; the state after the measurement is the ground state of the Hamiltonian with the integral of motion n X 2n serving as a global Z 2 symmetry.After appropriate spin flips to remove the signs and the circuit n CZ 2n,2n+2 , the Hamiltonian reads which describes two decoupled critical Ising chains.We thus confirm that we have physically implemented the KW transform on a gapless state.
Let us remark that this procedure does not rely on freefermion solvability of the XY chain and the Ising model.For example, the procedure still works in the presence of the XXZ deformation, which respects the Z 2 symmetry (albeit opening up a gap).

C. Non-Abelian topological order from sequentially gauging Abelian groups
Beyond cyclic groups Z n , cluster states and the corresponding KW dualities have been generalized to arbitrary finite groups [102][103][104], giving the potential to gauge non-Abelian groups by unitaries and measurement.However, unlike the Abelian case, which produces Abelian anyons depending on the measurement outcome, gauging non-Abelian groups can produce non-Abelian anyons that can only be paired up using linear depth string operators 8 .The intuition for this is that the string operators for moving such anyons consist of noncommuting operators which hence cannot be applied all at once9 .
Our implementation of the KW duality avoids this issue by a sequence of circuits and measurements, which can be interpreted as sequentially gauging Abelian groups.In such a method, the measurement outcomes in all intermediate states correspond to Abelian anyons, which can all be paired up in finite depth.In this way, all gauge theories whose gauge group is solvable (i.e., obtained by extending finite Abelian groups) can be constructed efficiently in this manner.For example, the S 3 quantum double can be obtained by gauging a Z 3 symmetry (i.e., measuring a Z 3 cluster state), which prepares a Z 3 toric code, followed by gauging the charge conjugation symmetry that permutes anyons e ↔ e 2 and m ↔ m 2 .We note that since S 3 is not nilpotent, it can be used for universal quantum computation [105].As a second example, the D 4 topological order can be obtained by first Jordan-Wigner transformation from finitedepth circuit and measurements.We show the process of entangling fermionic (red) and bosonic (blue) degrees of freedom and its relation to the JW transformation.Here ⟨0| corresponds to contracting with the empty state of fermions.We use Jordan-Wigner * to emphasize that this transformation differs from the usual JW by an additional KW transformation.Similar to Fig. 2, this can be utilized to implement the JW transformation via measurements (see main text).
preparing the 2D color code and gauging the Hadamard symmetry.In our companion paper we provide explicit finite-depth qubit-based circuits for these two examples [88].We note that sequentially gauging Abelian groups can also give rise to states beyond quantum doubles.For instance, the doubled Ising anyon theory can be obtained by gauging the e ↔ m symmetry of Z 2 topological order [96].Such a Kramers-Wannier transformation (implemented using our finite-depth circuit and single-site measurements) can indeed be performed since it is known that the Z 2 symmetry can be made on-site (for explicit models, see Refs.[106,107]).By definition, this state can be connected to any other state with Z 2 topological order through a finite-depth circuit, and we have already described how, e.g., the usual toric code can be obtained from the product state.

IV. JORDAN-WIGNER TRANSFORMATION FROM MEASURING FERMIONIC SPT PHASES
Analogous to the KW transformation, the Jordan-Wigner (JW) map is a nonlocal transformation which maps between fermionic and bosonic degrees of freedom [79,80].Similar to the KW transformation, here we can prepare and entangle bosonic and fermionic degrees of freedom as shown in Fig. 3.We can then perform either bosonization of an arbitrary input fermionic state by measuring the parity of all fermions, or fermionization of an arbitrary input bosonic state by measuring X on all the spins after the entangling step.

A. 1+1D bosonization
Let us demonstrate this case explicitly by preparing the Kitaev Majorana chain, which cannot be done in finite time with only unitary evolution [6].We start with N qubits on odd sites initialized in the |+⟩ state and N fermions on even sites initialized in the empty state P = −iγγ ′ = 1, where γ = c + c † and γ ′ = −i(c − c † ) are Majorana operators.Furthermore, we define the hopping operator S 2n = iγ ′ 2n−2 γ 2n , which hops a fermion from site 2n − 2 to 2n.We create a Z 2 × Z F 2 SPT [108-111] with the following circuit: where the operator is a hopping operator controlled by the qubit at 2n−1.In other words, a fermion is hopped if the spin at site 2n − 1 is down.We also remark that because all gates mutually commute, it can be implemented as a finite -epth circuit.
The resulting SPT (which we will call the Jordan-Wigner state) is the +1 eigenstate of the stabilizers Now, we measure all the spins with outcomes X 2n−1 = (−1) s2n−1 .The stabilizers of the measured state are (−1) s2n−1 γ ′ 2n−2 γ 2n and n Z 2n−1 P 2n Z 2n+1 = n P 2n , which after applying , gives the ground state of the Kitaev chain.We note that, alternatively, starting with the SPT, measuring the parity of all the fermions gives the GHZ state.

B. 2+1D bosonization
The recipe above extends to arbitrary dimensions.The generalization of the Jordan-Wigner transformation has been explored in a number of works including [81][82][83][84][85][86][87][112][113][114], and can be thought of in the context of this work as gauging the fermion parity symmetry.From this we can construct a particular state of fermions and spins which conserves fermion parity and a higher form Z 2 symmetry such that one can perform either bosonization, by measuring the parity of each fermion, or fermionization, by measuring the spins in the X-basis.Here, we demonstrate this for the 2D bosonization procedure of Ref. 81 on a square lattice.
As with the 2D KW transformation, we consider the square lattice with fermions initialized in the empty state P v = 1 on the vertices and spins are initialized in the |+⟩ state on the edges (X e = 1).We create an "SPT" state (see below for caveats) protected by fermion-parity symmetry and a global 1-form symmetry.The stabilizers of this "JW state" are given by Upon measuring the fermion parity of all fermions, the resulting state is described by the stabilizers which, up to a sign given by measurement outcomes, describe the 2D toric code.
To discuss the circuit required to prepare this SPT, we first define the fermion hopping operator for each edge as Then, we may define the controlled operator Here, the only novel subtlety-not present in the bosonic case or the 1D JW transformation-is that not all of the CS gates mutually commute and therefore must be applied sequentially.Nevertheless, it turns out that their ordering is irrelevant: Each choice of ordering gives a valid JW transformation [84,85], and moreover these choices only differ by phase gates.Thus, a given choice determines the spatial anisotropy of the stabilizers.
To obtain the stabilizers of the SPT in Eq. 27, the unitary that prepares it can be written as where e N (v) and e E(v) refer to the edges directly north and east of the vertex v, respectively.In other words, we have chosen to apply the control gates on all vertical edges (which mutually commute) followed by those on the horizontal edges; lastly, we apply appropriate CZ gates to obtain the desired form of the stabilizers.
The JW state has the property that if we form the open string operator associated to the 1-form symmetry, by taking a product of stabilizers, we will find a fermion operator at the end.Thus, it looks like a nontrivial SPT for fermion parity and the 1-form symmetry.However, if we consult the cobordism classification, we find there are no nontrivial SPTs in this symmetry class.In fact if we try to construct an SPT class with this property using the Atiyah-Hirzebruch spectral sequence, we find that the relevant class in H 2 (Z 2 [1], Ω 1 spin ) has a nonzero differential.It would be a supercohomology class but it does not satisfy the Gu-Wen equation [108] (also see [115]).
The puzzle is resolved by considering the cobordism classification as describing a torsor rather than a group, meaning that with this choice of 1-form symmetry, the associated open string must always end on a fermion, and in that sense there is only one SPT phase, but it is not quite trivial because the 1-form symmetry generator we've chosen is not completely "on-site".Indeed, in Refs.[112,113] it was stressed that the 1form symmetry in 2+1D bosonization has an anomaly Sq 2 B (unlike in 1+1D bosonization where we obtain an anomaly-free Z 2 symmetry upon bosonizing) and the kernel of the bosonization transformation gives a trivialization of this anomaly in the presence of fermions.In simple terms, the Sq 2 B anomaly says that the 1-form symmetry generator needs to obey fermionic statistics.Now, there is no issue with realizing such an anomalous symmetry in a not-on-site fashion, but because of the anomaly, it cannot be screened-there is no end-point operator that will give the open string long-range order.However, if physical fermions are present, we can have a short-range entangled state where the 1-form symmetry generator ends on these fermions, and we interpret this finding as a trivialization of the Sq 2 B anomaly, which is precisely what happens in the JW state.To trivialize the anomaly, the 1-form symmetry generator has to end on a fermion (which is essentially the Gu-Wen equation), so while it looks like a nontrivial SPT, there is really only one option, in harmony with the classification.
Similarly to the KW transformation, we can now apply the JW transformation to arbitrary states by measurements.For example, we can consider preparing the fermions in a 2+1D topological p + ip superconducting state with chiral Majorana edge modes.After coupling to the JW state and measuring fermion parity, the remaining spins will describe a chiral Ising topological order.Similarly, coupling ν stacks of p + ip superconductors to the SPT and performing the measurement can realize the topological orders in Kitaev's 16-fold way [13].
The generalization to higher dimensions [82,83] and to other types of fermionic gauge theories (including fracton models with fermionic statistics [84,85]) is straightforward by taking a sequential product of CS operators that mutually commute within each layer.

V. GENERALIZATIONS
Thus far, we have focused on two illustrative cases, where measuring sublattices of the cluster and JW states leads to LRE.In this last section, we generalize this approach in two directions.First, we make the case that the ability to produce LRE from measurements is indeed a property of the whole SPT phase, being robust to tuning away from a fixed-point limit.Second, we show that LRE is naturally obtained by measuring a broad class of SPT phases, of which the cluster and JW states are but two examples.
A. LRE generation as stable property of SPT phase

Intuition away from fixed-point limit
Let us first consider the 1D cluster SPT phase and ask whether one obtains a cat state upon measuring one of the sublattices starting with an arbitrary state in this phase.We present an intuitive argument, which holds away from the fixed-point limit.A key property of the cluster SPT phase in 1D is that it generically has longrange order for the following string operator [116]: where S 2m,2n := X 2m+1 X 2m+3 • • • X 2n−1 is a string operator consisting of the Z 2 symmetry of the odd sites.
The SPT invariant [117] is encoded in the fact that the string operator for one of the Z 2 symmetries only has long-range order if one includes an end-point operator that is charged under the other Z 2 symmetry (in this case Z 2n which is odd under m X 2m−1 ).Indeed, in the nontrivial SPT phase, one finds that the undressed string does not have long-range order: We would like to understand what happens if we measure all odd sites in the X-basis, which is a rather challenging many-body question, and Secs.V A 2-V A 5 will be devoted to addressing this issue.However, as a first encounter, and to build some intuition, let us imagine that instead of measuring all odd sites, we measure a single global observable, namely the string operator S 2m,2n for a fixed choice of m and n.Since all X measurements commute, we can indeed think of this as a first step in our measurement process, and we find that this first step indeed produces long-range entanglement.
To determine the result of measuring S 2m,2n , first note that Eq. ( 33) tells us that if we choose n and m far enough apart, then ⟨S 2m,2n ⟩ ≈ 0. Hence, both measurement outcomes S 2m,2n = ±1 = (−1) s are equally likely.The two possible postmeasurement states can thus be written as: ), we obtain Moreover, using the dual string operator, one can prove that ), such that for either measurement outcome, we have We thus find that measuring the string leads to longrange cat-state-like entanglement between the two endpoints!This result is consistent with the notion of SPT entanglement explored in Ref. [118], where the author showed that measuring a large connected block of sites leads to a Bell pair between the two end-points.
The above argument can be extended to higher dimensions.For instance, let us revisit the 2D case mentioned in Sec.II: the Lieb lattice with spins on the vertices (A sublattice) and bonds (B sublattice) of the square lattice.The cluster state on this lattice is an SPT phase protected by a global Z 2 symmetry U A = a∈A X a , as well as a "1-form symmetry," U B γ = b∈γ⊂B X b , meaning a symmetry defined for each closed curve γ on the bonds of the square lattice [98,119,120].
In the SPT phase, we have long-range order for the membrane operator S ∂R a∈A∩R X a where R is some region and S ∂R is a string operator on the boundary which "braids" with U B γ , meaning , where the exponent is the number of intersection points between the curves γ and γ ′ .For the fixed point cluster state, Upon measuring the membrane, we are left with longrange order for S γ (see Fig. 5).This quantity serves as an order parameter for spontaneously breaking the 1form symmetry, thereby implying topological order.In fact, this point of view naturally generalizes to other SPT phases, as we will discuss in Sec.V B.
However, while the above is intuitive and encouraging, it does not actually prove that the LRE persists upon measuring all (or a finite density of) sites.In particular, in the 1D case, we have thus far only measured S 2m,2n and not yet all odd sites.This calculation does not automatically guarantee that the long-range order in Eq. ( 36) persists after performing the other measurements10 since measurements can reduce entanglement.We now argue that, generically, it does indeed persist.

Conjecture and theorem: LRE from SPT
Having gained the above intuition, let us now try to formalize how and when long-range entanglement is produced by measuring SPT phases.To this end, we state a general conjecture, for which we give plausibility arguments.In addition, we provide a rigorous theorem for a slightly more constrained setting.
We consider a (short-range entangled) wave function |ψ⟩ in a nontrivial SPT phase protected by an Abelian symmetry group G × H.Moreover, we presume that the SPT phase is mixed, which means that explicitly breaking either G or H would trivialize the SPT phase.Note that the notion of an on-site symmetry automatically implies the notion of a unit cell, whereby a global symmetry U ∈ G × H can be decomposed as a tensor product over the unit cells: U = n U n .The physical act of measuring the G-charge (for a given unit cell n) means that, mathematically, we apply a projector where q is a charge labeling the (random) measurement outcome, and χ q is the corresponding character.For a given set of measurement outcomes {q n } n (one for each unit cell), we thus obtain the postmeasurement state The probability of obtaining a given measurement outcome (and thus the corresponding postmeasurement state) is, of course, given by Born's rule.For each given outcome, one can ask whether the postmeasurement state is long-range entangled.We generally expect that this is indeed the case.For concreteness, we will consider the one-dimensional case, although many of the arguments have higher-dimensional analogs.(We will discuss higher-dimensional examples in Sec.V B.) Conjecture.If the premeasurement state |ψ⟩ has a conventional SPT string order parameter 11 for a mixed Abelian G × H SPT phase, then the probability of the postmeasurement state being long-range entangled is unity.
We will give plausibility arguments for this conjecture in the next subsection.The above claim of unit probability allows for a 'measure zero' case where the postmeasurement state can be short-range entangled.Indeed, we will see examples of this in our numerical exploration in Sec.V A 5. However, if we slightly strengthen our assumptions, we can prove one always obtains long-range entanglement: Theorem.Let |ψ⟩ be in a nontrivial mixed SPT phase for Abelian symmetry group G × H.If it admits a finitebond dimension matrix product state (MPS) description, 11 In other words, the SPT wave function has long-range order in On⟩ ̸ = 0 for a certain U ∈ G and for a particular choice of end-point operator O that is supported on a single unit cell, or at the very least, that commutes with G in each individual unit cell.The nontrivial (mixed) SPT class implies that O will carry nontrivial charge under H. Ref. 117 proved there always exists an O that gives long-range order, although it does not guarantee the additional local properties.
then there exists a choice of unit cell such that measuring the G-charge for each unit cell produces a state with longrange entanglement for any measurement outcome.More precisely, the postmeasurement state is a cat state for the (partial) spontaneous symmetry breaking of H.
To phrase and prove this result, we use the notion of matrix product states (MPS).In fact, this same framework will provide an intuitive justification for our more general conjecture.We thus turn to an MPS-based description of our set-up.

Proof using matrix product states
For a review of MPS, we point the reader to Refs.92 or 121.The key idea of MPS is that a wave function is written in terms of finite-dimensional tensors: where N labels the number of unit cells, i = 1, • • • , d labels the states in each unit cell, and A i is a χ × χ matrix.(For convenience, we work with translationinvariant states, where the tensor is identical for all sites.)Here χ ∈ N is called the bond dimension, with χ = 1 corresponding to a product-state wave function.It is known that up to exponentially small errors in local quantities, ground states of gapped Hamiltonians are well approximated by such an MPS [122,123].In what follows, we will use the graphical notation.For instance, Eq. ( 39) becomes where we ignore boundary conditions, or equivalently, we work in the thermodynamic limit.
A key property that makes MPS such a useful framework, is that global symmetries, such as U = n U n , imply nice local properties on the MPS tensor.In particular, one can 'push' physical symmetries through to the 'virtual' level 12 [3, 22, 25, 92, 124]: There exists an operator V g such that In other words, we see that the physical operator U g is equivalent to acting with V g and V † g at the virtual level.As a sanity check, we indeed see that if we apply U g on each site, then each V g is canceled by a V † g , thereby confirming n (U g ) n is a global symmetry of |ψ⟩.
An interesting property of these virtual symmetry actions V g is that they only need to form a projective representation of the symmetry group.Thus, for any g, g ′ ∈ G × H, we have V g V g ′ = ω(g, g ′ )V gg ′ with a potentially nontrivial phase factor ω(g, g ′ ) ∈ U (1).A nontrivial SPT class is then equivalent to the statement that , where the nontrivial SPT phase corresponds to the projective representation where the two generators anticommute.More generally, a mixed SPT class implies that ω(g, h) ̸ = ω(h, g) for a certain choice of g ∈ G and h ∈ H, which we will use to derive long-range entanglement in the postmeasurement state.
As discussed, the act of measurement corresponds to applying a projector (37).The MPS tensor for the postmeasurement state ( 38) is simply: Since for any g ∈ G we have U g P G = χ q (g)P G (i.e., the symmetry operator acts like a number ) we thus have the following local tensor properties: for g ∈ G and h ∈ H. Eq. ( 44) tells us that H still acts like a physical symmetry on the postmeasurement state; however, Eq. ( 43) tells us that G now only acts on the virtual degrees of freedom, which we can interpret as a sort of higher symmetry.More concretely, as we will now argue, V g acts as an order parameter for the spontaneous breaking of H symmetry, such that the postmeasurement state is a long-range entangled cat state for symmetry breaking.
The key identity we will need is the ability to push V g from the virtual level to the physical level.In particular, the question is whether there exists an operator O g such that Let us temporarily earmark the question of whether O g exists and first explain how its existence is sufficient to prove that the postmeasurement state is long-range entangled.
From the projective group relations , one can straightforwardly prove that if O g exists, it must carry charge under H.In particular, in Appendix D we prove that Eq. ( 45) implies Since we are considering a mixed SPT phase, we know that this phase factor is nontrivial for certain g ∈ G and h ∈ H; let us henceforth fix those elements, such that α g,h ̸ = 1.One consequence of Eq. ( 45) is that in the postmeasurement state, the expectation value of O g must vanish.Indeed, taking the expectation value of both sides of Eq. ( 46) and using that U h is a symmetry, we obtain Since α g,h ̸ = 1, this implies that ⟨O g ⟩ postmeas = 0.However, the two-point correlation is nonzero.Indeed, combining Eq. ( 45) with Eq. ( 43) directly implies that We thus have long-range mutual information and thus long-range entanglement.In more physical terms, we see that the postmeasurement state can be interpreted as a cat state for the (partial) spontaneous symmetry breaking of H.
We have thus proven that the existence of O g , as defined in Eq. ( 45), is sufficient to prove long-range entanglement.The final issue is when we expect this to hold.One scenario where we can show that O g exists is when the conditions of the theorem in Sec.V A 2 are met.Indeed, it is known that short-range entangled MPS satisfy a certain injectivity condition [92] which means that after potentially blocking sites a finite number of times, the MPS tensor defines an injective map where we consider the virtual legs to be its input and the physical leg its output.Equivalently, there exists a tensor 13 C that functions as an inverse for A: where we will henceforth presume one has blocked the unit cell to achieve the injectivity condition.Using this, we can define the physical operator O g as follows: Using Eq. ( 49), one sees that this operator satisfies Eq. ( 45) for the A tensor.Moreover, one can prove that O g commutes with the G symmetry, i.e., U g O g U † g = O g for any g ∈ G (see Appendix D).Hence, O g commutes with the projection P G , such that we obtain Eq. ( 45) also for the B tensor.This concludes the proof of the theorem in Sec.V A 2.
Thus, if we are willing to block unit cells a finite 14 number of times, we can prove that LRE is obtained for any measurement outcome.In the absence of such blocking, we believe one can only make a probabilistic statement.In fact, while we do not offer a proof of the conjecture stated in Sec.V A 2, the above MPS arguments provide an intuitive justification.To see this case, let us first remark that to make probabilistic arguments, one only needs a weaker version of Eq. ( 45), namely, that there exists an O g such that one has finite overlap with the virtual V g action, i.e., for some λ ̸ = 0. Indeed, one can again show that this implies O g carries nontrivial charge under H.Moreover, the same argument as above still implies that one expects O g to have a long-range two-point function, since it picks up on the long-range order of V g (see Eq. ( 43)).
The only way this case can fail is if the multiple terms on the right-hand side of Eq. ( 51) conspire to exactly cancel out the long-range contributions, which this certainly can happen (we will give an example in the next subsection); however this requires a delicate balancing of 14 We emphasize the finiteness since if one is willing to block an unbounded number of times, we can effectively appeal to an RGbased argument whereby one flows to the fixed-point state with zero correlation length, which would be less interesting.
terms and is thus likely a measure zero case over the ensemble of all possible measurement outcomes.Lastly, we note that Eq. 51 can be expected to hold for SPT phases which admit a conventional SPT order parameter, as defined in footnote 11.Indeed, the very reason the string order parameters have nontrivial end-point operators is because they are able to cancel out the virtual V g action of the symmetry string or disorder operator [117].Commonly used string order operators have an end-point O g supported on a single unit cell and commute with the corresponding symmetry generator U g , such that if Eq. ( 51) applies to the A tensor it also automatically carriers over to the postmeasurement B tensor.In conclusion, for these reasons, we conjecture that only a measure zero of measurement outcomes can fail to give long-range entanglement.It would be interesting to sharpen this intuition into a rigorous proof of our conjecture.First, we consider a deformation of the cluster state: Here |ψ(0)⟩ is the cluster state of Eq. (1).For any β, this state admits a χ = 2 MPS representation [125] and one can show that for any finite β, this state is in the nontrivial SPT phase protected by Z 2 × Z 2 symmetry.Its correlation length ξ increases monotonically with β and diverges as β → ∞.The MPS tensor turns out to be injective without blocking, meaning that our theorem implies that measuring, say, X 2n+1 on odd sites produces a long-range entangled state on the remaining qubitsfor any possible measurement outcome.As a second example, we consider the paradigmatic spin-1 AKLT state [126], which is known to be described by a χ = 2 MPS and is an SPT phase protected by the Z 2 × Z 2 symmetry of π-rotations.As generators, we can choose R x = n e iπS x n and R z = n e iπS z n .If we block the spin-1s into two-site unit cells, then the MPS satisfies the aforementioned injectivity property.Hence, our MPS-based arguments prove that if one measures, say, R z 2n−1 R z 2n ∈ {−1, 1} charge on each two-site unit cell, then the postmeasurement state will always have long-range entanglement.
What if we did not block in the last example?If we measure R z n ∈ {−1, 1} in each single-site unit cell, then there is a measure-zero chance that we obtain R z n = 1 for all sites.In this case, the postmeasurement state is simply the product state |0⟩ N , where |0⟩ is the unique +1 eigenstate of R z = e iπS z .However, as long as a finite density of sites projects onto the −1 eigenstate of R z , the postmeasurement state is a long-range entangled of GHZ type, capturing the spontaneous symmetry-breaking of R x .This example is thus consistent with our conjecture and it illustrates the importance of making probabilistic statements in the cases where one does not block unit cells.
While both examples are illustrative, by definition they are analytically tractable.One might wonder about SPT phases of ground states that are not exactly solvable.For this reason, we now turn to a numerical exploration.

Numerics: Cat state from the spin-1 Heisenberg chain
To emphasize the generality of our claim that SPT phases can be used to generated LRE upon measurement, we consider the incarnation of the Haldane SPT phase in the spin-1 Heisenberg chain.Its Hamiltonian is a just nearest-neighbor antiferromagnetic coupling: This spin chain is known to be gapped [127], forming a nontrivial SPT phase for the Z 2 ×Z 2 group of π-rotations generated by R γ = n e iπS γ n with γ = x, y, z [21,126,128,129].Indeed, it has been argued to be in the same phase as the tractable AKLT state encountered in the previous section [126].
By our general proposal, we expect that measuring, say, the R z charge for every site, should result in a cat state for the remaining Z 2 symmetry.An interesting difference from the cluster chain is that the symmetries do not act on distinct sites.We thus measure R z n = e iπS z n on every single site.Effectively, this process comes down to measuring whether (S z n ) 2 is 0 or 1.For the first outcome, the site has no degree of freedom left, whereas for the latter, we still have a remaining qubit (S z n = ±1) which is toggled by R x .Hence, with the exception of there being no qubits left (which is of measure zero in the thermodynamic limit), we expect a cat state for the remaining chain of qubits.This is similar to the AKLT discussion in Sec.V A 4, although now we cannot rely on an exact solution.
To test this prediction, we numerically obtain the ground state of Eq. ( 53) using the density matrix renormalization group (DMRG) [121,130,131] for a variable system size L with periodic boundary conditions.We then project each site into (S z n ) 2 = 0 with probability 1/3 or (S z n ) 2 = 1 with probability 2/3.As a robust way of detecting whether the resulting state is a cat state, we calculate the Fisher information, which in this case is simply the variance of the total (staggered) magnetization: We consider the ground state of the spin-1 Heisenberg chain, which is in a nontrivial SPT phase for the Z2 × Z2 symmetry of π-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 iπS z n -charge on every site, the remaining state has Fisher information F ∼ 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 Z2 symmetry of the Haldane SPT phase creates a cat state for the remaining Z2 symmetry, even if one is not at a fine-tuned fixed-point limit.
This Fisher information is a quantitative measure for the use of the state for quantum metrology purposes [89,90].While SRE states obey a scaling F ∼ L, only nonlocal cat states have F ∼ L 2 .Our numerical results 15 are shown in Fig. 4.While the original ground state has F ∼ L, we find that the postmeasurement state indeed has F ∼ L 2 , confirming that it is a cat state.In addition, it is interesting to see that F (L) varies relatively continuously with L, despite each system size having a completely random measurement outcome (each red dot is computed for only a single measurement shot).
The above emergence of a cat state can actually be linked to the original interpretation of the Haldane SPT phase.Indeed, when the topological string order parameter was first introduced in 1989 [128], it was designed to pick up the 'hidden symmetry-breaking' of the state, where it was observed that if one imagines removing all S z n = 0 states, then the remaining S z n = ±1 states form long-range Néel order.However, since the S z n = 0 states are interspersed within the S z n = ±1 states and are allowed to have quantum fluctuations, they disorder this local order (which can now only be picked up with a string order parameter).Our above procedure can be interpreted as making this hidden order manifest: The measurement pins the location of S z n = 0, preventing them from disordering the Néel state.

B. Measuring general SPT phases
Here, we discuss how LRE arises upon measuring more general SPT states, even beyond 1D.As a natural starting point, we consider one of the simplest SPT phases (beyond 1D) which are protected by more than a single cyclic group-such that it is meaningful to measure one symmetry and preserve the other.Let us thus consider the Z 3  2 "cubic" SPT in 2+1D.One model for this phase [98,132] is given by placing spins on the sites of a triangular lattice, with each Z 2 acting as j∈A,B,C X j on each of three triangular sublattices A, B, C. For each site j, there is a stabilizer given by where the product is over triangles ⟨jqq ′ ⟩ with vertices j, q, q ′ .When we measure X j on the A sublattice, we are left with a state on a honeycomb sublattice with around each hexagon, for some fixed signs (determined by our measurement outcome s j ).The loop operators ij∈γ CZ i,j along a closed path γ of vertices can be considered as a Z 2 1-form symmetry of this state.Note that this acts as the cluster SPT entangler for Z B,C 2 along γ, which implies there is a mixed anomaly; therefore, the resulting state obtained from measurement cannot be short-range entangled.Note that this anomaly can be realized on the boundary of a lattice model of a 3D SPT protected by Z 2 2 × Z 2 [1] as studied in Ref. 98.
We believe that a similar conclusion holds generally when we measure SPT states, at least when the corresponding topological term is linear in the gauge field associated with the measured charge.Let G and H be (p − 1)-and (q − 1)-form symmetries where G and H are onsite symmetries that act only on subsystems A and B respectively.Denote the background gauge fields of G and H, A p and B q , respectively.Now, consider an SPT associated with the cohomology class A p F (B q ) ∈ H d+1 (G × H, U (1)), where d is the space dimension and F (B q ) ∈ H d+1−p (H, G ⋆ ) describes a topological G current made from B q where G ⋆ = Hom(G, U (1)).Physically, F (B q ) can be understood as an H SPT in d − p + 1 spatial dimensions, and the SPT A p F (B q ) corresponds to decorating fluctuating G-domain walls with this H SPT [133].
In this fixed-point model, if we now measure the G charges, we essentially project out the topological current F (B q ).Analogously to the CZ ring in Eq.( 56), we similarly obtain a p-form symmetry, the remnant of the G symmetry action by symmetry fractionalization of the parent SPT phase before measurement-applying the G symmetry in a region is equivalent to acting on the FIG. 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 × H SPT fixed point models of the linear form ApF (Bq), 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.
boundary of that region with the entangler of the H-SPT (see Fig. 5).
This anomaly can also be seen from studying the topological response of the G × H SPT. Projecting out G charges is equivalent to making the G gauge field A p dynamical.Measuring the G charges can be thought of as making A p dynamical with a charge background fixed by the measurement outcome.Since we began with a gapped phase, there are no fluctuating G charges at low energies.As a result, there is an emergent p-form symmetry that acts as A p → A p + λ, known as the center, or electric symmetry [119].This symmetry is the same as the p-form symmetry we defined above.From the form of the topological response, assumed to be A p F (B q ), we see that this global symmetry is broken when there is a nontrivial B q since it produces a variation of the effective action, namely λF (B q ).This variation is characteristic of an anomaly associated with a d + 1-dimensional topological response Ãp+1 F (B q ) [134], where Ãp+1 is the background p + 1-form gauge field (note the shift) associated with the center symmetry.
When the SPT class is not linear in A p , we will not be able to fractionalize the G symmetry so that the boundary operator commutes with the G charges [33].However, if it is the form F 1 (A p )F 2 (B q ), where F 1 (A p ) ∈ H j (G, K) and F 2 (B q ) ∈ H d+1−j (G, K * ), for some Abelian group K, then there will be a codimension j + 1 defect Poincaré dual to dF 1 (A p ) that can factorize, defining a j + 1form symmetry in the fixed point model postmeasurement corresponding to a field C j+2 .The anomaly will then be C j+2 F 2 (B q ) ∈ H d+2 (K[j + 1] × H, U (1)).For example, if we measure both Z 1 ), (2) 1 and F (B q ) = A 1 .Thus, the anomaly postmeasurement is 1  2 C 3 A 1 for a Z 2 2-form symmetry associated with C 3 .

VI. OUTLOOK
In this work, we have presented a general framework for which performing measurements of short-range entangled states produces long-range entanglement.We have given some intuitive arguments that this is a stable property of the SPT phase, as well as proven that this always holds if the measurements are performed in an appropriately large enough unit cell.We would also like to determine the nature of the long-range entangled states which appear.
We have also described how non-local transformations including Kramers-Wannier and Jordan-Wigner arise from coupling an arbitrary state with a symmetry to a cluster-like SPT and performing measurements.It would be interesting to see whether other SPTs define useful transformations this way.If so, what family of MPOs do they define?We note that, given a general MPO, it is not obvious how to implement it from finite-depth unitaries and measurements.
Sequential applications of our procedure even lead to non-Abelian topological order, including quantum doubles for solvable groups.A natural question is to find an analogue for nonsolvable groups-or to prove a no-go theorem.We also argued that non-Abelian states beyond quantum doubles can be obtained, such as the doubled Ising anyon theory, although we have left an explicit prescription of a circuit to future work.
Another feature of our method is that it can be performed in an arbitrary region, producing a duality defect on its boundary.We expect this defect to be topological [135][136][137].It might even be natural to consider moving it by measurements?
It is also interesting to note the similarities to quantum teleportation [138] and measurement-based quantum computation (MBQC) [139][140][141][142], where measurement effectively performs unitary operations on the input state.Here, the act of measurement instead performs a non-local transformation on the initial state.It would be interesting to make contact with similar notions of "computational phases of matter" in MBQC [143][144][145][146][147] .Exploring connections to the topological bootstrap [148] is also a promising future direction.
It may also be interesting to "soften" the projectors, considering either weak measurements or an open system weakly interacting with the environment by a subset of its degrees of freedom.
Note added -While finalizing the preprint of the present manuscript, Ref. [149] appeared, which overlaps with our section of KW duality.Moreover, after our preprint appeared, we learned of a parallel work preparing quantum double topological order via measurements [150].Our results agree with both of these works where they intersect.
Based on this finding, we show how to prepare the Wen plaquette model via measuring an appropriate cluster state.The A and B are the vertices of the square (red) and dual square (blue) sublattices, respectively.We create the cluster state given by the stabilizers Note that because of the couplings within the A sublattice, this cluster state is not bipartite.Now, let us measure the X operators on the A sublattice.The local product of stabilizers that commute with the measurements is and the non-local products are X along each x and y lines.
Thus, with measurement outcomes X = (−1) sv we have the stabilizers which, up to single site rotations, are the stabilizers of the Wen plaquette model.
Although the Wen plaquette model is in the same topological phase as the toric code, it has the advantage of treating the e and m anyons on equal footing.In particular, it naturally has a dislocation defect which permutes the e and m anyons that encircles the defect [153].In other words, the dislocation hosts a Majorana zero mode.Consider the cluster state given by the graph which features a dislocation on the B sublattice (dotted lines).Here the black lines connect AB sites, while the red lines connect AA sites.Performing measurements on the A sublattice, the stabilizers for each plaquette on the blue sites are given by It is argued that the three-Fermion Walker-Wang (3FWW) model [154] cannot be created from a circuit; it requires a quantum cellular automaton [155].Here, we argue that we can alternatively create this state by measuring an appropriate 3D cluster state.The preparation of such a state can prove useful for measurement-based quantum computation using such Walker-Wang models [156] by effectively evolving the two-dimensional topological order on the boundary using measurements [157,158].
The 3FWW model can be obtained by gauging a Z 2 2 1-form SPT [159].The response of this SPT to background Z 2 2-form gauge fields B 1 and B 2 is given by B 2 1 + B 2 2 + B 1 B 2 .The physical interpretation of the three terms is that they statistically transmute the anyons on the boundary to become that of fermions.Conveniently, the above SPT phase is itself a cluster state.Therefore, combining with the cluster state that implements the KW duality on each sublattice, the cluster state we would like to perform measurements on to obtain the 3FWW is a Z 4 2 1-form SPT.Its response to background gauge fields B i for i = 1, 2, 3, 4 is B 2 1 + B 2 2 + B 1 B 2 + B 1 B 4 + B 2 B 3 .The 3FWW is obtained by measuring the 1 and 2 sublattices.
Because it is a 1-form SPT, we define the cluster state on the edges of a cubic lattice, with four qubits placed per edge (i.e. 12 sites per unit cell).It is convenient to describe the cluster state using polynomials [160], which denote the connectivity of this cluster state.
Therefore, our desired cluster state is the +1 eigenstate of the stabilizers

FIG. 1 .
FIG.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.

FIG. 4 .
FIG.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 Z2 × Z2 symmetry of π-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 iπS z n -charge on every site, the remaining state has Fisher information F ∼ 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 Z2 symmetry of the Haldane SPT phase creates a cat state for the remaining Z2 symmetry, even if one is not at a fine-tuned fixed-point limit.
SPT, then in this case, we identify A p = (A (1) 1 , A
4. Analytics: Cat state from the deformed cluster state and AKLT state Let us illustrate our general theorem with two MPSbased examples.Both examples will be SPT phases with nonzero correlation length, i.e., away from the simple fixed-point cases studied in the earlier sections of this work.