Entanglement Transition in the Projective Transverse Field Ising Model
EEntanglement Transition in the Projective Transverse Field Ising Model
Nicolai Lang ∗ and Hans Peter B¨uchler Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology,University of Stuttgart, 70550 Stuttgart, Germany (Dated: June 18, 2020)Discrete quantum trajectories of systems under random unitary gates and projective measurementshave been shown to feature transitions in the entanglement scaling that are not encoded in thedensity matrix. In this paper, we study the projective transverse field Ising model , a stochasticmodel with two non-commuting projective measurements and no unitary dynamics. We numericallydemonstrate that their competition drives an entanglement transition between two distinct steadystates that both exhibit area law entanglement, and introduce a classical but non-local model thatcaptures the entanglement dynamics completely. Exploiting a map to bond percolation, we arguethat the critical system in one dimension is described by a conformal field theory, and derive theuniversal scaling of the entanglement entropy and the critical exponent for the scaling of the mutualinformation of two spins exactly. We conclude with an interpretation of the entanglement transitionin the context of quantum error correction.
I. INTRODUCTION
Entanglement has emerged as a powerful tool to charac-terize states of matter, such as ground states of quantumsystems [1], and access their topological properties [2, 3],but also to distinguish between generic thermal statesand the phenomenon of many-body localization [4]. Re-cently, a unique transition in random quantum circuitshas been identified where the entanglement entropy ofthe wave function is the key observable that character-izes two different steady states [5–9]. This entanglementtransition is driven by the competition between randomunitary operations and projective measurements appliedat discrete time steps on the wave function. Remarkably,the density matrix of both steady states is maximallymixed, and the transition is only visible in the average ofparticular properties of wave functions over ensembles ofquantum trajectories [10]. In this paper, we study a newtype of entanglement transition of quantum circuits thatis driven by random projective measurements only.Quantum circuits are an example of quantum dynami-cal maps where quantum operations are applied on qubitsat discrete time steps. Random unitary operations be-tween neighboring qubits spread entanglement and, ifthis is the dominating process, entail a volume law forthe entanglement entropy [11–13]. By contrast, randomprojective measurement of local observables remove en-tanglement from the system, and eventually lead to wavefunctions with area law entanglement. It is the compe-tition between these two processes that gives rise to theentanglement transition at a finite critical rate of the twoprocesses [5–9]. While the numerical simulation of genericquantum circuits is a computationally hard problem, itwas noticed that unitary gates restricted to the Cliffordgroup allow the time evolution of the quantum circuit tobe studied numerically even for large systems in the stabi-lizer formalism [14–17] (the formal statement is referred ∗ [email protected] to as Gottesmann-Knill theorem ). This approach allowedfor the precise characterization of the critical propertiesof the phase transition [18]. But also analytical methodshave been contrived to unravel the nature of the transi-tion [9, 19], such as descriptions in terms of conformalfield theories [20]. However, it is well established thatphase transitions with conventional symmetry breaking,characterized by an order parameter, as well as topolog-ical order in non-equilibrium steady states, can appearin driven quantum systems with competing dissipativeprocesses [21, 22]. This motivates the question whetherentanglement transitions can appear in random quantumcircuits with projective measurements only. A necessaryingredient are certainly non-commuting, competing mea-surements.In this paper, we present a detailed study of an entangle-ment transitions between two steady states (characterizedby quantum jump trajectories of wave functions) thatboth feature area law entanglement. The quantum cir-cuit is constructed from two non-commuting projectivemeasurements that are applied randomly; this model canbe viewed as the natural translation of the transversefield Ising model into a circuit of projective measurementsonly, and is therefore referred to as projective transversefield Ising model (PTIM). Also in this particular case,the transition is only visible in the average of certainentanglement measures of wave functions over ensemblesof quantum trajectories. While the quantum circuit canbe studied numerically using the stabilizer framework,we demonstrate that the entanglement dynamics can bemapped onto a simpler, classical model which can be sim-ulated more efficiently. In addition, this mapping providesa simple intuition for the spreading of entanglement inour model. We find that the transition exhibits a behav-ior similar to conventional second order quantum phasetransitions, where at the critical point long-range order isestablished. Here, the role of long-range order is playedby a finite mutual information between two separatedspins. Furthermore, the entanglement entropy divergeslogarithmically at the phase transition. These quantities a r X i v : . [ c ond - m a t . d i s - nn ] J un unveil two universal properties of the transition: Theprefactor ˜ c of the logarithmically diverging entanglemententropy, and the critical exponent κ of the algebraic de-cay of the mutual information. We demonstrate thatthe critical point is described by bond percolation anddetermine ˜ c = 3 √ / (2 π ) and κ = 2 / c is not the conformalcharge of the underlying conformal field theory, which onewould expect for ground states of critical one-dimensionalsystems [23, 24].The entanglement transition studied in this paper istightly related to quantum error correction of topolog-ically protected qubits encoded in the ground state ofa Majorana chain [25, 26]: The two projective measure-ments can be interpreted as syndrome measurements andlocal errors, respectively. In this context, the steady statecharacterized by finite mutual information correspondsto the regime where the quantum information of the codespace is preserved. By contrast, in the error-dominatedregime the quantum information is lost and the mutualinformation vanishes. While in the context of (active)quantum error correction it is well-established that suchphase transitions exist, it is remarkable that an entangle-ment transition appears even if the syndrome measure-ments are not recorded and no active error correctiontakes place. The encoded information is still contained inthe wave function of the quantum trajectory but it wouldrequire an exponential number of measurements to ex-tract this information. Thus, the entanglement transitionis in general hidden from our experimental observations.Finally, we would like to point out that during finaliz-ing our manuscript, we became aware of two recent andclosely related studies [27, 28] where also an entanglementtransition between different steady states with area lawentanglement was observed.The paper is organized as follows: We introduce thequantum circuit with two non-commuting projective mea-surements in Section II. Although this model can beefficiently simulated as a stabilizer circuit, we presentan exact mapping to a simpler, classical model whichdescribes the entanglement dynamics of the system com-pletely. The details of this mapping are presented inSection III, and the relevant observables are discussed inSection IV. In Section V we present numerical results:We focus on the one-dimensional chain, demonstrate thedivergence of the entanglement entropy at the criticalpoint and determine the prefactor ˜ c of this logarithmicdivergence. Then we discuss the mutual information asindicator of long-range entanglement and show that it ex-hibits the characteristic behavior of a second-order phasetransition. We determine its critical exponent κ and pro-vide an intuitive interpretation of the transition in termsof Bell clusters. The mapping to percolation is shown inSection VI which allows us to determine the position ofthe entanglement transition exactly. We conclude thissection with a comparison of numerically determined crit-ical points of the entanglement transition for different Figure 1.
Projective time evolution.
A few typical time stepsfor p ≈ . L = 10 spins with periodic bound-aries. Blue boxes on edges denote measurements M zze onadjacent spins e = ( i, i + 1), red circles measurements M xi ona single spin i . Each time step comprises one row of M zze -measurements followed by a row of M xi -measurements. Notethat the order of M zze -measurements does not affect the dy-namics as their projectors commute. The system is initializedin the product state | + · · · + (cid:105) . lattices in two-dimensions with known results for bondpercolation. In Section VII, we exploit the equivalence ofour model and bond percolation to determine the confor-mal field theory that describes the critical point of theone-dimensional system and derive ˜ c and κ analytically.We conclude with a discussion of the relation to quantumerror correction in Section VIII. II. THE MODEL
We start with a detailed description of our model. Con-sider a one-dimensional chain of spin-1 / i ∈ V L = { , . . . , L } . Each spin is repre-sented by Pauli matrices σ αi , α ∈ { x, y, z } , and the Hilbertspace is denoted as H = (cid:78) i C i . The quantum circuit isdefined by projective measurements of observables O , andthe action of such a measurement is denoted as M [ O ],i.e., M [ O ]( | Ψ (cid:105) ) = P λ | Ψ (cid:105) (cid:112) (cid:104) Ψ | P λ | Ψ (cid:105) (1)is the new state after measurement of the discrete eigen-value λ of O with probability Pr( λ ) = (cid:104) Ψ | P λ | Ψ (cid:105) ; P λ denotes the projector onto the corresponding eigenspace.Note that M [ O ]( | Ψ (cid:105) ) is a random variable with values in H that is parametrized by the input | Ψ (cid:105) ; M [ O ]( • ) is nota linear operator (hence the parentheses).Throughout this paper, we are interested in measure-ments of the observables σ xi and σ zi σ zi +1 , i.e., M xi ≡ M [ σ xi ] with P λ = 12 ( + λ σ xi ) (2a) M zze ≡ M [ σ zi σ zi +1 ] with P λ = 12 (cid:0) + λ σ zi σ zj (cid:1) (2b)for each site i and edge e = ( i, i + 1) between adjacentsites. The measurement results are λ ∈ {− , +1 } . Itis important to point out that σ xi and σ zi σ zi +1 do notcommute if they involve the same site so that repeatedmeasurements lead to a non-trivial quantum dynamics.This quantum dynamics is described as a stochasticprocess on H generated by the measurements (2). Incontrast to previous studies on entanglement transitions,we do not apply additional unitary operations. We startwith the initial product state | Ψ(0) (cid:105) = | + + · · · + (cid:105) (3)with |±(cid:105) = ( | (cid:105) ± | (cid:105) ) / √
2. Then, we evolve the systemiteratively as follows (Fig. 1): In each time step, we set thesite variable x i = 1 with probability p ( x i = 0 otherwise),and – independently – for each edge e = ( i, i + 1), we set z e = 1 with probability 1 − p and again z e = 0 otherwise.The vectors x = ( x i ) and z = ( z e ) determine the sites(edges) on which the observables σ xi ( σ zi σ zi +1 ) will bemeasured. Given the state | Ψ( t ) (cid:105) at time t , the new wavefunction at t + 1 is given by (more precisely: drawn fromthe distribution) | Ψ( t + 1) (cid:105) = M x x ◦ M zz z ( | Ψ( t ) (cid:105) ) (4)with measurements M x x = (cid:89) i : x i =1 M xi and M zz z = (cid:89) e : z e =1 M zze . (5)This defines a fully projective time evolution that yieldsa single quantum trajectory | Ψ( t ) (cid:105) at discrete times t =0 , , , . . . .We are interested in characteristic properties of suchwave functions along a quantum trajectory for a giventime t . Denote a generic quantity as X ( | Ψ( t ) (cid:105) ) with X : H → C . Examples are conventional observables suchas correlations X = (cid:104) Ψ | σ zi σ zj | Ψ (cid:105) , but also the entangle-ment entropy X = S ( A ) of a subsystem A ⊂ V L . Thesequantities are then averaged over many different quantumtrajectories, defining the sample averages X ≡ M (cid:88) Ψ ∈N X ( | Ψ( t ) (cid:105) ) . (6)Here, N = {| Ψ( • ) (cid:105)} denotes an ensemble of M randomlygenerated quantum trajectories.For fixed time t and M → ∞ , the above process definesa classical probability distribution P ( t ) on H . Assum-ing that there exists a stationary limit, we define the projective transverse field Ising model (PTIM) as being characterized by P ∞ = lim t →∞ P ( t ). Here, we are inter-ested in properties of P ∞ in dependence of the relativestrength of non-commuting measurements p ∈ [0 , U = (cid:81) i ∈ V L σ xi . In particular, the initial state (3) is U -invariant; this property is conserved under the projectivedynamics and restricts the accessible part of the Hilbertspace. As a consequence, the density matrix describingthe steady state of the PTIM is only maximally mixedup to this symmetry constraint. III. THE COLORED CLUSTER MODEL
To study the properties of the PTIM (projective trans-verse field Ising model), we start with a discussion of thenumerical approach that we use to generate samples N ofquantum trajectories | Ψ( t ) (cid:105) . It is important to point outthat measurements of Pauli operators can be describedin the stabilizer formalism [14, 15] (this is also true forthe initial state (3)); the projective time evolution of thequantum trajectories can therefore be efficiently simu-lated on a classical computer – despite the exponentiallygrowing dimension of H [16, 17]. Although simulations inthe stabilizer formalism are reasonably efficient, genericand well-understood to bootstrap trustworthy results, itis not the most efficient approach to study the PTIM.The numerical approach we leverage in this paper isbased on an equivalent classical process that can be sam-pled more efficiently. This process turns out to be non-local and, in addition, provides an intuitive picture of themechanism that drives the PTIM phase transition (seebelow). The derivation of this process exploits the spe-cial structure of the PTIM and is based on the followingobservations: • Measuring σ z σ z in the product state (we omit nor-malizing factors) | ++ (cid:105) = | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) (7)yields the entangled Bell pairs | (cid:105) + | (cid:105) or | (cid:105) + | (cid:105) with each 50% probability. We refer to bothstates as a 2-qubit Bell cluster (as they are equiva-lent under local unitary operations, they are identi-cal from an entanglement point of view). • Measuring σ z σ z in the product state( α | (cid:105) + β | (cid:105) ) ⊗ | + (cid:105) = α ( | (cid:105) + | (cid:105) ) + β ( | (cid:105) + | (cid:105) ) (8)yields the entangled states α | (cid:105) + β | (cid:105) or α | (cid:105) + β | (cid:105) with each 50% probability. Theresult is therefore an enlarged (3-qubit) Bell cluster.Note that the amplitudes survive. • Measuring σ x in the 3-qubit Bell cluster α | (cid:105) + β | (cid:105) = α ( | +00 (cid:105) + |− (cid:105) ) + β ( | +11 (cid:105) − |− (cid:105) ) (9) Figure 2.
Colored cluster model.
Single time step of thecolored cluster model, split into two substeps: (1) the appli-cation of Π zz z on edges (crosses on faces in spacetime) thatmerges/grows/nucleates clusters and (2) Π x x on sites (crosseson vertical edges in spacetime) that erodes clusters. Sites withdashed boundary represent the intermediate state between t and t + 1, gray sites denote single-site clusters. yields either | + (cid:105) ⊗ ( α | (cid:105) + β | (cid:105) ) or |−(cid:105) ⊗ ( α | (cid:105) − α | (cid:105) ) with 50% probability. The result is thereforea shrunk (2-qubit) Bell cluster. Note that again theamplitudes survive (up to a sign that depends onthe measurement outcome). • Measuring σ z σ z in a state with two 2-qubit Bellclusters( | (cid:105) + | (cid:105) ) ⊗ ( α | (cid:105) + β | (cid:105) )= α | (cid:105) + β | (cid:105) + α | (cid:105) + β | (cid:105) (10)yields the entangled states α | (cid:105) + β | (cid:105) or α | (cid:105) + β | (cid:105) with each 50% probability. Theresult is therefore a merged 4-qubit Bell cluster.Again the amplitudes survive.We conclude that the dynamics of the PTIM is essen-tially characterized by the nucleation, growth, decay andmerging of Bell clusters while phase coherence is preserved.This cluster dynamics does not depend on the specificrealization of Bell clusters (i.e., their amplitudes and theirspin patterns), for instance, |− (cid:105) + |− (cid:105)| +000 (cid:105) − | +111 (cid:105)|− (cid:105) + |− (cid:105)| +011 (cid:105) + | +100 (cid:105) (11)all give rise to the same entanglement dynamics and rep-resent the same entanglement structure. This motivatesthe following classical stochastic process (Fig. 2): • The states of the system are vectors s ∈ N L , so thatthe state of each site i is described by a non-negativeinteger s i ∈ N ; s i = 0 encodes that site i is in aproduct state and unentangled with the rest of thesystem. s i = n > n . Forexample, all four states in (11) can be describedcollectively by s = (0 , , ,
1) where n = 1 is thelabel of the only (3-qubit) Bell cluster. • The initial state of the process is s (0) = (0 , . . . , | Ψ(0) (cid:105) = | + · · · + (cid:105) in (3) (note that e.g. | Ψ(0) (cid:105) = |− + − − · · · + (cid:105) would not alter theentanglement dynamics and therefore correspondsto the same state s (0)). • Instead of measurements, the transformation s ( t + 1) = Π x x ◦ Π zz z ( s ( t )) (12)is applied iteratively withΠ x x = (cid:89) i : x i =1 Π xi and Π zz z = (cid:89) e : z e =1 Π zze . (13)The function Π xi acts locally and is defined by s (cid:48) =Π xi ( s ) with s (cid:48) j = s j for all j (cid:54) = i and s (cid:48) i = 0. Bycontrast, the function Π zze for e = ( i, j ) acts non-locally and is defined via s (cid:48) = Π zze ( s ) as follows(sites that are not mentioned remain unchanged): – Case 1: s i = 0 and s j = 0; then s (cid:48) i :=next( s ) =: s (cid:48) j . Here, next( s ) = min( n ∈ N \ s )returns the smallest integer that is not usedas a cluster label in s . This process creates anew, independent cluster with two spins. – Case 2a: s i (cid:54) = 0 and s j = 0; then s (cid:48) j := s i .This process joins site j to the cluster of site i . – Case 2b: s i = 0 and s j (cid:54) = 0; then s (cid:48) i := s j .This process joins site i to the cluster of site j . – Case 3: s i (cid:54) = 0 and s j (cid:54) = 0. Let s = min( s i , s j );then set s l := s for all sites l with s l = s i or s l = s j . This process merges two clusters andthereby reduces the number of independentclusters by one without reducing the numberof spins that belong to clusters.The last case defines a non-local transformation – aconsequence of the quantumness of the PTIM wherethe non-locality of clusters is naturally realized byentanglement.Note that one can interpret the PTIM as a local quan-tum simulator for this non-local classical process. In thefollowing, we color the sites i according to their state s i and refer to this model as the colored cluster model(CCM) . It is this simpler but equivalent model that weevolve and sample numerically. We also cross-checkedour results numerically for the PTIM using the stabilizerformalism [14–17]. IV. ENTANGLEMENT MEASURES
By simulating the CCM (colored cluster model), welose access to some properties of the PTIM. In particu-lar, conventional expectation values of the wave functionalong the quantum trajectory are no longer accessible.However, the entanglement transition cannot be detectedby observables anyway as the density matrix in the steadystate is maximally mixed (up to symmetry constraints).Indeed, the appropriate quantities that characterize theentanglement transition are the entanglement entropy andthe mutual information – both of which can be efficientlycomputed using the CCM.The entanglement entropy of a subsystem A ⊂ V L fora wave function along a quantum trajectory is defined as S ( A ) ≡ − Tr [ ρ A log ρ A ] (14)with ρ A = Tr V L \ A [ ρ ] the reduced density matrix of thesubsystem. In terms of Bell clusters, S ( A ) counts thenumber of clusters with support both in A and A ≡ V L \ A ;a quantity that can be easily computed from CCM states s ( t ). Note that here we define the entanglement entropywith the binary logarithm log such that each Bell clustercontributes 1 instead of ln 2.The actual quantity of interest is the entanglemententropy S ( A ) averaged over many, randomly sampledquantum trajectories. Therefore we denote by S ( A ) thesample-averaged entanglement entropy as defined in (6).We also define S L ( l ) as the (sample-averaged) entangle-ment entropy of l contiguous spins in the center of a chainwith L sites, i.e., A comprises the l sites in the interval[ L/ − l/ , . . . , L/ l/ A, B ⊂ V L . The mutual information I ( A, B ) between A and B is definedas I ( A, B ) ≡ S ( A ) + S ( B ) − S ( A ∪ B ) . (15)A non-vanishing value of I ( A, B ) indicates entanglementbetween the subsystems A and B for wave functionsalong a quantum trajectory [29, 30]. Here we are mainlyinterested in the mutual information between two spinsat sites i, j ∈ V L , that is I ( i, j ) ≡ S ( { i } ) + S ( { j } ) − S ( { i, j } ) . (16)For a state | Ψ (cid:105) along a quantum trajectory of the PTIM, I ( i, j ) is a L × L matrix that encodes the structure ofBell clusters in the system completely. For example, | Ψ (cid:105) = | , , , (cid:105) but also | Ψ (cid:105) = ( | , (cid:105) + | , (cid:105) ) ⊗ ( | , (cid:105) + | , (cid:105) ) yields I (1 ,
4) = 0 since spins 1 and 4are not part of a common Bell cluster. By contrast, | Ψ (cid:105) = ( | , (cid:105) + | , (cid:105) ) ⊗ ( | , (cid:105) + | , (cid:105) ) yields I (1 ,
4) =2 and for | Ψ (cid:105) = | , , , (cid:105) + | , , , (cid:105) we have I (1 ,
4) = 1; in both cases, spins 1 and 4 belong the sameBell cluster. If we consider I ( i, j ) as adjacency matrix ofa graph, the connected components of this graph are inone-to-one correspondence with the Bell clusters of | Ψ (cid:105) .In the language of the CCM, it is I ( i, j ) = 0 for sites ofdifferent color s i (cid:54) = s j (or s i = 0 = s j ) and I ( i, j ) = 1for sites of the same color s i = s j (cid:54) = 0; for clusters thatcontain only the two sites i and j , it is I ( i, j ) = 2. V. NUMERICAL RESULTS
For the following results, we sampled typically M ∼ − trajectories for sufficiently long time t ∼ × M u t ua l i n f o r m a t i on Rate E n t ang l e m en t en t r op y PBC
Figure 3.
Phase transition.
Numerical results for the entan-glement entropy S L ( L/
2) (red) and the mutual information I (1 , L/
2) (blue) as functions of the rate p for systemsof length L = 100 , ,
500 (circles, squares, diamonds) withperiodic boundary conditions (PBC). At the critical point p = 0 . p c , the entanglement entropy grows logarithmicallywith the system size (black markers). Each point is based on10 sampled trajectories. such that all quantities of interest reached their equilib-rium values in the steady state. The quantities of interestare the sample-averaged entanglement entropy S L ( l ) andthe mutual information I ( i, j ). A. Entanglement Entropy
In Fig. 3 we show the entanglement entropy S L ( L/ p for chains of lengths L = 100 , , p → M zze ), theentanglement entropy saturates at 1 since the systemapproaches a global Bell cluster | m (cid:105) + | m (cid:105) where | m (cid:105) ≡U | m (cid:105) with m ∈ Z L a random spin-pattern in the z -basis; here again U = (cid:81) i σ xi denotes the global symmetrywhich is conserved along the quantum trajectory. For p → M xi ), the system approaches the unentangledproduct state | + · · · + (cid:105) so that S L ( L/
2) vanishes smoothly.For 0 < p < p c ≈ . S L ( L/
2) for L → ∞ . Below, we willdemonstrate analytically that the transition indeed takesplace at the critical point p c = 0 .
5. The two regimes canbe understood intuitively in the context of Bell clusters ifwe recall that S L ( L/
2) counts the number of independentBell clusters with support in both halves of the system: • For p (cid:29) p c , the projections onto |±(cid:105) dominate andmake the clusters decay rapidly; they cannot growto extensive size and cross the two boundaries ofthe subsystem rarely. • For p ≈ p c , nucleation and growth of clusters onone side, and annihilation and decay on the otherside are balanced. Clusters become deconfined andspread throughout the system. Typically, two in-dependent clusters connect the two halves of thesystem: one located at each of the two boundariesof the subsystem. In rare cases, additional, inde-pendent clusters contribute entanglement, so that S L ( L/ (cid:38) • For p (cid:28) p c , the growth of clusters dominates. Sincethe probability that two independent clusters mergegrows exponentially with their surface, the proba-bility for two or more extensive clusters vanishesexponentially. This is a condensation mechanismwhere newly created clusters (“condensation nuclei”)quickly get absorbed by the macroscopic cluster (the“condensate”). This explains why S L ( L/ → p → p →
0. For p →
1, thisis due to the destructive force of the M xi -measurementsthat dissolve the clusters. For p →
0, the clusters donot dissolve but get absorbed by the condensate and thismechanism becomes more efficient when the density ofthe condensate increases.In Fig. 4(a) we show the behavior of S L ( l ) for 0 ≤ l ≤ L for different parameters p . As expected, we observe thatfor p (cid:54) = p c the entanglement entropy saturates quickly,indicating area law entanglement. However, this behavioris modified at the critical point by a logarithmic contri-bution. The slow divergence of S L ( l ) at criticality for L → ∞ with l/L = const (recall Fig. 3), and for l → ∞ with l (cid:28) L , is a well-known feature of critical systems inone dimension that can be described by a conformal fieldtheory: The scaling law for ground states of conformallyinvariant systems with periodic boundaries at the criticalpoint is asymptotically described by [23, 24] S L ( l ) ∼ c (cid:20) Lπ sin (cid:18) π lL (cid:19)(cid:21) l (cid:28) L ≈ c ( l ) (17)with the central charge c (up to a non-universal constant).In the following, we analyze whether the critical entan-glement scaling of the PTIM exhibits the same behavior.To this end, we consider the normalized entanglemententropy∆ S ( l/L ) ≡ S L ( l ) − S L ( L/ ∼ ˜ c (cid:20) sin (cid:18) π lL (cid:19)(cid:21) (18)that is expected to be independent of the system size L .In Fig. 5 we plot this quantity for system sizes L =100 , . . . ,
300 at criticality. The collapse of data for differ-ent L is remarkable, and fitting Eq. (18) to our numericaldata, we find the prefactor ˜ c ≈ .
57. This observationsuggest that the critical properties of the PTIM are de-scribed by a conformal field theory. Indeed, we will arguebelow that the critical properties are determined by aconformal field theory with central charge c = 0, whilethe prefactor of the entanglement entropy takes the exactvalue ˜ c = 3 √ / (2 π ) ≈ . -4 -3 -2 -1 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 PBC PBC (a) (b)(c) (d) P r obab ili t y den s i t y OBCOBC OBCOBC OBCOBC P r obab ili t y den s i t y Relative weight Relative diameter
Weight distribution Diameter distribution M u t ua l i n f o r m a t i on Length Distance E n t ang l e m en t en t r op y Figure 4.
Entanglement structure.
Various quantities char-acterizing the entanglement structure of a chain with length L = 500 above, at and below the critical point; all resultsare based on 10 sampled trajectories: (a) Entanglemententropy S L ( l ) as a function of the length l/L of the sub-system for a chain with periodic boundary conditions. Thesystem obeys an area law in both phases, p = 0 . < p c and p = 0 . > p c , with a logarithmic contribution at the criticalpoint p = 0 . p c . (b) Mutual information I ( i, j ) = I ( | i − j | )as a function of the distance | i − j | /L for a chain with peri-odic boundary conditions. I ( | i − j | ) vanishes exponentiallyfor p > p c and saturates at a finite value for p < p c . At thecritical point p = p c , it is well described by the algebraic decay I ( | i − j | ) = α | i − j | − κ with fitted exponent κ ≈ .
66 and non-universal fit parameter α (dashed black line). (c) Probabilitydensity W ( n ) × L of the relative weight n/L of Bell clustersfor a chain with open boundary conditions. At the phasetransition, clusters of extensive weight emerge. (d) Probabilitydensity D ( n ) × L of the relative diameter n/L of Bell clustersfor a chain with open boundary conditions. At the phasetransition, clusters with diameters on all length scales exist. states of critical one-dimensional quantum systems, theprefactor of the entanglement entropy is not the centralcharge. B. Mutual Information
Next, we analyze the sample-averaged mutual informa-tion I ( i, j ) in the steady state. A finite value indicatesa finite probability to find a Bell cluster that encom-passes sites i and j . In Fig. 3, the mutual information I = I (1 , L/
2) as a function of p for a chain with peri- -0.6-0.4-0.20.00.0 0.1 0.2 0.3 0.4 0.5100150200250300Fit PBC E n t ang l e m en t en t r op y Length
Figure 5.
Critical scaling.
Entanglement entropy ∆ S ( l/L ) asa function of the subsystem size l/L for chains of differentlength L with periodic boundary conditions at the criticalpoint p = 0 . p c . The data collapse is almost perfect andthe prediction for critical systems of the form ∆ S ( l/L ) =˜ c/ ξ ( l/L ) with ξ ( x ) = sin( πx ) describes the dependencyremarkably well for the fit parameter ˜ c ≈ .
57 (dashed blackline). Each curve is based on 10 sampled trajectories. odic boundaries is shown. Clearly the system undergoesa continuous entanglement transition at the critical value p c ≈ . I > p < p c and I = 0 for p > p c .This is consistent with our interpretation above: Belowthe critical value, there is a “condensate” of Bell clus-ters, that is, a single, macroscopic cluster that permeatesthe whole system, creating entanglement between spinsthat are far apart. The limit I ∞ = lim | i − j |→∞ I ( i, j ) isa measure for the density of the macroscopic cluster andcontinuously converges to 1 for p →
0. This behavior isin close analogy to a conventional second-order quantumphase transition where, at the critical point, long-rangeorder is established. Similarly, at the entanglement tran-sition, the mutual information for distant spins attains afinite value.In Fig. 4(b) we show the behavior of I ( i, j ) for increas-ing distance between the two spins (note that I ( i, j ) = I ( | i − j | ) because of translation invariance). While for p > p c the mutual information I ( i, j ) vanishes exponen-tially with the distance, it saturates at a finite value for p < p c . However, at the critical point p c , it exhibits analgebraic decay with critical exponent κ , I ( i, j ) = I ( | i − j | ) ∼ α | i − j | κ , (19)with non-universal parameter α . From the numericalresults, we find the critical exponent κ ≈ .
66. Below, wedetermine this critical exponent by arguments based onthe mapping of the critical regime to a conformal fieldtheory and find the exact value κ = 2 / C. Distribution of Bell clusters
To quantify the emergence of a macroscopic cluster atthe entanglement transition, we define the diameter d ( B )of a Bell cluster B ⊂ V L as d ( B ) = max {| i − j | | i, j ∈ B } (20)for systems with open boundaries; this is just | i ← − i → | with the leftmost (rightmost) spin i ← ( i → ) that belongsto B . In addition, we define the weight | B | as the numberof spins that make up the cluster B . Let B Ψ denote theset of all Bell clusters of a given wave function | Ψ (cid:105) along aquantum trajectory (we count single spins as 1-qubit Bellclusters). The distribution D ( n ) with n = 0 , , . . . , L − D ( n ) = |{ B ∈ B Ψ | d ( B ) = n }| / |B Ψ | , (21)and similarly for the weight W ( n ) = |{ B ∈ B Ψ | | B | = n }| / |B Ψ | . (22)In Fig. 4(d), we show the diameter distribution D ( n ),i.e., the average over quantum trajectories of the distri-bution D ( n ) as a function of the relative diameter n/L .In Fig. 4(c), we show the averaged distribution W ( n ) asa function of the relative weight n/L . Note that at thephase transition, the distribution D ( n ) features a longtail (demonstrating the existence of clusters on all lengthscales) while for p < p c it becomes bimodal with con-siderable contributions for diameters n ∼ L , indicatingthe emergence of clusters that permeate the system. Bycontrast, the weight distribution W ( n ) evolves towards asaddle point at p c which then leads to a second maximumat weights n ∼ L/ n ∼ L for p →
0. These observations illustrate that the extensivecluster is sparse close to the critical point and grows indensity for p → z -polarizedspins | . . . ↑ . . . ↓ . . . (cid:105) in the initial state (3) (“ z -poisoning”)alters the entanglement dynamics dramatically. In partic-ular, for p → collapse Bell clusters for purely x -polarized initial states, z -polarized spins trigger an avalanche of cluster collapsesby M zze -measurements that (for small p ) quickly drivethe system into a product state. Interestingly, even with z -poisoning, there are still critical fluctuations at p c asindicated by a sharp peak of the entanglement entropy.We do not consider effects of z -poisoning in this paper. VI. PERCOLATION
The PTIM and the CCM (which captures the entan-glement properties of the PTIM) are intimately linked to (a) T i m e (b) Figure 6.
Percolation. (a) The same measurement patter as in Fig. 1. Here we mark horizontal edges with M zze -measurementsand vertical edges without M xi -measurements as active (bold black lines). Then, the probability for both horizontal and verticaledges to be active is 1 − p and the projective dynamics gives rise to isotropic bond percolation on a square lattice in spacetime.(b) A possible history for the CCM state in Fig. 2. Sites at time t have the same color if and only if they are connected viaactive edges in spacetime. bond percolation . The mapping is illustrated in Fig. 6(a)and described in the following: The discrete time stepsgive rise to a square lattice in spacetime where each timestep comprises one horizontal row of edges and all verticaledges that connect it to the next row. If we mark horizon-tal edges with z e = 1 and vertical edges with x i = 0 as“active” (bold black edges), the probability for activity isin both cases 1 − p . Thus every sequence of measurementson the spins on a one-dimensional PTIM is in one-to-one correspondence with a pattern of active bonds onits spacetime square lattice. Following our observationsthat led to the construction of the CCM, it is easy to seethat at a given time t , two spins i and j are entangled(belong to the same Bell cluster; in the CCM: have thesame color) if and only if the two sites are connected by apath of active edges on the spacetime lattice in the past,see Fig. 6(b). This observation immediately implies thatthe critical point for the entanglement transition of thePTIM coincides with the transition for bond percolation;on a square lattice, this transition takes place at p c = 0 . L where the measurements M xi act with probability p on vertices i ∈ V ( L ) and the measurements M zze withprobability 1 − p on edges e ∈ E ( L ). The correspondingCCM is then induced by bond percolation on the half-infinite stack of lattices L with vertical edges connectingvertices of adjacent layers (for instance, the PTIM onthe two-dimensional square lattice is described by bondpercolation on the 2+1-dimensional cubic lattice). Weestimated the critical values p c for square, Kagome, hon-eycomb, and triangular lattice from simulations with up Table I.
Critical values.
Estimates of critical values p c forvarious lattices. In 2D, we compare them with numericalvalues ˜ p c from Ref. 32 for three-dimensional bond percolationon stacks of the corresponding 2D lattices. Values markedwith ∗ are exact. Our estimates are based on lattices up to50 ×
50 spins.Dimension Lattice p c ˜ p c Percolation lattice . . ∗ Square .
75 0 . Cubic .
74 0 . stacked Kagome .
70 0 . stacked Honeycomb .
83 0 . stacked Triangular to 50 ×
50 spins. In Table I we compare these values withknown (numerical) results for the corresponding bondpercolation problems in three dimensions [32] and findreasonable agreement between them.Furthermore, we would like to briefly comment on themore generic case where M xi and M zze occur with inde-pendent probabilities p x and p z , respectively. Then theconnectivity on the spacetime lattice is determined by anisotropic bond percolation [33] with probability p ⊥ = p z for horizontal edges and p (cid:107) = 1 − p x for vertical edges.In two dimensions it can be shown by duality argumentsthat the system is critical for p ⊥ + p (cid:107) = 1, or, equivalently, p x = p z [31]. If we choose the parametrization p z = q and p x = rq , the entanglement transition occurs for r c = 1and is independent of q (which quantifies the overall mea-surement rate). We verified this numerically and found nodependence of the entanglement transition of the PTIMin one dimension on the measurement rate q . However, inhigher dimensions, the relation between the critical values p ⊥ ,c and p (cid:107) ,c is no longer linear [33]. As a consequence,we expect the critical ratio r c = p xc /p zc for the PTIM onthe square lattice to depend on the measurement rate q . We checked this numerically and indeed found a shifttowards larger ratios r c in the limit q → q → M xi - and M zze -measurements with rate parameters λ x /λ z = r . Then,the order of measurements becomes irrelevant – in con-trast to the PTIM where in each time step we first apply M zze and subsequently M xi , cf. Eq. (4). In this limit, theprocess belongs to the family of continuum random clus-ter models [34–37] in d + 1 dimensions which are knownto describe quantum Q -state Potts models [36–38] in d dimensions. In particular, percolation is described by the Q → VII. CONFORMAL FIELD THEORY AT THECRITICAL POINT
The close relation between the PTIM in one dimensionand bond percolation on a two-dimensional square latticeallows us to derive the conformal field theory describingthe critical point of the entanglement transition. First,note that bond percolation on the square lattice is thesimplest random cluster model with cluster weight Q = 1(more generally, cluster models are equivalent to classical Q -state Potts models [35]). Planar random cluster mod-els can be mapped to 6-vertex models [39] which, at thecritical point, have an equivalent description as a densegas of oriented loops with weight √ Q [40]. Interpretingthe oriented loops as contour lines of a discrete “height”field φ ( x ) ∈ π Z establishes an equivalent description interms of a solid-on-solid (SOS) model [41, 42]. At largedistances and after coarse-graining, the height field canbe approximated by a continuous field Φ( x ) ∈ R andthe solid-on-solid model renormalizes to a Gaussian fixedpoint with coupling g = 1 − e where √ Q = 2 cos( πe ) [43–45]. If defined on a cylinder (corresponding to a periodicPTIM in one dimension), the correct weighting of non-contractible loops makes it necessary to put charges ± e on the two boundaries of the cylinder [39, 45, 46] by in-serting vertex operators V ± = exp( ± ie φ ). This modifiesthe vacuum energy on the cylinder and shifts the cen-tral charge to c = 1 − e / (1 − e ) [45]. For percolation,we have e = 1 / c = 0; therefore, the prefactor ˜ c for the entanglemententropy (18) must play another role.In the following, we derive the prefactor ˜ c for the asymp-totic behavior of the entanglement entropy in the confor-mal field theory. The approach is motivated by recentresults on the valence bond entanglement entropy in theground state of an antiferromagnetic spin chain [47]. Westart by considering a half-infinite cylinder by shiftingthe lower boundary to infinity. Such a half cylinder is (a) (b)(c) SimulationCFT
PBC E n t ang l e m en t en t r op y M u t ua l i n f o r m a t i on Length Distance/ -1.0-0.8-0.6-0.4-0.20.0 0.01 0.03 0.05 0.1 0.3 0.50.020.050.100.200.30
Figure 7.
Conformal field theory. (a,b) Field configura-tions on the half-infinite cylinder mapped to the complexplane. Vertex operator insertions are indicated by ⊗ . (a) Scal-ing of the entanglement entropy S ( A ). Field configurationswith contours that connect A (red segment) with the envi-ronment (black boundary) contribute to the entanglementbetween them (light red domains). (b) Scaling of the mu-tual information I ( x , x ). Field configurations that connectthe points x and x (light blue domain) contribute to themutual information between them. (c) Comparison of nu-merical results (squares and bullets) and theoretical predic-tions (solid lines) for the PTIM at criticality. The simula-tions are based on 10 trajectories for a system of length L = 500 with periodic boundary conditions. The CFT pre-dictions are of the form ∆ S ( l/L ) = ˜ c/ ξ ( l/L ) + α and I ( i, j ) = I ( | i − j | ) = β ξ − κ ( | i − j | ) + γ with ˜ c and κ asgiven in the text and α , β , γ non-universal fit parameters; ξ ( x ) = sin( πx ) accounts for the finite size and the periodicboundaries. Note that the plot for ∆ S is logarithmic on the l/L -axis; the plot for I is logarithmic on both axes. conveniently mapped to the complex plane, which leavesus with a disc where V − is inserted at the origin and V somewhere on the boundary. We follow now thelines of Ref. 47 and split V into two vertex operators V e ± = exp[ i ( ± e + e / h = e − ( e / − e (23)and e ∈ R a free parameter, see Fig. 7(a). The pairof vertex operators V e ± , inserted at x and x on theboundary, modifies the weight of loops that connect theboundary segment A = [ x , x ] of length ∆ x = | x − x | with the rest of the boundary A . Therefore, thecorrelation function of the vertex operators can be written0as V A ( w ) = (cid:104) V e + ( x ) V e − ( x ) (cid:105) = (cid:80) ˜ w ˜ N w N A (cid:80) ˜ w ˜ N + N A ∼ x h (24)where the sums go over all allowed loop configurations. N A is the number of loops connecting A and A , whereas ˜ N counts the loops attached with both ends either to A or to A . The weights are ˜ w = 2 cos( πe /
2) and w = 2 cos( πe ).Relation (24) allows us to derive the entanglemententropy S ( A ) of segment A , see Fig. 7(a). Each inde-pendent Bell cluster of the PTIM that lives both in A and A increases S ( A ) by one. In the picture of discretebond percolation, such Bell clusters derive from clusters ofedges in spacetime that connect A with A . Since the loopsof the continuum model essentially describe the bound-aries of these clusters, we conclude that S ( A ) ∼ (cid:104) N A (cid:105) / (cid:104) N A (cid:105) derives from (24) bythe relation (cid:104) N A (cid:105) = ˜ w [ ∂ w V A ( w )] w = ˜ w . Given the scalingdimension h (23) for the vertex operators, we find thelogarithmic divergence of the entanglement entropy S ( A ) ∼ (cid:104) N A (cid:105) ∼ √ π log ∆ x . (25)A comparison with Eq. (17) implies the exact value ofthe prefactor for the entanglement entropy at the criticalpoint ˜ c = 3 √ π . (26)It matches the numerical results in Fig. 7(c) remarkablywell.Next, we focus on the mutual information I ( x , x )between the two boundary points x and x of A ,see Fig. 7(b). For I ( x , x ) = 1, a percolation cluster thatconnects the sites x and x is required. This precludesclusters connecting the interior of A with the interior of A .We therefore argue that I ( x , x ) ∼ (cid:104) δ N A =0 (cid:105) . Note thatwe expect I ( x , x ) = 2 (indicating monogamous entan-glement between the two sites) to be irrelevant in the con-tinuum limit. Using (24), we find (cid:104) δ N A =0 (cid:105) = V A ( w → w = 2 cos( πe ) = 0, we choose e = 1 / h = 1 / I ( x , x ) ∼ (cid:104) δ N A =0 (cid:105) ∼ x ) κ (27)with κ = 2 h = 2 /
3, again consistent with numericalresults to a remarkable degree, see Fig. 7(c).
VIII. RELATION TO QUANTUM ERRORCORRECTION
Here we reinterpret the PTIM dynamics as a compe-tition between projective errors and syndrome measure-ments on a topological quantum memory; an approach also successfully applied for the entanglement phase tran-sition in the unitary regime [48]. To this end, consideran open chain of L spinless fermions c i and define theMajorana modes γ i − = c i + c † i and γ i = i ( c i − c † i ) (28)with { γ i , γ j } = 2 δ ij , γ † i = γ i , and γ i = . Define stabi-lizer operators S e =( i,i +1) = iγ i γ i +1 for i = 1 , . . . , L − S † e = S e , S e = and [ S e , S e (cid:48) ] = 0. We areinterested in the two-fold degenerate ground state spaceof the quadratic fermion Hamiltonian [25] H = − (cid:88) e S e = − L − (cid:88) i =1 iγ i γ i +1 (29)characterized by S e = 1 on all edges e . Let {| g (cid:105) , | g (cid:105)} be a basis of the ground state space, the code space ofthe Majorana chain quantum code [26]. A logical qubitwith amplitudes α and β is then encoded as | Φ (cid:105) = α | g (cid:105) + β | g (cid:105) .The elementary errors of the code are generated bythe Hermitian on-site operators E i = iγ i − γ i and, dueto S e E i = − E i S e for i ∈ e , lead to excitations of theHamiltonian (29). In the following, we assume that theenvironment measures E i projectively with probability p per site and time step. To detect and correct these errors,we are allowed to measure the stabilizers S e projectivelyand use the measurement outcomes, the so called errorsyndrome . Common schemes to protect the qubit | Φ (cid:105) from decoherence employ time-periodic measurementsof all stabilizers and then use majority voting on thesyndromes to decide on unitary corrections in each timestep [26, 49]. Here we modify this scheme and performstabilizer measurements randomly with probability 1 − p per edge and time step.To reveal the connection to the PTIM, we need a spin-1/2 representation of the fermion operators. We opt forthe slightly unconventional Jordan-Wigner transformation γ i − = (cid:89) j
1. Inthis representation, the qubit is encoded as a global Bellcluster | Φ (cid:105) = α | . . . (cid:105) + β | . . . (cid:105) , (32)the random errors E i correspond to measurements of σ xi ,and the random measurements of stabilizers S e to mea-surements of σ zi σ zi +1 . We end up with a new interpretationof the PTIM with open boundaries in one dimension, de-scribing the competition between errors and stabilizermeasurements on a Majorana chain quantum code.1
10 100 1000 C l u s t e r li f e t i m e (a) (b) (c) Size expalglog
OBC
Figure 8.
Quantum error correction. (a,b) Time evolution with random stabilizer measurements S e between adjacent qubits(vertical bars) and random errors E i on qubits (circles); time runs upwards. The system is initialized in the code space with aglobal Bell cluster (bold black vertical lines). (a) The initial cluster survives and the amplitudes are preserved. (b) Adding afew additional errors (black discs) makes the black cluster die out, loosing all information about the encoded qubit. The new,independent cluster (red) is trivial and does not carry quantum information. (c) Average cluster lifetime τ of a global Bellcluster as function of system size L for p ∈ { . , . , . } . For p = 0 . < p c , τ grows exponentially so that the amplitudes ofthe initial cluster are retained almost indefinitely. At criticality p = 0 . p c , τ grows algebraically with τ ∼ L β and β ≈ p = 0 . > p c , the growth is only logarithmic. The solid lines are fits of the form αe βL (blue), αL β + γ (red), α log( L ) + β (yellow). Each point is based on 10 sampled trajectories on a chain with open boundary conditions and a cutoff simulationtime t max = 5 × . In our previous studies of the PTIM, we initializedthe system in the unentangled product state | Ψ(0) (cid:105) = | + · · · + (cid:105) and used the PTIM to build up entanglement.Our discussion of the Majorana chain quantum code sug-gests as initial state the global Bell cluster (32). Howlong does the system retain information about this qubitin the presence of projective errors and random stabilizermeasurements? In our analysis of the PTIM dynam-ics, we found that the amplitudes α and β survive themerging, growth, and shrinking of clusters. Thereforethe quantum information disappears irretrievably only ifthe initial cluster is completely degraded. In Fig. 8 wesketch two quantum trajectories: one where the initialcluster survives (a) and one where it decays (b). Foreach quantum trajectory | Φ( t ) (cid:105) with | Φ(0) (cid:105) = | Φ (cid:105) , wecan define the time τ Φ at which the initial cluster diesout. We then define the average cluster lifetime τ byaveraging τ Φ over many quantum trajectories. This timescale defines the characteristic decay time of the storedquantum information in the system.In Fig. 8(c) we plot the scaling of this time scale τ with the system size L for different probabilities p . Again,the entanglement phase transition is clearly visible atthe critical value p c = 0 .
5: For p < p c , we find an expo-nentially diverging lifetime for increasing L , indicatingthat quantum information can be robustly stored in largesystems. By contrast, numerics suggests that the growthof τ is only logarithmic for p > p c , while at criticality p = p c we find an algebraic behavior τ ∼ L β with β ≈ | Φ (cid:105) = α | (cid:105) + β | (cid:105) , letthe system evolve under conditions such that the initialcluster survives [as in Fig. 8(a)], and finally measure thestabilizers on all edges, the system ends up in the state | Φ (cid:48) (cid:105) = α | m (cid:105) + β | m (cid:105) where again | m (cid:105) = U | m (cid:105) with U = (cid:81) i σ xi . Note that along the time evolution, the signbetween the two amplitudes of the initial cluster maychange. However, the last stabilizer measurements thatcondense all clusters into one always reproduce the correctsign between the amplitudes [to see this, use Eqs. (7)-(10)and generalizations thereof]. The final spin configuration m in | Φ (cid:48) (cid:105) depends on the measurement outcomes duringthe evolution and is only known if all measurements arerecorded. This implies that the quantum informationis still stored in the system – but to access the qubit(mandatory for a useful quantum memory), one has todeduce the configuration m from the collected syndromemeasurements in an efficient way. This decoding of thequantum memory (and its efficiency) is beyond the scopeof this paper.Conversely, for quantum trajectories where the initialcluster dies out and the quantum information is lost [asin Fig. 8(b)], the final state is | m (cid:105) + | m (cid:105) with probability | α + β | / | m (cid:105) − | m (cid:105) with probability | α − β | / σ xi that eventually removesthe initial cluster determines the sign and performs a“measurement” of the observable U on the stored qubit,projecting the system either into the symmetric or theantisymmetric eigenspace of the symmetry U .2 IX. SUMMARY AND OUTLOOK
In this paper, we introduced and studied the dissipativetransverse field Ising model , a random circuit model wherein each time step the non-commuting observables σ xi and σ zi σ zi +1 are measured randomly with probabilities p and1 − p on sites and edges, respectively. When averagedover many quantum trajectories, the mutual informationbetween far apart spins behaves like a correlation functionin conventional second-order quantum phase transitions:while zero above a critical point p c , it is finite for p < p c .This emergence of long-range entanglement between spinsis only visible in averages over quantum trajectories andnot in the (up to symmetries, maximally mixed) densitymatrix of the system. Using a classical model for theentanglement dynamics – the colored cluster model – weperformed extensive numerical simulations and presentedan intuitive picture of the entanglement transition whichcan be understood as the condensation of colored clusters.We would like to point out that this entanglementtransition is not necessarily linked to conventional phasetransitions of non-equilibrium steady states in driven dis-sipative systems [21, 22]. Indeed, if one generalizes theprojective transverse field Ising model to higher dimen-sions and adds feedback to the process (say, spin flipsthat are conditioned on the measurement outcomes), thenlong-range spin-correlations are possible and spontaneoussymmetry breaking can occur in the steady state. Suchnon-equilibrium phase transitions are reflected in thedensity matrix and seem to be unrelated to the entangle-ment transition studied in this paper. For instance, in onedimension, no long-range order is possible – but the entan-glement transition is still visible in ensembles of quantumtrajectories. In higher dimensions, both long-range orderand the entanglement transition can be observed; however,the critical points of these transitions are not necessarilythe same.In a next step, we related the projective transversefield Ising model to bond percolation on the spacetimelattice of the process. This allowed us to infer the critical point of the one dimensional system exactly; we verifiedthis relation also for various lattices in two dimensions.Switching to the continuum paved then the way for a con-formal field theory of the critical one-dimensional system.With this machinery, we derived the universal prefactor˜ c = 3 √ / (2 π ) that describes the scaling of the en-tanglement entropy, and the critical exponent κ = 2 / ACKNOWLEDGMENTS
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