--- title: Measurement-Induced Long-Range Order url: https://www.emergentmind.com/topics/measurement-induced-long-range-order type: topic --- # Measurement-Induced Long-Range Order Searching arXiv for recent and foundational papers on measurement-induced long-range order. arXiv.search query="measurement induced long range order" max_results=10 arXiv.search query="measurement protected quantum phases 2004.09509" max_results=5 arXiv.search query="Local measurements and the entanglement transition in quantum spin chains 2602.05914" max_results=5 Measurement-induced long-range order denotes regimes in which local, structured, or global measurements generate, reveal, or protect correlations that remain nonzero or decay only algebraically at arbitrarily large separations. In the literature, this includes symmetry-breaking order, spin-glass order, string order descended from symmetry-protected topological phases, topological order, and long-range multipartite entanglement. The phenomenon appears in hybrid circuits with competing unitaries and projective measurements, in measurement-only circuits, in single-round post-measurement states of gapless systems, in mixed-state channels with feedback, and in continuously monitored many-body systems [2004.09509][2602.05914][2412.07830][2604.22022][2303.15507]. ## 1. Architectural paradigms and basic mechanism A central architecture is the hybrid quantum circuit, in which measurements collapse degrees of freedom while unitaries regenerate entanglement. In the one-dimensional Ising-symmetric construction of "Measurement Protected Quantum Phases" [2004.09509], qubits live on a periodic chain of length \(L\) with brick-wall two-qubit gates. At each gate location, one either measures with probability \(p\) or applies a random unitary with probability \(1-p\). The unitaries are two-qubit Clifford gates commuting with the global Ising parity \[ T=\prod_{i=1}^L X_i, \] equivalently satisfying \[ U(X_iX_{i+1})U^\dagger=X_iX_{i+1}. \] Measurements are of two kinds: \[ M_1:\ Z_iZ_{i+1}\quad\text{at rate }pr,\qquad M_2:\ X_i\quad\text{at rate }p(1-r). \] The basic mechanism is competition between entangling unitaries and projective measurements. In the measurement-dominated regime, the steady state obeys an area law for the second Rényi entropy, \[ S_A=-\log \mathrm{Tr}\,\rho_A^2\sim O(1), \] while appropriately chosen commuting measurements can simultaneously carve out large correlated clusters. In the Ising example, a percolating network of \(Z_iZ_{i+1}\) measurements produces GHZ-like cat-state ensembles with \(\langle Z_iZ_j\rangle=\pm1\) for all \(i,j\), so long-range order is compatible with area-law entanglement [2004.09509]. A distinct paradigm removes unitaries altogether. Long-range measurement-only Clifford circuits with two-qubit parity checks and measurement-only circuits with competing three-qubit cluster and long-range \(ZZ\) measurements show that repeated projections alone can generate steady states with symmetry breaking, symmetry-protected topological order, critical algebraic correlations, and entanglement growth beyond area law [2604.22022][2605.25845]. Single-round measurement protocols provide a third paradigm: one local measurement layer on a gapless parent state can induce long-range order or a boundary transition controlled by boundary conformal field theory [2412.07830]. Continuous weak monitoring of collective observables gives yet another route, as in ultracold Fermi gases where spatially structured backaction competes with Hubbard dynamics to produce antiferromagnetic order or density modulations [1510.04883]. ## 2. Symmetry, protected phases, and steady-state order In the Ising-symmetric hybrid circuit, the long-range order is characterized by the squared correlator \[ q_{ij}\equiv \langle Z_iZ_j\rangle^2, \] and the global order parameter \[ O\equiv \frac{1}{L}\sum_{i,j=1}^L q_{ij}. \] In a trivial product-like or paramagnetic state, \(q_{ij}\) decays exponentially and \(O\to\mathrm{const}\) as \(L\to\infty\). In the long-range cat-state ensemble, \(O\sim L\). For \(r=1\), the model exhibits a direct transition at \[ p_c\simeq 0.38 \] between an area-law spin-glass phase for \(p>p_c\), with \(O\sim L\) and \(S_A\sim O(1)\), and a volume-law paramagnet for \(p0\), and moderate \(p_{ZZ}\) with moderate to large \(1/\alpha\) produces an intermediate entanglement-rich regime in which both \(O_{\rm SSB}\) and \(O_{\rm SPT}\) are approximately zero, while correlations decay algebraically and the half-chain entropy is enhanced beyond area law for \(\alpha\lesssim2\). At \(\alpha=2\), one obtains \(S_{\rm half}(L)\sim c\log L\); for smaller \(\alpha\), \(S_{\rm half}(L)\sim L^\gamma\) with \(\gamma\lesssim1\) [2605.25845]. These constructions also generalize beyond one dimension. In \((2+1)\)D, a brick wall of four-qubit gates with symmetric Clifford unitaries and three \(ZZ\) measurements admits a regime where both the measurement cluster and the unitary cluster percolate, yielding simultaneous volume-law entanglement, \[ S_A\sim |A|, \] and long-range spin-glass order, \[ O\sim L_xL_y. \] Replacing the measurement layers by toric-code plaquette and star measurements can lock in a random topological state without any symmetry restriction [2004.09509]. ## 3. Measurement-only circuits and single-round boundary transitions Long-range measurement-only Clifford circuits provide a minimal setting in which repeated parity checks induce phases with or without genuine long-range order [2604.22022]. On a ring of \(N\) qubits, each layer performs \(M_2\) disjoint two-qubit parity-check measurements with density \(p=M_2/N\). The pair range \(r\) is drawn from \[ P(r)\propto r^{-\alpha},\qquad r=1,2,\dots,N/2, \] and the projector is \[ P_{ij}^\beta=\frac12(1+\sigma_i^\beta\sigma_j^\beta),\qquad \beta\in\{X,Y,Z\}. \] In the random-basis design, each measurement independently chooses \(\beta\in\{X,Y,Z\}\). In the single-basis design, the entire layer uses one basis \(\beta\), chosen uniformly between layers. A replica mapping yields an effective long-range XX Hamiltonian in the continuous-time limit, \[ H_{\mathrm{eff}}=-\sum_{i\alpha_c\), there is a critical XY phase with quasi-long-range order, \(C_x(r)\sim r^{-\eta}\), and \[ S_2(L)\sim \frac{c_{\rm eff}}{3}\log L,\qquad c_{\rm eff}=1. \] The single-basis design additionally exhibits a regime with simultaneous volume-law entanglement, long-range entanglement, ancilla purification time \(\tau=O(1)\), and absence of scrambling as detected by \(I_3\ge 0\) [2604.22022]. Single-round measurements on gapless parent states produce a different form of measurement-induced long-range order. In the gapless parent of the one-dimensional cluster state, measuring all sites on one chain in the \(X\) basis and post-selecting the uniform outcome sector yields \[ \langle Z_{j,2}Z_{k,2}\rangle_{\rm uni}\to \mathrm{const}\qquad (|j-k|\to\infty), \] while other correlations remain power law, for example \[ \langle X_{j,2}X_{k,2}\rangle_{\rm uni}\sim |j-k|^{-\min(4,4K)}. \] The resulting post-measurement state therefore combines true long-range order with residual gapless structure, and its entanglement entropy becomes area law due to relevant boundary pinning [2412.07830]. Rotating the measurement basis to \[ O_j=\cos\omega\,X_{j,1}+\sin\omega\,Z_{j,1} \] induces a measurement-induced boundary transition. For \(0<\omega<\pi/4\), both boundary cosine perturbations pin and lower-chain \(Z\)-order persists; for \(\pi/4<\omega<\pi/2\), complementary pinning yields disorder-string order; at \[ \omega_c=\pi/4, \] one mode remains free, producing an intermediate BCFT with effective \(c_{\rm eff}=1\). Similar measurement-induced boundary transitions occur in tricritical Ising and three-state Potts critical theories [2412.07830]. ## 4. Rigorous and mixed-state formulations A rigorous \(C^*\)-algebraic version of measurement-induced long-range order is given for infinite quantum spin chains initially in a non-trivial mixed SPT phase with symmetry \(\mathcal G=G\times H\), where \(G\) is Abelian [2602.05914]. Measuring the local \(G\)-charge on intervals \(X_n=[-n,n]\) produces post-measurement states \[ \omega_{n,\mathbf q_n}(A) = \frac{\omega(M_{X_n}(\mathbf q_n)\,A\,M_{X_n}(\mathbf q_n))} {\omega(M_{X_n}(\mathbf q_n))}. \] Although each \(\omega_n\) is still SRE in isolation, the required locality bounds deteriorate with \(n\). In the infinite-volume limit \(\omega_\infty\), there exist almost-local operators \(W^i\) and \(W^j\) such that \[ \bigl|\omega_n(W^iW^j)-\omega_n(W^i)\omega_n(W^j)\bigr|=1, \] and hence \[ C_W(i,j)=\pm1\neq0 \qquad \forall\, |i-j|\gg1. \] Theorem 4.3 shows that there is no uniform Lieb-Robinson bound for the split automorphisms mapping all \(\omega_n\) back to a product state. The hidden SPT order is thus converted into explicit classical long-range order by local measurements [2602.05914]. Mixed-state channels with measurement and feedback extend the mechanism beyond pure trajectories. In the general construction of [2303.15507], one measures subsystem \(A\) with rank-one projectors \(P_\alpha\), applies an outcome-dependent unitary \(U_\alpha\) on subsystem \(B\), and obtains \[ \mathcal E[\rho_0] = \sum_\alpha U_\alpha P_\alpha \rho_0 P_\alpha U_\alpha^\dagger. \] For the one-dimensional cluster-state SPT, measuring every \(A\)-site in the \(X\) basis and feeding forward on \(B\) converts string order into GHZ-type order, \[ \mathrm{Tr}[\rho_B\,Z_{b,i}Z_{b,j}] \to c>0. \] At the SPT-to-critical point, the output instead has algebraic correlations \[ \langle Z_{b,i}Z_{b,j}\rangle\sim |i-j|^{-1/4},\qquad \langle X_{b,i}\cdots X_{b,j}\rangle\sim |i-j|^{-1/2}, \] and logarithmic negativity \[ E_N(\ell)\approx \frac{c_{\rm eff}}{4}\ln \ell, \qquad c_{\rm eff}\approx0.3\sim\tfrac12. \] The same framework converts spinful free fermions into a mixed state with spin correlations enhanced from \(|i-j|^{-2}\) to \(|i-j|^{-1}\), and converts Chern insulators into mixed states with critical bulk spin correlations [2303.15507]. Adaptive circuits with local measurements, local unitaries, and non-local classical communication further show that long-range entangled quantum matter can be prepared in constant or logarithmic depth, bypassing the unitary-only bound \(\xi\le c\cdot D\) [2206.13527]. The same work gives a constant-depth GHZ example with \(\langle X_iX_j\rangle=+1\) for all \(i,j\) after measuring a one-dimensional cluster SPT, and constant-depth or \(O(\log L)\) constructions for topological orders, critical CFT states, arbitrary CSS codes, and symmetry-enriched topological order [2206.13527]. ## 5. Diagnostics, scaling laws, and entanglement structure The diagnostics of measurement-induced long-range order vary across constructions but share a common function: they distinguish genuine long-distance order from short-range entanglement or mere entropy suppression. In the Ising-symmetric hybrid circuit, the single-site mutual information \[ I(i:j)=S_i+S_j-S_{ij} \] saturates to a nonzero constant at large separation in the ordered area-law phase, decays exponentially in the disordered volume-law phase, and decays algebraically, \[ I(i:j)\sim |i-j|^{-\alpha}, \] at criticality, with \(\alpha\approx1.4\) at \(r=1\) [2004.09509]. For hybrid Haar circuits with single-site \(Z\) measurements, multipartite entanglement itself becomes the order parameter [2404.16095]. The model undergoes a measurement-induced transition at \[ p_c\simeq 0.17(1) \] for chains up to \(L=24\). At criticality, the logarithmic negativity and tripartite entanglement indicators decay algebraically with separation: \[ \langle E_2(x)\rangle\propto x^{-\alpha_2},\qquad \alpha_2\simeq 7.1(1), \] \[ \langle W(x)\rangle\propto x^{-\alpha_3},\qquad \alpha_3\simeq 8.8(2), \] with \(\Delta_3>\Delta_2\). The correlation length obeys \(\xi(p)\sim |p-p_c|^{-\nu}\) with \(\nu\approx4/3\), and genuine four-party entanglement persists up to half the chain length at intermediate measurement rate [2404.16095]. Measurement-only long-range circuits require additional probes beyond entanglement entropy: mutual information, tripartite mutual information, purification from an ancilla, and Bell-cluster statistics [2604.22022]. In that setting, the sign of \(I_3\) distinguishes strong scrambling \((I_3\ll0)\), marginal scrambling \((I_3\simeq0)\), and no scrambling \((I_3>0)\). The ancilla purification time further separates phases: \(\tau\sim O(e^N)\) in the random-basis continuous-symmetry-broken regime, \(\tau=O(1)\) in the dense single-basis regime, and \(\tau\sim O(N)\) or logarithmic in sub-volume-law phases [2604.22022]. A recurrent misconception is that measurement-induced order must coincide with low entanglement. The examples above show otherwise. Area-law entanglement can coexist with spin-glass order [2004.09509], volume-law entanglement can coexist with long-range entanglement and no scrambling [2604.22022], and mixed-state long-range order or criticality can coexist with volume-law entropy [2303.15507]. A second misconception is that measurement-induced order is necessarily conventional symmetry breaking: the literature includes SPT string order, Bell-pair order, topological order, and multipartite entanglement as equally central manifestations [2605.25845][2410.13844]. ## 6. Experimental platforms, numerical methods, and fragility under averaging In ultracold Fermi gases in optical lattices, weak continuous monitoring of a spatially structured collective observable can build long-range antiferromagnetic or density-wave order without requiring strong interactions [1510.04883]. For a one-dimensional Fermi-Hubbard chain, \[ H_0=-\hbar J\sum_{\sigma=\uparrow,\downarrow}\sum_{\langle i,j\rangle} f_{j,\sigma}^\dagger f_{i,\sigma} +\hbar U\sum_{i=1}^L n_{i\uparrow}n_{i\downarrow}, \] the measurement operator can be chosen proportional to the staggered magnetization \[ M_s=\sum_i(-1)^i m_i,\qquad m_i=n_{i\uparrow}-n_{i\downarrow}. \] Since \(c^\dagger c\propto M_s^2\), photodetections amplify \(\langle M_s^2\rangle\). In the strong-measurement regime \(\gamma\gg J\), a trajectory selects a value of \(|M_s|\), producing a macroscopic superposition of the two Néel patterns; in the weak regime \(\gamma\ll J\), the order grows oscillatory with frequency set by \(J\). The magnetic structure factor develops a strong peak at \(q=\pi/d\), and the escaped photon flux directly tracks the ordering process [1510.04883]. For equilibrium parent states with many measurements, "Post-measurement Quantum Monte Carlo" develops a generalized stochastic series expansion for the measured density matrix and applies it to the spin-\(\tfrac12\) Heisenberg antiferromagnet on the square lattice [2410.13844]. Measuring bond-spin projectors \[ P_{(jk)}^{0}=\tfrac14-\mathbf S_j\!\cdot\mathbf S_k,\qquad P_{(jk)}^{1}=\tfrac34+\mathbf S_j\!\cdot\mathbf S_k \] allows efficient evaluation of post-measurement correlators in sign-problem-free ensembles. The method demonstrates deterministic creation of long-range Bell pairs, measurement-induced enhancement of Néel correlations, and measurement-induced SPT order. At the \(O(3)\) quantum critical point, the measured correlations are consistent with the extraordinary-log form \[ C(r)\sim [\ln(r/r_0)]^{-q},\qquad q\approx0.63 \] for large measurement strength [2410.13844]. The principal limitation emphasized by solvable trajectory-averaged models is fragility to information loss. In the monitored Kitaev chain with jump operators \(L_j=\sqrt{\gamma}\,c_j\), perfect postselection of no-click trajectories \((q=1)\) yields algebraic correlations in the critical window \(|\mu|<1\) and \(\gamma<\gamma_c(\mu)=4\sqrt{1-\mu^2}\). For any \(q<1\), however, partial averaging over missed clicks introduces a finite correlation length, a nonzero Liouvillian gap, and saturation of the entanglement negativity. The real-space covariance obeys an exponential bound, and numerics confirm \(\xi(q)<\infty\) for every \(q<1\) [2407.13837]. This shows that in some monitored systems the long-range order is a property of sharply resolved trajectories rather than of trajectory-averaged mixed states. Taken together, these results establish measurement-induced long-range order as a broad non-equilibrium phenomenon rather than a single mechanism. Measurements can protect order against entangling dynamics, generate order without any unitaries, convert hidden nonlocal order into explicit long-range correlations, and even prepare long-range entangled matter in low depth. At the same time, the dependence on conditioning, post-selection, feedback, symmetry, and measurement range means that the precise notion of “order” is architecture-specific, spanning steady-state spin glasses, symmetry-broken and SPT phases, topological order, critical mixed states, and long-range multipartite entanglement [2004.09509][2604.22022][2605.25845][2404.16095]. Source: https://www.emergentmind.com/topics/measurement-induced-long-range-order