---
title: Measurement-Induced Phase Transitions
url: https://www.emergentmind.com/topics/measurement-induced-phase-transition
type: topic
---

# Measurement-Induced Phase Transitions

Measurement-induced phase transition denotes a nonequilibrium transition generated by the competition between entangling quantum dynamics and measurements that locally collapse the wavefunction. In the canonical monitored many-body setting, tuning the measurement rate or strength produces a change in the scaling of trajectory entanglement, most commonly from a volume law to an area law, with logarithmic entanglement growth and scale-invariant correlations at criticality in several one-dimensional settings [1908.11253][2405.08861]. The term is also used more broadly for other non-analytic phenomena produced by measurement–dynamics feedback, including first-detection transitions in periodically measured Ising chains, teleportation transitions in monitored Sachdev-Ye-Kitaev protocols, and entropy-production transitions in state estimation of classical chaotic systems [1603.03561][2210.03083][2501.00547].

## 1. Canonical monitored dynamics and phase structure

The standard formulation considers a many-body system evolving under unitary gates or a Hamiltonian while local projective or weak measurements are performed repeatedly. In this setting, unitary dynamics tends to generate nonlocal entanglement, whereas measurements disentangle by collapsing local degrees of freedom. A sharp transition then appears at a critical measurement rate or strength. In one-dimensional random circuits with local unitary scrambling and projective measurements at rate \(p\), typical individual quantum trajectories undergo a transition between a phase that sustains volume-law entanglement at low \(p\) and one that exhibits area-law entanglement at high \(p\) [2405.08861].

Concrete realizations span both Hamiltonian and circuit models. In a nonintegrable one-dimensional Bose-Hubbard chain, a product Mott state with \(n_i=1\) is quenched into the superfluid regime \(U/J_0=0.14\), while local projective measurements \(\Pi_i=\lvert 1\rangle\langle 1\rvert\) are applied at random in space and time. The resulting phase diagram in the spatial measurement probability \(P_x\) and temporal measurement-layer density \(N_t\) shows a volume-to-area law entanglement transition [1908.11253]. In a monitored transverse-field Ising chain, each spin is continuously monitored in the \(\sigma^z\)-basis with jump operators \(n_i=(1+\sigma_i^z)/2\) at rate \(\gamma\); postselecting the no-click trajectories yields pure-state evolution under the non-Hermitian effective Hamiltonian \(H_{\rm eff}=H-(i\gamma/2)\sum_i n_i\), and the steady-state entanglement changes at \(\gamma_c=4\sqrt{1-h^2}\) [2302.10132].

The entanglement structure is not always limited to a strict volume-law versus area-law dichotomy. In monitored free bosons with long-range couplings, local measurements do not produce a \(\gamma\)-driven transition, while nonlocal measurements lead instead to a subvolume-to-area law transition [2405.19768]. In monitored free fermions in spatial dimensions \(d>1\), the weak-measurement phase exhibits \(L^{d-1}\ln L\) scaling rather than a volume law, followed by an area-law phase at stronger monitoring [2605.05306]. These cases indicate that the defining feature is not a single universal pair of scaling laws, but a measurement-controlled change between distinct long-time entanglement structures.

## 2. Diagnostics and order parameters

The primary diagnostic is the entanglement entropy of a subsystem \(A\). In the Bose-Hubbard study, the long-time steady-state entropy satisfies \(S_A\sim c_1\lvert A\rvert\) in the volume-law phase and saturates at \(O(1)\) in the area-law phase; at fixed \(N_t=50\), the critical spatial measurement rate is \(P_{x,c}\approx 0.06\) [1908.11253]. In strongly disordered spin chains, the order parameter is the tripartite mutual information
$$
I_3(t)=S_A+S_B+S_C+S_{ABC}-S_{AB}-S_{BC}-S_{AC},
$$
with \(A,B,C\) each occupying one quarter of the chain; in the long-time limit, \(I_3^s\) saturates to an \(O(1)\) negative constant in the volume-law phase and to zero in the area-law phase [2512.02100].

Several works emphasize more refined diagnostics. The Bose-Hubbard calculation introduces the single-site entropy
$$
s_i=-{\rm Tr}\,\rho_i\log\rho_i,
$$
whose probability distribution \(P(s)\) has two peaks, one at \(s\approx 0\) and one at \(s\approx 1.25\), deep in the volume-law phase; the finite-entropy peak broadens and loses weight near criticality, and only the \(s=0\) peak remains above the transition. The same study also uses the two-site mutual information \(I(i,j)=S_i+S_j-S_{i\cup j}\) as a probe of critical spatial structure [1908.11253].

In the non-Hermitian Ising setting, the quantum Fisher information (QFI) provides both a metrological and an entanglement diagnostic. For a pure steady state \(\lvert\psi_\infty\rangle\) probed by a small twist \(e^{-i\phi S_x}\), the QFI is \(F=4\Delta^2S_x\), and the chain QFI obeys
$$
F_N\propto N^\eta.
$$
The exponent changes sharply across the transition: \(\eta\approx 1.5\) in the entangling phase and \(\eta=1\) in the disentangling phase. Since \(F/N>m\) certifies at least \((m+1)\)-partite entanglement, the super-extensive scaling in the entangling phase witnesses large-scale multipartite entanglement [2302.10132].

When trajectory discrimination is imperfect, mixed-state diagnostics become necessary. In the solvable partially postselected Kitaev-chain Liouvillian, the fermionic entanglement negativity grows as \(\ln \ell\) only for perfect postselection \(q=1\) and \(\gamma<\gamma_c\); for any \(q<1\), it crosses over and saturates beyond a finite length scale \(\xi\) [2407.13837]. For noisy monitored circuits, the conditional entanglement entropy
$$
\overline{S(M|A)}=\overline{S(M,A)}-\overline{S(A)}
$$
is used as the central probe; under the symmetry conditions specified below, it becomes a valid entanglement measure and supports a genuine entanglement transition [2406.14109].

## 3. Critical scaling, universality, and field-theoretic descriptions

A central result in the nonintegrable Bose-Hubbard chain is that both temporal and spatial criticality are logarithmic. For subsystem size \(L=8\), the entanglement growth at the critical spatial measurement rate satisfies
$$
S_A(t)\simeq \beta \log t + {\rm const},\qquad \beta\approx 0.22,
$$
while in the long-time steady state at \(P_x=P_{x,c}\),
$$
S_A(L)\simeq \alpha \log L + {\rm const},\qquad \alpha\approx 0.21.
$$
The same work reports the one-parameter scaling form
$$
S_A(P_x)-S_A(P_{x,c})=F[(P_x-P_{x,c})L^{1/\nu}],
$$
with \(P_{x,c}=0.060(4)\) and \(\nu=2.00(15)\). At criticality the mutual information exhibits a power-law tail \(I(i,j)\sim r^{-\Delta}\), with \(\Delta\approx 1.3(36)\to 1.6(48)\), tending toward \(\Delta=2\) in the large-\(L\) limit; both \(\nu\approx 2\) and \(\Delta\approx 2\) are described as consistent with a boundary-percolation/CFT picture [1908.11253].

For the standard \(1+1\) dimensional circuit transition without conserved densities, the large-local-Hilbert-space description is a nonunitary CFT / percolation theory, with correlation-length exponent \(\nu\approx 1.3\) and dynamical exponent \(z=1\). The half-chain entanglement obeys the finite-size scaling form
$$
S_{N/2}(L,p)=L^\alpha F\!\bigl((p-p_c)L^{1/\nu}\bigr),
$$
and criticality corresponds to a logarithmic violation of area-law scaling [2405.08861].

Coupling the measurement process to a diffusing conserved density changes the universality class. In a random Clifford circuit monitored by classically diffusing particles, the diffusive space-time correlations in the measurement density are a relevant perturbation to the usual random MIPT fixed point. The resulting “diffusive MIPT” has dynamical exponent \(z'=2\), correlation-length exponent \(\nu'\approx 1.6(5)\), and critical entanglement scaling
$$
S(\ell,t)\simeq \ell^{\alpha'}\mathcal G\!\bigl(t/\ell^{z'}\bigr),
$$
with \(\alpha'\approx 0.4\). The modified finite-size scaling form
$$
S_{N/2}(L,p)=L^{\alpha'}\,g\!\bigl((p-p_c)L^{1/\nu'},L^{-1/2}\bigr)
$$
contains an explicit \(L^{-1/2}\) argument encoding the diffusive fluctuation amplitude of the measurer density [2405.08861].

Long-range couplings likewise alter criticality. In a one-dimensional hybrid circuit with two-qubit Clifford gates acting at distance \(r\) with probability \(P(r)\propto 1/r^\alpha\), the transition is short-range-like for \(\alpha\gtrsim 3\), with \(z=1\) and \(\nu\approx 1.3\), but for \(\alpha\lesssim 3\) the critical exponents vary continuously with \(\alpha\), giving a continuum of non-conformal universality classes. For \(\alpha<2\), no true area-law phase survives; the large-\(p\) regime shows sub-volume scaling \(S_{L/2}\sim L^{2-\alpha}\) [2104.13372].

Not all reported critical structures are unambiguous thermodynamic transitions. In periodically driven monitored free fermions, a high-frequency renormalization-group analysis of the non-Hermitian sine-Gordon model predicts that a symmetric drive with \(\epsilon=0\) always favors the area-law phase, while an asymmetric drive can exhibit a potential signature of a Berezinskii-Kosterlitz-Thouless transition between a logarithmic phase and an area-law phase. The finite-size collapse takes the form
$$
S(L/2,L,\gamma)-S(L/2,L,\gamma_c)=F\bigl[(\gamma-\gamma_c)(\ln L)^2\bigr],
$$
but the work explicitly states that it is almost impossible to rule out a finite-size crossover rather than an actual thermodynamic transition [2412.01917].

## 4. Dependence on interactions, disorder, conservation laws, and measurement geometry

Interacting, nonintegrable dynamics supports the canonical trajectory entanglement transition. The Bose-Hubbard chain studied by matrix product state and time-evolving block decimation demonstrates that a generic nonintegrable many-body system can exhibit a sharp measurement-induced entanglement transition with volume-law and area-law phases, logarithmic entanglement at criticality, and mutual-information scaling consistent with a single universality class [1908.11253].

Strong disorder changes the phase structure in a qualitatively different way from the clean Bose-Hubbard example. In the disordered transverse-field Ising chain and in the effective \(l\)-bit Hamiltonian, any infinitesimal measurement rate \(p>0\) destroys the purely many-body-localized logarithmic growth \(S(t)\sim \ln t\) and instead induces fast entanglement growth \(S(t)\sim t\) followed by a volume law \(S\sim \alpha L\). A second transition then occurs at \(p_c\), above which repeated measurements produce an area-law state. The critical rate is exponentially small in the disorder strength \(W\) and the overlap \(O\) between the measurement operator and the local integrals of motion,
$$
p_c(W,\pi/2)\sim \exp[-\alpha W/(1-O^2)],
$$
and the saturation time changes from \(t_s(p=0)\sim \exp[aL]\) in the unmeasured MBL regime to \(t_s\sim L\) in the measurement-induced volume-law phase [2512.02100].

For monitored free fermions, quenched disorder has a different status. In \(d\)-dimensional noninteracting fermions with static random on-site potential and continuous weak measurements of local density, the long-time universal behavior is governed by the same nonlinear sigma model as in the clean case, with disorder entering only through a modification of model parameters. This implies that the presence or absence of a transition is unchanged by disorder: in \(d>1\), a transition occurs between an \(L^{d-1}\log L\) phase and an area law, while in \(d=1\) there is only an area law and no genuine transition [2605.05306].

Measurement geometry can itself determine whether a transition exists. In continuously monitored free bosons with long-range couplings, local measurements \(\hat{\mathcal O}_n=\sqrt{\gamma}\,\hat x_n\) do not induce a \(\gamma\)-driven transition because the long-range Hamiltonian entanglement generation overcomes local entanglement destruction. By contrast, nonlocal measurements
$$
\hat{\mathcal O}_{j,r,\pm}=\sqrt{\frac{\gamma}{r^\alpha}}\;(\hat x_j\pm \hat x_{j+r})
$$
directly suppress long-range entanglement generation and produce a subvolume-to-area law transition at a critical \(\gamma_c(\alpha)\) for \(\alpha<1\) [2405.19768].

Topological input can also generate distinct measurement-induced phases. Random single-qubit Pauli measurements on a subsystem of the toric-code ground state map, via a parton construction, to a completely-packed loop model with crossings. Varying \((p_x,p_y,p_z)\) generates a surface of continuous transitions between short-loop and long-loop phases. When all but a one-dimensional boundary is measured, the unmeasured boundary realizes a \(1+1\) dimensional hybrid circuit whose half-chain entanglement changes from an area law to a logarithmic form in the long-loop phase; with adaptive boundary unitaries conditioned on the bulk measurement outcomes, the resulting mixed state can display experimentally accessible linear ferromagnetic order [2307.02292].

## 5. Postselection, feedback, noise, and experimental access

A recurring obstacle is that the entanglement transition is defined at the level of quantum trajectories, whereas ensemble averaging over outcomes typically masks it. One proposed solution is pre-selection, in which the measurement outcome randomness is explicitly broken by steering the system toward a representative state. The modified dynamics introduces a unique dark or absorbing state with macroscopic order, replacing the maximally mixed stationary state of the unconditioned ensemble. Because the added terms are short-range, symmetry-allowed, and RG-irrelevant, the transition retains the original critical exponents and entanglement scaling, while standard local observables can diagnose it without trajectory postselection [2208.10506].

A related experimental route uses deterministic feedback after projective measurements. In an attractive Bose-Hubbard model for a transmon array, local measurements of boson number are probabilistically interleaved with unitary evolution, producing a measurement-induced entanglement transition in the steady-state trajectory ensemble. In the feedback version, the post-measurement state is reset to a predetermined local state \(\lvert\alpha_\ell\rangle\), so that the single-site occupation histogram becomes approximately uniform in the volume-law phase and \(\delta\)-peaked in the area-law phase. The crossing point of distances to the uniform and \(\delta\)-peaked distributions gives an estimate of \(p_c\) consistent with the entanglement-based transition, offering a simple-observable probe that avoids full trajectory postselection [2302.02934].

Continuous-time formulations make postselection explicit at the level of the evolution equation. A generalized Lindblad master equation for the double-space density matrix \(\rho^D=\rho\otimes\rho\) implements postselection as an EPR-state projection and directly tracks the second Rényi entropy \(S_A^{(2)}=-\log {\rm Tr}\,\rho_A^2\). The construction preserves Hermiticity, unit trace, and positive definiteness, and a hard-core Bose-Hubbard example shows a transition from volume-law to area-law scaling as the measurement rate is increased [2204.09049].

Environmental noise complicates experimental observation because generic decoherence destroys the volume-law phase. One proposal counteracts this by inserting quantum-enhanced operations that supply a random field opposite to the noise-induced one in the effective statistical-mechanics mapping. The relevant symmetry-restoring condition is
$$
q_n=q_e,\qquad U_{SE}=U_{SA},\qquad \rho_E=\rho_A,
$$
which yields zero net external field and an average apparatus–environment exchange symmetry. Under this condition, the conditional entanglement entropy is again a valid entanglement probe, and in a \((2+1)\)-dimensional Clifford circuit with dephasing one finds, at \(q=0.1\), a collapse consistent with
$$
p_c=0.214(2),\qquad \nu=0.9(1)
$$
for the conditional tripartite mutual information [2406.14109].

The opposite limit is information loss rather than extra control. In the exactly solvable monitored Kitaev chain with partial postselection, missing each click with probability \(1-q\) produces a state-dependent but quasi-free Liouvillian. For any \(q<1\), averaging over multiple realizations introduces an effective finite length scale, beyond which long-range correlations are exponentially suppressed, the Liouvillian gap remains strictly positive, and the entanglement negativity saturates. The critical phase survives only at perfect postselection \(q=1\), where \(\gamma_c(\mu)=4\sqrt{1-\mu^2}\) [2407.13837]. This establishes that imperfect trajectory discrimination does not merely broaden the transition; it can eliminate the critical regime altogether.

## 6. Related phenomena, conceptual links, and interpretive frameworks

Several analytical descriptions recur across the literature. In the Bose-Hubbard study, the observed critical exponents and mutual-information decay are described as consistent with a boundary-percolation/CFT picture, and the parallels to many-body localization suggest a unified effective description in terms of replica field theories or random tensor-network models [1908.11253]. In the diffusive-measurer problem, a generalized Harris criterion diagnoses the relevance of correlated measurement-rate fluctuations and leads to a new fixed point with \(z'=2\) and weak Griffiths-like effects [2405.08861]. In monitored free fermions, the long-time behavior is captured by a nonlinear sigma model whose target manifold and RG structure coincide with the clean monitored case, while disorder simply renormalizes the stiffness and decay rate [2605.05306].

Replica symmetry breaking provides another route to entanglement criticality. In Brownian SYK chains with continuous monitoring, the noninteracting SYK\(_2\) model has a continuous \(O(2)\) symmetry between replicas. The low-measurement phase spontaneously breaks that symmetry, generating a Goldstone mode and logarithmic entanglement entropy attributed to the free energy of vortices; the high-measurement phase is replica-symmetric and area-law. In the interacting case, the symmetry is reduced to \(C_4\), and the symmetry-broken phase acquires volume-law entanglement because domain walls have a linear free-energy cost [2104.08270].

The term “measurement-induced phase transition” is not restricted to trajectory entanglement. In a periodically measured transverse Ising chain, repeated global measurements of whether the transverse magnetization per site satisfies \(M_z\neq 1\) yield an exponentially decaying first-detection probability \(p_n\sim e^{-\beta n}\) with decay constant
$$
\alpha(\Gamma,\tau)=-\frac{1}{2\pi\tau^2}\int_0^\pi dk\,\ln[1-g_k^2].
$$
The function \(\alpha(\Gamma,\tau)\) is continuous, but its first derivatives have finite jumps across the critical manifold
$$
\tau_0\sqrt{1-\Gamma_0^2}=\frac{\pi}{4},\qquad \Gamma_0<1,
$$
producing a non-analyticity that is explicitly distinct from the equilibrium ground-state quantum phase transition at \(\Gamma=1\) [1603.03561]. A later study of periodic projective measurements onto the all-up state in a quantum Ising chain finds a finite-size transition at \(\tau_c\sim 0.2\) for \(h=1/2\) and \(L\sim 26\), but derives evidence that \(\tau_c\sim 1/\sqrt{L}\), so the transition recedes to \(\tau_c=0\) in the thermodynamic limit [2501.17606].

Beyond entanglement, monitored dynamics can reorganize information transfer and estimation. In Kitaev-Yoshida and Gao-Jafferis-Wall teleportation protocols for SYK, large projection rate \(\gamma\equiv\kappa\) or large coupling \(\mu\) drives a transition from a regime where teleportation is possible only within a finite time window to a steady state where teleportation can occur at any moment. In dual Jackiw-Teitelboim gravity, this is interpreted as the formation of an eternal traversable wormhole, with the late-time two-sided correlator acting as the order parameter [2210.03083]. In a classical chaotic system represented by a branching tree with \(K=e^{\lambda\Delta t}\), repeated noisy measurements lead to a transition in the Shannon entropy production rate,
$$
s=\lim_{t\to\infty}\frac{\langle S_t\rangle}{t}
=
\begin{cases}
\lambda-\frac{1}{\Delta t}D_{\rm KL}(P_1\parallel P_0), & D_{\rm KL}\le \lambda\Delta t,\\[4pt]
0, & D_{\rm KL}>\lambda\Delta t,
\end{cases}
$$
with the critical condition \(D_{\rm KL}(P_1\parallel P_0)=\lambda\Delta t\). The mapping to a directed polymer on a Cayley tree shows that the transition location coincides with the polymer freezing point, although the critical behavior differs [2501.00547].

Taken together, these results show that “measurement-induced phase transition” now names a family of nonequilibrium non-analytic phenomena generated by the feedback between dynamics and information extraction. The most developed branch concerns trajectory entanglement transitions in monitored many-body systems, but the same language also encompasses first-detection singularities, teleportation transitions, and state-estimation thresholds. What unifies them is not a single microscopic mechanism, but the emergence of sharp dynamical phase structure once measurement backaction competes with coherent evolution.

Source: https://www.emergentmind.com/topics/measurement-induced-phase-transition