Measurement-Induced Phase Transitions

• Measurement-induced phase transitions are nonequilibrium phenomena where the competition between entanglement-generating unitary dynamics and local measurements drives the system from volume-law to area-law entanglement.
• Key studies using Bose-Hubbard, Ising, and free-fermion models demonstrate sharp transitions with logarithmic scaling and distinctive critical exponents, elucidating the role of measurement feedback.
• This topic offers practical insights into experimental diagnostics, utilizing tools such as mutual information and quantum Fisher information to probe phase transitions in quantum many-body systems.

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 (Tang et al., 2019, Ha et al., 2024). 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 (Dhar et al., 2016, Milekhin et al., 2022, Gerbino et al., 2024).

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 $p$ (Ha et al., 2024).

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 (Tang et al., 2019). 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 $p$0; postselecting the no-click trajectories yields pure-state evolution under the non-Hermitian effective Hamiltonian $p$1, and the steady-state entanglement changes at $p$2 (Fresco et al., 2023).

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 $p$3-driven transition, while nonlocal measurements lead instead to a subvolume-to-area law transition (Yokomizo et al., 2024). In monitored free fermions in spatial dimensions $p$4, the weak-measurement phase exhibits $p$5 scaling rather than a volume law, followed by an area-law phase at stronger monitoring (Liao et al., 6 May 2026). 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 $p$6. In the Bose-Hubbard study, the long-time steady-state entropy satisfies $p$7 in the volume-law phase and saturates at $p$8 in the area-law phase; at fixed $p$9, the critical spatial measurement rate is $p$0 (Tang et al., 2019). In strongly disordered spin chains, the order parameter is the tripartite mutual information

$p$1

with $p$2 each occupying one quarter of the chain; in the long-time limit, $p$3 saturates to an $p$4 negative constant in the volume-law phase and to zero in the area-law phase (Tang et al., 1 Dec 2025).

Several works emphasize more refined diagnostics. The Bose-Hubbard calculation introduces the single-site entropy

$p$5

whose probability distribution $p$6 has two peaks, one at $p$7 and one at $p$8, deep in the volume-law phase; the finite-entropy peak broadens and loses weight near criticality, and only the $p$9 peak remains above the transition. The same study also uses the two-site mutual information $n_i=1$0 as a probe of critical spatial structure (Tang et al., 2019).

In the non-Hermitian Ising setting, the quantum Fisher information (QFI) provides both a metrological and an entanglement diagnostic. For a pure steady state $n_i=1$1 probed by a small twist $n_i=1$2, the QFI is $n_i=1$3, and the chain QFI obeys

$n_i=1$4

The exponent changes sharply across the transition: $n_i=1$5 in the entangling phase and $n_i=1$6 in the disentangling phase. Since $n_i=1$7 certifies at least $n_i=1$8-partite entanglement, the super-extensive scaling in the entangling phase witnesses large-scale multipartite entanglement (Fresco et al., 2023).

When trajectory discrimination is imperfect, mixed-state diagnostics become necessary. In the solvable partially postselected Kitaev-chain Liouvillian, the fermionic entanglement negativity grows as $n_i=1$9 only for perfect postselection $U/J_0=0.14$0 and $U/J_0=0.14$1; for any $U/J_0=0.14$2, it crosses over and saturates beyond a finite length scale $U/J_0=0.14$3 (Paviglianiti et al., 2024). For noisy monitored circuits, the conditional entanglement entropy

$U/J_0=0.14$4

is used as the central probe; under the symmetry conditions specified below, it becomes a valid entanglement measure and supports a genuine entanglement transition (Qian et al., 2024).

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 $U/J_0=0.14$5, the entanglement growth at the critical spatial measurement rate satisfies

$U/J_0=0.14$6

while in the long-time steady state at $U/J_0=0.14$7,

$U/J_0=0.14$8

The same work reports the one-parameter scaling form

$U/J_0=0.14$9

with $\Pi_i=\lvert 1\rangle\langle 1\rvert$0 and $\Pi_i=\lvert 1\rangle\langle 1\rvert$1. At criticality the mutual information exhibits a power-law tail $\Pi_i=\lvert 1\rangle\langle 1\rvert$2, with $\Pi_i=\lvert 1\rangle\langle 1\rvert$3, tending toward $\Pi_i=\lvert 1\rangle\langle 1\rvert$4 in the large-$\Pi_i=\lvert 1\rangle\langle 1\rvert$5 limit; both $\Pi_i=\lvert 1\rangle\langle 1\rvert$6 and $\Pi_i=\lvert 1\rangle\langle 1\rvert$7 are described as consistent with a boundary-percolation/CFT picture (Tang et al., 2019).

For the standard $\Pi_i=\lvert 1\rangle\langle 1\rvert$8 dimensional circuit transition without conserved densities, the large-local-Hilbert-space description is a nonunitary CFT / percolation theory, with correlation-length exponent $\Pi_i=\lvert 1\rangle\langle 1\rvert$9 and dynamical exponent $P_x$0. The half-chain entanglement obeys the finite-size scaling form

$P_x$1

and criticality corresponds to a logarithmic violation of area-law scaling (Ha et al., 2024).

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 $P_x$2, correlation-length exponent $P_x$3, and critical entanglement scaling

$P_x$4

with $P_x$5. The modified finite-size scaling form

$P_x$6

contains an explicit $P_x$7 argument encoding the diffusive fluctuation amplitude of the measurer density (Ha et al., 2024).

Long-range couplings likewise alter criticality. In a one-dimensional hybrid circuit with two-qubit Clifford gates acting at distance $P_x$8 with probability $P_x$9, the transition is short-range-like for $N_t$0, with $N_t$1 and $N_t$2, but for $N_t$3 the critical exponents vary continuously with $N_t$4, giving a continuum of non-conformal universality classes. For $N_t$5, no true area-law phase survives; the large-$N_t$6 regime shows sub-volume scaling $N_t$7 (Block et al., 2021).

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 $N_t$8 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

$N_t$9

but the work explicitly states that it is almost impossible to rule out a finite-size crossover rather than an actual thermodynamic transition (Chatterjee et al., 2024).

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 (Tang et al., 2019).

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 $\sigma^z$0-bit Hamiltonian, any infinitesimal measurement rate $\sigma^z$1 destroys the purely many-body-localized logarithmic growth $\sigma^z$2 and instead induces fast entanglement growth $\sigma^z$3 followed by a volume law $\sigma^z$4. A second transition then occurs at $\sigma^z$5, above which repeated measurements produce an area-law state. The critical rate is exponentially small in the disorder strength $\sigma^z$6 and the overlap $\sigma^z$7 between the measurement operator and the local integrals of motion,

$\sigma^z$8

and the saturation time changes from $\sigma^z$9 in the unmeasured MBL regime to $n_i=(1+\sigma_i^z)/2$0 in the measurement-induced volume-law phase (Tang et al., 1 Dec 2025).

For monitored free fermions, quenched disorder has a different status. In $n_i=(1+\sigma_i^z)/2$1-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 $n_i=(1+\sigma_i^z)/2$2, a transition occurs between an $n_i=(1+\sigma_i^z)/2$3 phase and an area law, while in $n_i=(1+\sigma_i^z)/2$4 there is only an area law and no genuine transition (Liao et al., 6 May 2026).

Measurement geometry can itself determine whether a transition exists. In continuously monitored free bosons with long-range couplings, local measurements $n_i=(1+\sigma_i^z)/2$5 do not induce a $n_i=(1+\sigma_i^z)/2$6-driven transition because the long-range Hamiltonian entanglement generation overcomes local entanglement destruction. By contrast, nonlocal measurements

$n_i=(1+\sigma_i^z)/2$7

directly suppress long-range entanglement generation and produce a subvolume-to-area law transition at a critical $n_i=(1+\sigma_i^z)/2$8 for $n_i=(1+\sigma_i^z)/2$9 (Yokomizo et al., 2024).

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$00 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 $p$01 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 (Negari et al., 2023).

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 (Buchhold et al., 2022).

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 $p$02, so that the single-site occupation histogram becomes approximately uniform in the volume-law phase and $p$03-peaked in the area-law phase. The crossing point of distances to the uniform and $p$04-peaked distributions gives an estimate of $p$05 consistent with the entanglement-based transition, offering a simple-observable probe that avoids full trajectory postselection (Martín-Vázquez et al., 2023).

Continuous-time formulations make postselection explicit at the level of the evolution equation. A generalized Lindblad master equation for the double-space density matrix $p$06 implements postselection as an EPR-state projection and directly tracks the second Rényi entropy $p$07. 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 (Zhou, 2022).

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

$p$08

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 $p$09-dimensional Clifford circuit with dephasing one finds, at $p$10, a collapse consistent with

$p$11

for the conditional tripartite mutual information (Qian et al., 2024).

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 $p$12 produces a state-dependent but quasi-free Liouvillian. For any $p$13, 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 $p$14, where $p$15 (Paviglianiti et al., 2024). 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 (Tang et al., 2019). 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 $p$16 and weak Griffiths-like effects (Ha et al., 2024). 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 (Liao et al., 6 May 2026).

Replica symmetry breaking provides another route to entanglement criticality. In Brownian SYK chains with continuous monitoring, the noninteracting SYK$p$17 model has a continuous $p$18 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 $p$19, and the symmetry-broken phase acquires volume-law entanglement because domain walls have a linear free-energy cost (Jian et al., 2021).

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 $p$20 yield an exponentially decaying first-detection probability $p$21 with decay constant

$p$22

The function $p$23 is continuous, but its first derivatives have finite jumps across the critical manifold

$p$24

producing a non-analyticity that is explicitly distinct from the equilibrium ground-state quantum phase transition at $p$25 (Dhar et al., 2016). A later study of periodic projective measurements onto the all-up state in a quantum Ising chain finds a finite-size transition at $p$26 for $p$27 and $p$28, but derives evidence that $p$29, so the transition recedes to $p$30 in the thermodynamic limit (Chaki et al., 29 Jan 2025).

Beyond entanglement, monitored dynamics can reorganize information transfer and estimation. In Kitaev-Yoshida and Gao-Jafferis-Wall teleportation protocols for SYK, large projection rate $p$31 or large coupling $p$32 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 (Milekhin et al., 2022). In a classical chaotic system represented by a branching tree with $p$33, repeated noisy measurements lead to a transition in the Shannon entropy production rate,

$p$34

with the critical condition $p$35. 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 (Gerbino et al., 2024).

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.