Measurement-Induced Phase Transitions (MIPT)

• Measurement-induced phase transitions (MIPTs) are nonequilibrium transitions in monitored quantum systems where competing unitary dynamics and measurements switch entanglement from volume law to area law scaling.
• They are characterized by universal critical scaling, with key metrics like von Neumann entropy and Rényi entropies, and are often described by nonunitary conformal field theories.
• MIPTs offer practical insights for quantum error correction and information processing by enabling experimental protocols that probe distinct universality classes in quantum circuits.

Measurement-induced phase transitions (MIPTs) are nonequilibrium transitions that emerge in quantum many-body systems subject to both intrinsic unitary dynamics and quantum measurements. As the rate or strength of measurements is tuned, the system can transition abruptly between an "entangled" phase—where quantum correlations and entanglement scale extensively—and a "disentangled" or purified phase, where entanglement is suppressed. Unlike conventional thermal or symmetry-breaking transitions, MIPTs are defined by the scaling behavior of nonlinear entanglement measures (e.g., the von Neumann or Rényi entropies), as the system's wavefunction is conditioned on stochastic measurement outcomes. The critical properties of MIPTs can often be mapped to nonunitary conformal field theories, and their universality classes are sensitive to the microscopic details of the quantum circuit, the measurement protocols, and any spatial correlations in the measurement process.

1. Fundamentals: Competing Dynamics and Entanglement Scaling

At the core of MIPT is the competition between the entanglement-generating effect of unitary time evolution and the disentangling effect of local or global quantum measurements. In generic monitored circuits, unitary gates scramble quantum information, leading to rapid growth of bipartite or multipartite entanglement: for a subsystem of size $\ell$ in an $L$-site chain, the entanglement entropy $S_A$ typically follows a volume law, $S_A \sim \ell$. Projective measurements, in contrast, collapse local degrees of freedom, suppressing correlations and driving the system toward a pure product state (area law, $S_A \sim \mathcal{O}(1)$).

MIPTs separate these regimes: as the measurement rate $p$ increases past a critical threshold $p_c$, the scaling of the steady-state entanglement entropy transitions from volume law to area law. At $p=p_c$, critical scaling—often logarithmic in system size—emerges. This scenario is quantitatively mirrored in models such as random unitary circuits with interleaved projective measurements, monitored Clifford and Haar-random circuits, and measurement-only circuits.

Universal scaling forms for the entanglement entropy are observed,

$$S_A(p, cN) \sim N^\gamma F\left[(p-p_c)N^{1/\nu}\right]$$

where $F$ is a universal scaling function, $\gamma$ and $\nu$ are critical exponents, and $c=\ell/N$. At criticality, $S_A \sim \log N$ in one dimension—directly analogous to scaling at equilibrium conformal critical points (Christopoulos et al., 10 Sep 2024).

2. Conformal Field Theory and Universality

The critical point of the MIPT is governed by an emergent nonunitary conformal field theory (CFT), but with effective central charge $c_{\mathrm{eff}}\neq 0$ even though the underlying theory is nonunitary and formally $c=0$ in the replica limit. Using a transfer-matrix approach, both the effective central charge and operator scaling dimensions can be extracted via Lyapunov exponents,

$$f(L) = f(\infty) - \frac{\pi c_{\mathrm{eff}}}{6L^2} + ...$$

and

$$f_i(L) - f(L) = \frac{2\pi x_i^{\mathrm{typ}}}{L^2}$$

where $f(L)$ is the free energy density, and $x_i^{\mathrm{typ}}$ are typical scaling dimensions (Zabalo et al., 2021).

MIPTs in Haar-random and Clifford circuits belong to distinct universality classes, with Clifford circuits (stabilizer dynamics) exhibiting $c_{\mathrm{eff}} \approx 0.37$ and generic Haar circuits $c_{\mathrm{eff}} \approx 0.25$. At large local Hilbert space dimension, the MIPT maps to a classical percolation transition, yet for $d=2$ (qubits), the observed universality classes diverge from percolation. Moreover, the generic critical point displays multifractal scaling—correlation functions exhibit a continuum of exponents rather than a single monofractal decay law; this is reflected in scaling of moments of order-parameter correlation functions and a spectrum of scaling dimensions.

Boundary effects—studied via transfer-matrix with boundary-condition changing operators—yield additional invariants, such as dissipator and entanglement scaling dimensions $h_{f|a}, h_{a|b}$, further distinguishing universality classes (Kumar et al., 2023).

3. Measurement Protocols, Noise, and Symmetry Constraints

MIPT phenomenology generalizes to a broad class of measurement protocols beyond local projective measurements. Protocols where quantum information from the system is transduced to an apparatus (or environment) in a symmetric or balanced manner exhibit a similar entanglement transition. This is formalized via “information exchange symmetry” (IE symmetry): transitions occur as spontaneous symmetry breaking of the replica subgroup $D_n \rtimes \mathbb{Z}_2$, where $D_n$ is the dihedral group. Explicitly, the IE symmetry ensures the conditional Rényi entropies for conditioning on apparatus or environment are equal,

$$S^{(n)}(P_S, P_{S'}; \mathcal{A}) = S^{(n)}(P_S, P_{S'}; \mathbb{E})$$

and its spontaneous breaking demarcates the transition from area- to volume-law entanglement scaling. These ideas extend to quantum-enhanced and noisy hybrid protocols, with distinct universality classes if the cluster weights in the associated random bond cluster models differ ($h(n)<n!$ for $n>2$ is not percolation) (Kelly et al., 16 Feb 2024).

In realistic devices, decoherence noise can destroy the volume-law phase, but recent work demonstrates that quantum-enhanced operations—incorporating ancilla-assisted coding—can protect the MIPT. Conditional entanglement entropy (CEE), averaged over ancilla/environment, remains a good probe for the transition if an average apparatus-environment symmetry (aAEE) is restored, operationally achievable by tuning noise and QE rates to restore a zero external field in a dual statistical mechanics mapping (Qian et al., 20 Jun 2024).

4. Dynamical and Critical Features

The temporal dynamics of the entanglement entropy near MIPT can be characterized by critical exponents associated with scaling collapse. Importantly, the transition is not always revealed in ensemble-averaged observables, but is a property of the underlying distribution of quantum trajectories—ensemble averaging or information loss (imperfect post-selection) generically destroys the critical scaling of long-range correlations, introducing a finite correlation length and Liouvillian gap, and saturates the entanglement negativity (Paviglianiti et al., 18 Jul 2024).

The critical properties of MIPT can further be probed using relaxation and driven dynamics scaling. For example, when driving the measurement rate linearly across $p_c$ (Kibble–Zurek mechanism), the entanglement entropy at $p=p_c$ scales as $S\sim\ln R$ (from the area-law side) or $S\sim R^{1/r}$ (from the volume-law side), with $r = z + 1/\nu$ (Wang et al., 11 Nov 2024). Relaxation after a quench from a product state shows $S\sim\ln t$ at $p=p_c$, while from a volume-law state $S\sim t^{-1}$. These regimes are unified by a universal dynamical scaling function,

$$S(t, g, L) = \alpha \ln L + G\left(tL^{-z}, gL^{1/\nu}, X_0\right)$$

where $z$ is the dynamic exponent, and $X_0$ encodes information about the initial state (Wang et al., 19 Feb 2025).

Multipartite entanglement is detectable via Quantum Fisher Information (QFI), which shifts its scaling from super-extensive ($F\sim N^{1.5}$) to extensive ($F\sim N$) as measurement strength crosses $p_c$. The non-analyticity and divergence of QFI with respect to measurement strength serve as further witnesses of the MIPT (Fresco et al., 2023).

5. Circuit Architectures, Classical Stochastic Mappings, and Applications

A broad spectrum of monitored quantum circuits exhibits MIPTs: random unitary circuits with Haar or Clifford gates, tree-like circuits with weak to strong measurements, and even circuits with diffusive or correlated measurement processes. For certain architectures, e.g., binary quantum trees, recursive decoding and analytical methods allow for linear-complexity extraction of the MIPT critical point and scaling laws—these results have been experimentally demonstrated on quantum hardware (e.g., Quantinuum H1-1) (Feng et al., 3 Feb 2025).

Classical models, such as the directed polymer problem or percolation, provide a mapping for understanding the scaling and entropy growth at the transition. In classical chaotic systems, repeated imperfect measurements induce an MIPT in the observer’s uncertainty, with the transition point coinciding with the freezing transition of the directed polymer, albeit with distinct scaling behavior arising from the replica structure of the measurement problem (Gerbino et al., 31 Dec 2024).

Applications of MIPT extend to quantum error correction, quantum information theory, and the paper of learnability transitions in monitored dynamics. For instance, robust quantum-enhanced protocols use QE operations to protect entanglement in noisy environments. Attention-based machine learning methods (Quantum Attention Networks) can efficiently infer the phase boundary for MIPTs from measurement data alone, even in the absence of post-selection or direct entanglement diagnostics; these methods achieve efficient, noise-tolerant upper bounds on the location of the MIPT (Kim et al., 21 Aug 2025).

6. Experimental Challenges and Advances

The primary experimental barrier to studying MIPT is the post-selection problem: the need to distinguish single quantum trajectories, which incurs an exponential cost in measurement records. Recent progress has introduced circuit architectures (e.g., tree circuits with recursive structure) and protocols (e.g., cross-correlation and learnability-based machine learning approaches) that circumvent these limitations, greatly improving scalability and feasibility in near-term quantum devices (Feng et al., 3 Feb 2025, Kim et al., 21 Aug 2025).

Feedback-based measurement protocols and hybrid classical-quantum simulation strategies (e.g., for transmon arrays and Bose–Hubbard models) provide practical pathways to detect phase transitions using simple observables—such as local number distributions—thus bypassing full state tomography or exponential post-selection (Martín-Vázquez et al., 2023).

7. Outlook and Generalizations

Recent theoretical developments extend the paradigm of MIPT to generalized symmetry-breaking transitions, such as information exchange symmetry breaking, where the transition is defined by the equality of entropic measures conditioned on two complementary baths. In nonstandard physical contexts—including bosonic systems with long-range interactions (Yokomizo et al., 30 May 2024), systems with correlated/diffusive measurements (Ha et al., 14 May 2024), and even in classical chaotic inference—MIPT remains a robust concept, but universality and critical scaling can differ markedly from the canonical models.

A plausible implication is that MIPT-type transitions and their generalizations may play foundational roles in quantum information processing, error correcting architectures, and potentially even in emergent behaviors in networks (such as cognitive or neural networks via “learnability transitions” (Gorsky, 5 Jun 2025)).

Table: Key Theoretical Constructs in MIPT

Construct	Equation/Description	Reference
Entanglement entropy scaling	$S_A \sim \ell$ (volume law) or $S_A \sim \mathcal{O}(1)$ (area law)	(Zabalo et al., 2021, Christopoulos et al., 10 Sep 2024)
Free energy scaling, central charge extraction	$$f(L) = f(\infty) - \frac{\pi c_{\mathrm{eff}}}{6L^2}$$	(Zabalo et al., 2021)
Operator scaling dimensions	$$f_i(L) - f(L) = \frac{2\pi x^{\mathrm{typ}}_i}{L^2}$$	(Zabalo et al., 2021)
Information exchange symmetry	$$S^{(n)}(P_S,P_{S'}; \mathcal{A}) = S^{(n)}(P_S,P_{S'}; \mathbb{E})$$	(Kelly et al., 16 Feb 2024)
Relaxation scaling form	$$S(t,g,L) = \alpha \ln L + G(tL^{-z}, gL^{1/\nu}, X_0)$$	(Wang et al., 19 Feb 2025)
Kibble–Zurek scaling at MIPT	$S\propto \ln R$ or $S\propto R^{1/r}$; $r = z + 1/\nu$	(Wang et al., 11 Nov 2024)

These results collectively provide a comprehensive framework for describing, diagnosing, and probing measurement-induced phase transitions in monitored quantum systems and their generalizations, with wide-ranging implications for both theory and experiment.