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arXiv:2603.15744 · Physical Review Letters

Post-selected Criticality in Measurement-induced Phase Transitions

Dolly Nambi, Kabir Khanna, Andrew Allocca, Thomas Iadecola, Ciarán Hickey, Romain Vasseur, Justin H. Wilson

quant-phcond-mat.dis-nncond-mat.stat-mechcond-mat.str-elSubmitted Mar 16, 2026• Announced Mar 18, 2026• 0 citations• Revised Oct 2, 2026 (v2)
v1 · Mar 16, 2026v2 · Oct 2, 2026
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Abstract

Information-theoretic phase transitions, such as the measurement-induced phase transition (MIPT), characterize the robustness of quantum dynamics to local monitoring and are naturally formulated in terms of trajectories conditioned on typical measurement outcomes, which are naively accessible only through postselection. Here, we implement forced measurements to investigate how explicit postselection alters the nature of the transition. We find that postselection fundamentally alters the universality class by reweighting trajectories that are otherwise rare. In particular, we obtain a correlation-length exponent $ν\approx 2.1$ larger than that of the standard MIPT and a negative effective central charge $c_\mathrm{eff}\approx -0.35$. We also compare the postselected MIPT to the entanglement transition of random tensor networks, and demonstrate that their universality class is the same. This setup further allows time-periodic, translationally invariant circuits with postselected weak measurements. In both models, we find that an on-site dimension of at least 3 (qutrits but not qubits) is necessary to induce a transition.

Journal reference
Phys. Rev. Lett. 137, 130403 (2026)
DOI
10.1103/gx2q-9xz1
Comments
5+3 pages, 3+6 figures

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Information-theoretic phase transitions, such as the measurement-induced phase transition (MIPT), characterize the robustness of quantum dynamics to local monitoring and are naturally formulated in terms of trajectories conditioned on typical measurement outcomes, which are naively accessible only through postselection. Here, we implement forced measurements to investigate how explicit postselection alters the nature of the transition. We find that postselection fundamentally alters the universality class by reweighting trajectories that are otherwise rare. In particular, we obtain a correlation-length exponent ν≈2.1ν\approx 2.1ν≈2.1 larger than that of the standard MIPT and a negative effective central charge ceff≈−0.35c_\mathrm{eff}\approx -0.35ceff​≈−0.35. We also compare the postselected MIPT to the entanglement transition of random tensor networks, and demonstrate that their universality class is the same. This setup further allows time-periodic, translationally invariant circuits with postselected weak measurements. In both models, we find that an on-site dimension of at least 3 (qutrits but not qubits) is necessary to induce a transition.
The effects of different forms of weak measurements on the nature of the measurement induced phase transition are theoretically studied in hybrid random quantum circuits of qubits. We use a combination of entanglement measures, ancilla purification dynamics, and a transfer matrix approach to compute the critical exponents, the effective central charge, and the multifractal spectrum of the measurement induced transitions. We compare weak measurements with an infinite number of discrete outcomes to a protocol with only a pair of outcomes and find that to within our numerical accuracy the universal critical properties are unaffected by the weak measurement protocols and are consistent with the universality class found for strong projective measurements.
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Repeated local measurements of quantum many-body systems can induce a phase transition in their entanglement structure. These measurement-induced phase transitions (MIPTs) have been studied for various types of dynamics, yet most cases yield quantitatively similar critical exponents, making it unclear how many distinct universality classes are present. Here, we probe the properties of the conformal field theories governing these MIPTs using a numerical transfer-matrix method, which allows us to extract the effective central charge, as well as the first few low-lying scaling dimensions of operators at these critical points for (1+1)-dimensional systems. Our results provide convincing evidence that the generic and Clifford MIPTs for qubits lie in different universality classes and that both are distinct from the percolation transition for qudits in the limit of large on-site Hilbert space dimension. For the generic case, we find strong evidence of multifractal scaling of correlation functions at the critical point, reflected in a continuous spectrum of scaling dimensions.
Monitored quantum systems undergo measurement-induced phase transitions (MiPTs) stemming from the interplay between measurements and unitary dynamics. When the detector readout is postselected to match a given value, the dynamics is generated by a non-Hermitian Hamiltonian with MiPTs characterized by different universal features. Here, we derive a stochastic Schrödinger equation based on a microscopic description of continuous weak measurement. This formalism connects the monitored and postselected dynamics to a broader family of stochastic evolution. We apply the formalism to a chain of free fermions subject to partial postselected monitoring of local fermion parities. Within a two-replica approach, we obtain an effective bosonized Hamiltonian in the strong postselected limit. Using a renormalization group analysis, we find that the universality of the non-Hermitian MiPT is stable against a finite (weak) amount of stochasticity. We further show that the passage to the monitored universality occurs abruptly at finite partial postselection, which we confirm from the numerical finite size scaling of the MiPT. Our approach establishes a way to study MiPTs for arbitrary subsets of quantum trajectories and provides a potential route to tackle the experimental postselected problem.
Monitored many-body quantum systems can exhibit a measurement-induced phase transition (MIPT) between entangling and disentangling dynamical phases. Proposed approaches to study the MIPT experimentally typically rely on a classical decoding process, but the complexity of this decoding generally grows exponentially with the system size unless the dynamics is restricted to a fine-tuned set of unitary operators. In this work we overcome this difficulty in the context of tree-shaped quantum circuits. We construct a hybrid circuit with Haar-random unitary operators, and we show that the MIPT can be detected without postselection using a simple decoding process whose complexity grows linearly with the number of qubits. The tree structure also enables a complete theoretical description of the MIPT and all its critical properties. We experimentally realize the MIPT on a trapped-ion quantum computer and show that the results are precisely described by theory without the need for error mitigation. To overcome the exponential complexity of measurement induced phase transition (MIPT), the authors propose a protocol which can detect MIPT easily on a tree-like quantum circuit. They experimentally realize the MIPT on a trapped-ion quantum computer and show that the results are precisely described by theory without the need for error mitigation.
We study the critical properties of random quantum circuits with a U(1)U(1)U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)U(1)U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)U(1)U(1)-symmetric Haar and stabilizer random circuits.