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Measurement-induced criticality in random quantum circuits
Chao-Ming Jian, Yi-Zhuang You, Romain Vasseur, Andreas W. W. Ludwig
TL;DR
The paper asks how projective measurements drive critical transitions between area- and volume-law entanglement in Haar-random unitary circuits. It uses replicas to map Rényi-entropy calculations to a two-dimensional statistical-mechanics model, interpreting the transition as ordering and deriving conformal scaling. At infinite on-site dimension, the transition is percolation criticality with an exact logarithmic coefficient 1/6 for n ≥1 Rényi entropies, while finite dimension is accessed as a crossover from this limit.
Problem
Measurement-induced entanglement transitions were known largely from numerical results, leaving their critical theory, scaling, and relation to percolation and tensor-network transitions unresolved.
Source: §I · p. 1Method
A replica approach maps averaged Rényi entropies in Haar-random circuits with projective measurements onto a two-dimensional statistical-mechanics model.
Source: §I · p. 1, §II · p. 3Results
At d = ∞, the transition is in the percolation universality class and the universal logarithmic coefficient is 1/6 for n ≥1 Rényi entropies of a half-system with open boundaries.
Source: §I · p. 2Takeaways & Limitations
Conformal invariance supplies universal entanglement and mutual-information scaling, while finite-dimension transitions arise from a crossover away from the analytically tractable percolation limit.
Source: §I · pp. 1–2, §IV · p. 7, §VI · p. 9
Abstract
from arXiv · showhide
We investigate the critical behavior of the entanglement transition induced by projective measurements in (Haar) random unitary quantum circuits. Using a replica approach, we map the calculation of the entanglement entropies in such circuits onto a two-dimensional statistical mechanics model. In this language, the area- to volume-law entanglement transition can be interpreted as an ordering transition in the statistical mechanics model. We derive the general scaling properties of the entanglement entropies and mutual information near the transition using conformal invariance. We analyze in detail the limit of infinite on-site Hilbert space dimension in which the statistical mechanics model maps onto percolation. In particular, we compute the exact value of the universal coefficient of the logarithm of subsystem size in the $n$th R\'enyi entropies for $n \geq 1$ in this limit using relatively recent results for conformal field theory describing the critical theory of 2D percolation, and we discuss how to access the generic transition at finite on-site Hilbert space dimension from this limit, which is in a universality class different from 2D percolation. We also comment on the relation to the entanglement transition in Random Tensor Networks, studied previously in Ref. 1.
I. INTRODUCTION
The paper develops a theory of measurement-induced entanglement transitions in random unitary circuits and analyzes their critical behavior through a statistical-mechanics mapping. In the infinite-dimensional limit, it identifies percolation criticality and an exact Rényi-entropy logarithmic coefficient, while distinguishing this behavior from finite-dimensional transitions and zeroth Rényi results.
Theory and mapping
- A replica mapping turns the projective measurement-induced area-to-volume-law transition into an ordering transition in a two-dimensional statistical-mechanics model.The mapping also yields universal scaling forms for entanglement entropy and mutual information through conformal invariance.Source: §I · p. 1
Context and prior work
Infinite-dimensional limit
- At d = ∞, the transition belongs to the percolation universality class, with logarithmic coefficient 1/6 for n ≥1 Rényi entropies of a half-system with open boundaries.This coefficient is obtained from the exactly known percolation conformal field theory.Source: §I · p. 2
Rényi-index dependence
- The n ≥1 Rényi entropies and zeroth Rényi entropy have different universal coefficients, 1/6 versus approximately 0.27, despite coinciding in critical measurement probability at d = ∞.The paper interprets this difference as evidence that the two quantities probe distinct properties, consistent with their differing critical points at finite d.Source: §I · pp. 1–2
Finite-dimensional transition
- For finite on-site dimension, relaxing the fine-tuned percolation limit produces a crossover to a distinct universality class that is not yet analytically tractable.The paper frames the finite-dimensional transition as a crossover from the percolation theory.Source: §I · pp. 1–2
II. RANDOM QUANTUM CIRCUITS
The model is a brick-wall circuit of Haar-random two-site gates and randomly selected one-site generalized measurements acting on a 1D qudit chain. Quantum-channel and replica constructions define trajectory averages and convert Rényi entropies into statistical-mechanics partition-function ratios.
Circuit model
- The circuit evolves a 1D qudit chain with local dimension d using independently Haar-random two-site gates arranged in a brick-wall geometry.The gates act on neighboring sites, while the vertical circuit direction represents discrete time.Source: §II · p. 2, Figure 1 · p. 2
Quantum channels
- Generalized measurements are represented by localized Kraus-operator sets with nonnegative weights satisfying a weighted completeness condition.The associated quantum channel maps a density matrix through the weighted Kraus sum.Source: §II · p. 2
Measurement choices
- Standard projective measurement uses basis projectors P_i, while the probabilistic channel M_p applies a measurement with probability p and leaves the state intact with probability 1 − p.The projectors are P_i = |i⟩⟨i|, and the channel includes the identity alongside these projectors.Source: §II · p. 3
Trajectory averaging
- Each circuit realization defines a quantum trajectory whose probability combines the final-state norm, Kraus weights, and Haar measures over the random gates.Averaging integrates independently over the unitary Haar measures and sums over weighted measurement operators.Source: §II · p. 3
Rényi observables
- Rényi entropies are expressed using permutation operators acting on replicated states, with cyclic permutations imposed on the subsystem and identity permutations on its complement.The averaged entropy is evaluated after averaging replicated circuit and conjugate-circuit tensors.Source: §II · pp. 3–4
Replica mapping
- A double-replica trick resolves the averaged logarithm, producing Q = nm + 1 replicas and the limit m → 0, while the circuit average maps to a 2+0-dimensional statistical-mechanics model.Different permutation boundary conditions produce partition functions whose replica-limit ratios yield the averaged Rényi entropy.Source: §II · p. 4
Statistical-mechanics observable
- In the statistical-mechanics representation, the averaged Rényi entropy becomes a domain-wall free-energy cost between entanglement-region and trivial boundary conditions.For sufficiently deep circuits, the initial state is not essential from this perspective.Source: §II · p. 4
III. STATISTICAL MECHANICS MODEL
A replica calculation maps entanglement entropies in measured Haar random circuits to an SQ statistical mechanics model, where the measurement-driven transition becomes symmetry breaking. At d = ∞, this model reduces to a Q!-state Potts model, while finite-d corrections favor transposition domain walls and reduce the symmetry.
Replica statistical mechanics model
- The replica-averaged circuit becomes an SQ statistical mechanics model on an anisotropic honeycomb lattice, with permutation elements as spins and measurement contributions on solid links.Vertical dotted links encode Weingarten functions from unitary averaging, while zigzag solid links encode generalized measurements.Source: §III · p. 5
Measurement-induced transition
- Tuning the measurement probability p changes the solid-link couplings and drives an ordering transition from a small-p ordered phase to a large-p disordered phase.The ordered phase corresponds to ferromagnetic alignment of neighboring permutation spins, whereas increasing measurement density produces disorder.Source: §III · pp. 5–6
Infinite-dimension limit
- At d = ∞, the SQ model reduces to a Q!-state Potts model on a square lattice because its triangle weights factorize into bond weights.Each bond has weight 1 when neighboring group elements match and p otherwise, reproducing the Potts-model interaction.Source: §III · p. 6
Finite-d corrections
- The statistical mechanics weights can be nonpositive at finite d, making the model non-unitary, consistently with the non-unitary field theory expected from the replica limit.For example, a displayed triangle weight is negative for realistic d ≥ 2.Source: §III · p. 6
- Finite-d corrections preserve the square-lattice bond factorization to leading order but energetically favor transposition domain walls.These corrections break the artificially large SQ! symmetry of the d = ∞ weights to SQ × SQ.Source: §III · p. 6
IV. CONFORMAL INVARIANCE
Conformal invariance identifies the measurement-induced entanglement transition with a two-dimensional c = 0 CFT and fixes universal scaling forms. It predicts a common transition point for all n ≥ 1 Rényi entropies, logarithmic critical entanglement, and cross-ratio dependence of mutual information.
Critical theory
- The entanglement transition is an ordering transition described, when continuous, by a non-unitary two-dimensional CFT with central charge c = 0 in the replica limit.The c = 0 value follows because the replica-limit partition function Z∅ is trivial.Source: §IV · p. 7
Universal transition
- All Rényi entropies with n ≥ 1 share the same transition point p = pc and the same bulk critical exponents in the replica limit.The Rényi index enters through boundary observables, while bulk properties depend only on the replica number Q.Source: §IV · p. 7
Critical scaling
- Conformal invariance fixes the dynamical critical exponent to z = 1, so time and space scale identically at the transition.The correlation length diverges as ξ ∼ |p − pc|^-ν near criticality.Source: §IV · p. 7
Rényi entropy scaling
- At criticality, the nth Rényi entropy scales as log LA with a universal prefactor depending on n, up to additive non-universal constants.The scaling follows from the two-point function of boundary-condition-changing operators and agrees with numerical observations cited in the paper.Source: §IV · p. 7
Mutual information
- At criticality, the mutual information of two intervals depends only on their conformal cross ratio through a four-point function of boundary-condition-changing operators.This scaling form was checked numerically for Clifford unitary circuits in Ref. 44.Source: §IV · p. 7
V. PERCOLATION LIMIT d = ∞
At d →∞, the replica statistical mechanics model becomes a Q!-state Potts model and, in the replica limit, bond percolation. CFT analysis then yields an exact logarithmic coefficient for all n ≥1 Rényi entropies, beyond the minimal-cut description of S0.
Percolation mapping
- In the replica limit Q →1, the Q!-state Potts model becomes bond percolation with critical point p_c = 1/2, correlation length exponent 4/3, and central charge c = 0.The partition function is Z∅ = 1, and occupied bonds have probability 1 − p.Source: §V · p. 7
Beyond minimal cuts
- The percolation connection was anticipated from a minimal-cut picture, but universal quantities for n ≥1 Rényi entropies require the replica formulation and detailed percolation CFT.The relevant result depends on how the percolation CFT is approached in the replica limit, not only on geometric optimization.Source: §V · pp. 7–8
Replica boundary analysis
- A boundary-layer regularization splits the sharp identity-to-SWAP wall into m(n −1) elementary transposition walls, whose total cost is lower in the replica limit.Finite-1/d boundary corrections favor single-transposition walls and produce the split boundary-condition-changing operator.Source: §V · p. 8
- The split boundary-condition-changing operator is identified as Φ_BCC = Φ_{2ℓ−1,4ℓ−1}, with ℓ = m(n −1), and its scaling dimension gives lim_{m→0} h/(n−1)m = 1/6.This vanishing-dimension replica limit produces the universal logarithmic coefficient.Source: §V · p. 8
Rényi entropies
- For all Rényi entropies with n ≥1 at criticality in the limit d →∞, the universal prefactor in eq. (28) goes beyond the geometric “minimal cut” picture of Ref. 41.The universal prefactor is not purely a property of the 2D percolation CFT, because it depends on how that CFT is approached in the replica limit m →0.Source: §V · p. 8
Relation to tensor networks
- The same universal logarithmic coefficient arises in the Random Tensor Network model of Ref. 1 when it is fine-tuned to the percolation critical point.Source: §V · p. 8
VI. GENERIC ENTANGLEMENT TRANSITION AT FINITE HILBERT SPACE DIMENSION d
At finite d, the percolation Potts theory has an enlarged S_{Q!} symmetry that must be broken to S_Q × S_Q. The associated relevant two-hull perturbation is expected to flow to a different, currently unidentified universality class.
Finite-d universality
- Finite-d transitions are expected to differ from percolation because the S_{Q!} symmetry of the d = ∞ Potts model is reduced to S_Q × S_Q.The symmetry-breaking operator has scaling dimension Δ_{2-hull} = 5/4 < 2 and is relevant in the replica percolation limit.Source: §VI · p. 9
Symmetry breaking
- The leading symmetry-breaking perturbation is expressed through a class function W(a^{-1}b) of the permutation group S_Q.Only W(a^{-1}b) = δ_{a,b} preserves the larger S_{Q!} symmetry, whereas general class functions respect S_Q × S_Q.Source: §VI · p. 9
Open problem
- The infrared fate of the perturbed Potts model is unknown, so the conformal field theory of the generic finite-d transition remains unidentified.Source: §VI · p. 9
VII. DISCUSSION
The paper derives an exact replica statistical mechanics description of measurement-induced entanglement transitions and obtains conformal scaling and an exact d = ∞ result. It uses the percolation limit as a starting point for finite d, while identifying the generic finite-d CFT as an open problem.
Main contributions
- The replica trick yields an exact statistical mechanics model for random unitary circuits with projective measurements and predicts conformal scaling forms for entanglement entropy and mutual information.Source: §VII · p. 9
Main results
- At d = ∞, the transition lies in the percolation universality class, with an exact universal logarithmic coefficient for n ≥1 Rényi entropies.Source: §VII · p. 9
Open direction
- Finite-d transitions are framed as percolation perturbed by a 2-hull operator, but identifying the generic conformal field theory remains a future challenge.Source: §VII · p. 9
Related work
- A related work on weak-measurement transitions reported apparently different conclusions about the d →∞ and finite-d limits.Source: §VII · p. 9
Appendix A: The 1/d Expansion
The appendix derives the 1/d expansion of the triangle weight and rewrites it as local link Boltzmann weights on the square lattice. The resulting expansion agrees with the expressions stated in the main text.
Appendix A: The 1/d Expansion
- The expansion of d^|g| and the measurement weight W_p(g) supplies the ingredients for the triangle-weight calculation.Source: Appendix A · p. 11
- The triangle weight J(g_i, g_j; g_k) is evaluated using expansions of the relevant group and Weingarten functions.The derivation uses the 1/d expansion of the triangle weight together with the Weingarten-function expansion.Source: Appendix A · pp. 11–12
- The derived result matches Eqs. (22) and (23) of the main text.Source: Appendix A · p. 12
- To order O(d^-2), the triangle weight factorizes into Boltzmann weights on links of the square lattice.The factorized form is written using an energy function E(g_i,g_k).Source: Appendix A · p. 12
- The link-factorized form is verified by explicitly expanding the product and applying permutation identities.Source: Appendix A · p. 12