Monitored many-body quantum systems can undergo sharp learnability transitions characterized by how much information can be learned by the observer. When the dynamics conserves a non-Abelian charge, such as an
SU(2) spin, understanding how the observer learns the total charge remains an outstanding problem. Unlike the Abelian case, where charge measurements on distinct sites commute, the
SU(2)-symmetric readouts are noncommuting fusion measurements, making learning a genuinely quantum inference problem. In this work, we propose a theory of
1+1d monitored quantum dynamics with
SU(2) symmetry, and show that it can be described by an effective replicated loop model comprised of a replica-pairing field and a diffusive (
z=2) background sector that carries the
SU(2) charge and remains gapless throughout the phase diagram. Our theory predicts that the "spin-sharpening'' and entanglement transitions coincide as a single transition. Ordering of the pairing field produces volume-law entanglement and hides the background sector from measurements, leading to a learning time of
t∼L3 for the total spin. When the pairing field disorders, the background sector alone gives logarithmic entanglement and a diffusive learning time
t∼L2. Our analysis is controlled by a large-loop-fugacity expansion.
Monitored quantum circuits can exhibit an entanglement transition as a function of the rate of measurements, stemming from the competition between scrambling unitary dynamics and disentangling projective measurements.We study how entanglement dynamics in nonunitary quantum circuits can be enriched in the presence of charge conservation, using a combination of exact numerics and a mapping onto a statistical mechanics model of constrained hard-core random walkers.We uncover a charge-sharpening transition that separates different scrambling phases with volume-law scaling of entanglement, distinguished by whether measurements can efficiently reveal the total charge of the system.We find that while Rényi entropies grow sub-ballistically as ffiffi t p in the absence of measurement, for even an infinitesimal rate of measurements, all average Rényi entropies grow ballistically with time ∼t.We study numerically the critical behavior of the charge-sharpening and entanglement transitions in U(1) circuits, and show that they exhibit emergent Lorentz invariance and can also be diagnosed using scalable local ancilla probes.Our statistical mechanical mapping technique readily generalizes to arbitrary Abelian groups, and offers a general framework for studying dissipatively stabilized symmetry-breaking and topological orders.
The dynamics of monitored systems can exhibit a measurement-induced phase transition (MIPT) between entangling and disentangling phases, tuned by the measurement rate. When the dynamics obeys a continuous symmetry, the entangling phase further splits into a fuzzy phase and a sharp phase based on the scaling of fluctuations of the symmetry charge. While the sharpening transition for Abelian symmetries is well understood analytically, no such understanding exists for the non- Abelian case. In this work, building on a recent analytical solution of the MIPT on tree-like circuit architectures (where qubits are repatedly added or removed from the system in a recursive pattern), we study entanglement and sharpening transitions in monitored dynamical quantum trees obeying U(1) and SU(2) symmetries. The recursive structure of tree tensor networks enables powerful analytical and numerical methods to determine the phase diagrams in both cases. In the U(1) case, we analytically derive a Fisher-KPP-like differential equation that allows us to locate the critical point and identify its properties. We find that the entanglement/purification and sharpening transitions generically occur at distinct measurement rates. In the SU(2) case, we find that the fuzzy phase is generic, and a sharp phase is possible only in the limit of maximal measurement rate. In this limit, we analytically solve the boundaries separating the fuzzy and sharp phases, and find them to be in agreement with exact numerical simulations.
Monitored quantum circuits (MRCs) exhibit a measurement-induced phase transition between area-law and volume-law entanglement scaling. MRCs with a conserved charge additionally exhibit two distinct volume-law entangled phases that cannot be characterized by equilibrium notions of symmetry-breaking or topological order, but rather by the nonequilibrium dynamics and steady-state distribution of charge fluctuations. These include a charge-fuzzy phase in which charge information is rapidly scrambled leading to slowly decaying spatial fluctuations of charge in the steady state, and a charge-sharp phase in which measurements collapse quantum fluctuations of charge without destroying the volume-law entanglement of neutral degrees of freedom. By taking a continuous-time, weak-measurement limit, we construct a controlled replica field theory description of these phases and their intervening charge-sharpening transition in one spatial dimension. We find that the charge fuzzy phase is a critical phase with continuously evolving critical exponents that terminates in a modified Kosterlitz-Thouless transition to the short-range correlated charge-sharp phase. We numerically corroborate these scaling predictions also hold for discrete-time projective-measurement circuit models using large-scale matrix-product state simulations, and discuss generalizations to higher dimensions.
We study the critical properties of random quantum circuits with a
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)-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)-symmetric Haar and stabilizer random circuits.
A quantum circuit interrupted by random measurements undergoes a well-known entanglement transition; when it also conserves a charge, it undergoes a second, distinct "charge-sharpening" transition at which the measurement record comes to pin down the total charge. This paper shows that charge sharpening is an objectivity transition — the quantum-Darwinism transition of the conserved charge. For a charge-conserving monitored circuit the paper establishes several results. Objectivity onset is exactly charge sharpening, and is generically a different transition from the entanglement one. In the sharp phase the charge is redundantly broadcast into the record, with redundancy growing as the system size, and the sharp-phase state has broadcast structure whose center is the charge. A per-fragment information measure rises from zero to one bit across the transition. And the objective sector is exactly the conserved charge, while its conjugate stays hidden. The paper further shows that objectivity — many observers each agreeing — is strictly stronger than mere learnability by a single decoder reading the whole record, and that the picture extends from the simplest symmetry to any compact symmetry group, including non-abelian spin sharpening. The transition carries a falsifiable signature: the redundancy is an observable order parameter, few-shot readable from a small fragment in the sharp phase and not in the fuzzy phase, testable on today's symmetric monitored simulators; every structural result is proved analytically and cross-checked by exact computation. In short: charge sharpening is the moment a monitored circuit's conserved charge becomes objective — many witnesses, one classical charge.