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arXiv:2608.19325

Statistical Mechanics of Non-Abelian Learnability Transitions

Ruochen Ma, Romain Vasseur

quant-phcond-mat.stat-mechcond-mat.str-elSubmitted Aug 19, 2026• Announced Aug 21, 2026• 0 citations
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Abstract

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 $t\sim L^{3}$ for the total spin. When the pairing field disorders, the background sector alone gives logarithmic entanglement and a diffusive learning time $t\sim L^{2}$. Our analysis is controlled by a large-loop-fugacity expansion.

Comments
32 pages, 3 figures

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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)SU(2)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)SU(2)SU(2)-symmetric readouts are noncommuting fusion measurements, making learning a genuinely quantum inference problem. In this work, we propose a theory of 1+1d1+1d1+1d monitored quantum dynamics with SU(2)SU(2)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=2z=2z=2) background sector that carries the SU(2)SU(2)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∼L3t\sim L^{3}t∼L3 for the total spin. When the pairing field disorders, the background sector alone gives logarithmic entanglement and a diffusive learning time t∼L2t\sim L^{2}t∼L2. Our analysis is controlled by a large-loop-fugacity expansion.
Monitored quantum circuits exhibit entanglement transitions at certain measurement rates. Such a transition separates phases characterized by how much information an observer can learn from the measurement outcomes. We study SU(2)-symmetric monitored quantum circuits, using exact numerics and a mapping onto an effective statistical-mechanics model. Due to the symmetry's non-Abelian nature, measuring qubit pairs allows for nontrivial entanglement scaling even in the measurement-only limit. We find a transition between a volume-law entangled phase and a critical phase whose diffusive purification dynamics emerge from the non-Abelian symmetry. Additionally, we identify a ``spin-sharpening transition.'' Across the transition, the rate at which measurements reveal information about the total spin quantum number changes parametrically with system size.
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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.
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