Statistical Mechanics of Non-Abelian Learnability Transitions

arXiv:2608.19325v1 Announce Type: new 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.

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.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el)

arXiv:2608.19325 [quant-ph]

(or arXiv:2608.19325v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.19325 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Statistical Mechanics of Non-Abelian Learnability Transitions

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