非阿贝尔可学习性转变的统计力学
Statistical Mechanics of Non-Abelian Learnability Transitions
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中文总结 AI 辅助
本研究建立具有SU(2)对称的1+1维受监控量子动力学理论，揭示非阿贝尔荷系统中可学习性转变与纠缠转变的关联，明确配对场有序/无序时的学习时间与纠缠标度规律。
中文摘要 AI 辅助
受监控的多体量子系统可发生尖锐的可学习性转变，其特征取决于观察者能学习到的信息量。当动力学守恒非阿贝尔荷（如SU(2)自旋）时，观察者对总荷的学习仍是待解决问题。与阿贝尔情形中不同格点上的荷测量对易不同，SU(2)对称读出为非对易融合测量，使学习成为真正的量子推断问题。本研究提出具有SU(2)对称的1+1维受监控量子动力学理论，表明其可由有效复制圈模型描述，该模型包含复制配对场与携带SU(2)荷且在整个相图中保持无能隙的扩散（z=2）背景区。理论预测“自旋锐化”转变与纠缠转变重合为单一转变：配对场有序时产生体积定律纠缠，使背景区在测量中隐藏，总自旋学习时间为t~L³；配对场无序时，仅背景区产生对数纠缠，扩散学习时间为t~L²。分析由大圈逸度展开控制。
英文摘要
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.
