Geometry of Noisy Quantum Many-Body Dynamics with Continuous Symmetries: Entanglement and Correlations

What the math gives to ML

The paper identifies a transferable mechanism: after averaging Brownian dynamics, late-time observables are governed by a positive semidefinite replica Hamiltonian whose low-energy ground-state manifold is the symmetry-defined k-commutant. Singularities of this manifold, caused by symmetry-protected frozen void states, produce anomalous rather than purely exponential relaxation and sub-ballistic Rényi-entanglement growth. A neural-network analogue is to make stochastic hidden-state or parameter dynamics explicitly symmetry-preserving, compute the low-dimensional replica commutant, and use its smallest nonzero eigenvalues and conditioning as stability and noise-control signals. This yields falsifiable schedules based on empirical commutant-gap closure and void-associated condition-number divergence.