Speaker
Description
Understanding the complexity of quantum many-body states requires control not only over entanglement entropy but also over the distribution of entanglement across the Schmidt spectrum. The Small-Incremental-Entangling (SIE) theorem provides a remarkably general bound on the rate of change of bipartite von Neumann entanglement entropy, yet leaves this finer spectral structure unresolved. In this work, we introduce the Spectral-Entangling Strength and establish a spectral SIE theorem that bounds the rate of Rényi entanglement growth for all $\alpha \geq 1/2$. This implies a universal tail bound on the entanglement spectrum, with the threshold $\alpha = 1/2$ being optimal. This spectral control yields rigorous Schmidt-truncation bounds, thereby constraining the complexity of tensor networks. As applications, we establish a generalized entanglement area law along adiabatic paths beyond geometric locality and show that one-dimensional systems with long-range interactions admit polynomial-bond-dimension approximations for ground states, time-evolved states, and thermal states. Our results also provide an a priori precision guarantee for time-dependent density matrix renormalization group (tDMRG) simulations.