Hugo Mackay

How the detailed entanglement structure emerges from quantum dynamics remains a fundamental challenge, motivated by recent advances in quantum simulators and information processing. As a central milestone, the Small-Incremental-Entangling (SIE) theorem bounds the entanglement-entropy growth rate, but does not control the entanglement spectrum itself. In this work, we define the spectral-entangling strength, which quantifies how strongly an operator can reshape the distribution of Schmidt coefficients across a bipartition. We then prove a spectral SIE theorem: for R\'enyi index $\alpha \ge 1/2$, the growth rate of R\'enyi entanglement entropies admits a universal bound. Remarkably, our bound at $\alpha=1/2$ is both qualitatively and quantitatively optimal; below this threshold ($\alpha<1/2$), no universal speed limit on entanglement growth can exist. This result yields a sharp $1/s^2$ threshold in the tail of the ordered Schmidt coefficients (with $s$ the Schmidt index), enabling rigorous truncation-based error control and establishing a quantitative link between entanglement-spectrum structure and computational complexity. As a practical highlight, for one-dimensional power-law interactions $1/r^{\eta}$ with $\eta>2$, this implies matrix-product-state approximations with bond dimension polynomial in $(n/\varepsilon)$ for ground states, real-time evolved states, and Gibbs states, thereby closing the quasi-polynomial gap. By controlling R\'enyi entanglement, we further obtain a rigorous \emph{a priori} bound on truncation error for time-dependent DMRG/TEBD-type simulations. Overall, we extend the SIE paradigm from bounding entanglement entropies to constraining the entanglement spectrum itself.