How the detailed structure of quantum complexity emerges from quantum dynamics remains a fundamental challenge highlighted by advances in quantum simulators and information processing. The celebrated Small-Incremental-Entangling (SIE) theorem provides a universal constraint on the rate of entanglement generation, yet it leaves open the problem of fully characterizing fine entanglement structures. Here we introduce the concept of Spectral-Entangling strength, which captures the structural entangling power of an operator, and establish a spectral SIE theorem: a universal speed limit for R'enyi entanglement growth at
α≥1/2, revealing a robust
1/s2 decay threshold in the entanglement spectrum. Remarkably, our bound at
α=1/2 is both qualitatively and quantitatively optimal, defining the universal threshold beyond which entanglement growth becomes unbounded. This exposes the detailed structure of Schmidt coefficients and enables rigorous truncation-based error control, linking entanglement structure to computational complexity. Building on this, we derive a generalized entanglement area law under an adiabatic-path condition, extending a central principle of quantum many-body physics to general interactions. As a concrete application, we show that one-dimensional long-range interacting systems admit polynomial bond-dimension approximations for ground, time-evolved, and thermal states, thereby closing the long-standing quasi-polynomial gap and demonstrating that such systems can be simulated efficiently with tensor-network methods. By explicitly controlling R'enyi entanglement, we obtain a rigorous, a priori error guarantee for the time-dependent density-matrix renormalization-group algorithm. Overall, our results extend the SIE theorem to the spectral domain and establish a unified framework that unveils the detailed and universal structure underlying quantum complexity.
We prove that weak long-range many-body interactions preserve the bulk gap of a free fermion system and that a quasi-local spectral flow transports the free ground state to ground states of the interacting system. Let
Φh be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator
h has no spectrum in
(−g,g), and let
V be an even self-adjoint many-body interaction. For sufficiently small
∣λ∣, we construct pure ground states
ωλ of the interacting bulk system
Φλ=Φh+λV and quasi-local automorphisms
β0,λ such that
ωλ=ω0∘β0,λ, where
ω0 is the free ground state of
Φh. Each
ωλ has a GNS Hamiltonian with gap at least
g−C∗∣λ∣∥V∥∗, where
∥⋅∥∗ is the relevant interaction norm.
Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set
Γ⊂Rd, a sufficient condition is decay as
(1+∣x−y∣)−ϖ with
ϖ>2d, for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If
h and
V have stretched-exponential decay, so does the spectral-flow generator.
The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.
The dynamics of quantum systems strongly depends on the local structure of the Hamiltonian. For short-range interacting systems, the well-known Lieb–Robinson bound defines the effective light cone with an exponentially small error with respect to the spatial distance, whereas we can obtain only polynomially small errors for distance in long-range interacting systems. In this paper, we derive a qualitatively new bound for quantum dynamics by considering how many spins can correlate with each other after time evolution. Our bound characterizes the number of spins which support the many-body entanglement with exponentially small errors and is valid for a large class of Hamiltonians including long-range interacting systems. To demonstrate the advantage of our approach in quantum many-body systems, we apply our bound to prove several fundamental properties which have not been derived from the Lieb–Robinson bound.