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arXiv:2408.00743 · Physical Review X

Dynamical α -Rényi Entropies of Local Hamiltonians Grow at Most Linearly in Time

Daniele Toniolo, Sougato Bose

Published Jun 17, 2025• 8 citations
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Abstract

We consider a generic one-dimensional spin system of length L , arbitrarily large, with strictly local interactions, for example, nearest neighbor, and prove that the dynamical α -Rényi entropies, 0 < α ≤ 1 , of an initial product state grow at most linearly in time. This result arises from a general relation among dynamical α -Rényi entropies and Lieb-Robinson bounds. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of α close enough to 1, and, moreover, to initial pure states with low entanglement, of the order of log L , that are typically represented by critical states. We establish that low entanglement states have an efficient matrix product state representation that persists at least up to times of the order of log L . The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on α -Rényi entropies, and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest.

DOI
10.1103/ycdh-z8zf

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We consider a generic one-dimensional spin system of length L , arbitrarily large, with strictly local interactions, for example, nearest neighbor, and prove that the dynamical α -Rényi entropies, 0 < α ≤ 1 , of an initial product state grow at most linearly in time. This result arises from a general relation among dynamical α -Rényi entropies and Lieb-Robinson bounds. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of α close enough to 1, and, moreover, to initial pure states with low entanglement, of the order of log L , that are typically represented by critical states. We establish that low entanglement states have an efficient matrix product state representation that persists at least up to times of the order of log L . The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on α -Rényi entropies, and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest.
Dynamics of many-body localized systems: logarithmic lightcones and log⁡ t\log \, tlogt-law of ααα-Rényi entropies
In the context of the Many-Body-Localization phenomenology we consider arbitrarily large one-dimensional local spin systems, the XXZ model with random magnetic field is a prototypical example. Without assuming the existence of exponentially localized integrals of motion (LIOM), but assuming instead that the system's dynamics gives rise to a Lieb-Robinson bound (L-R) with a logarithmic lightcone, we rigorously evaluate the dynamical generation, starting from a generic product state, of $ α$-Rényi entropies, with $ α$ close to one, obtaining a log⁡ t\log \, tlogt-law, that denotes a slow spread of entanglement. This is in sharp contrast with Anderson localized phases that show no dynamically generated entanglement. To prove this result we apply a general theory recently developed by us in arXiv:2408.00743 that quantitatively relates the L-R bounds of a local Hamiltonian with the dynamical generation of entanglement. Assuming instead the existence of LIOM we provide new independent proofs of the known facts that the L-R bound of the system's dynamics has a logarithmic lightcone and show that the dynamical generation of the von Neumann entropy has for large times a $ \log \, t$-shape. L-R bounds, that quantify the dynamical spreading of local operators, may be easier to measure in experiments in comparison to global quantities such as entanglement.
Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.
We prove rigorous bounds on the growth of $\ensuremath{\alpha}$-R\'enyi entropies ${S}_{\ensuremath{\alpha}}(t)$ (the von Neumann entropy being the special case $\ensuremath{\alpha}=1$) associated with any subsystem AAA of a general lattice quantum many-body system with finite on-site Hilbert space dimension. For completely nonlocal Hamiltonians, we show that the instantaneous growth rates $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ (with $\ensuremath{\alpha}\ensuremath{\ne}1$) can be exponentially larger than $|{S}_{1}^{\ensuremath{'}}(t)|$ as a function of the subsystem size ∣A∣|A|∣A∣. For DDD-dimensional systems with geometric locality, we prove bounds on $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ that depend on the decay rate of interactions with distance. When $\ensuremath{\alpha}=1$, the bound is ∣A∣|A|∣A∣ independent for all power-law decaying interactions $V(r)\ensuremath{\sim}{r}^{\ensuremath{-}w}$ with w>2D+1w>2D+1w>2D+1. However, for $\ensuremath{\alpha}>1$, the bound is ∣A∣|A|∣A∣ independent only when the interactions are finite-range or decay faster than $V(r)\ensuremath{\sim}{e}^{\ensuremath{-}c\phantom{\rule{0.16em}{0ex}}{r}^{D}}$ for some ccc depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on kkk-local systems with or without geometric locality. A central theme of this work is that the value of $\ensuremath{\alpha}$ strongly influences the interplay between locality and instantaneous entanglement growth. In other words, the von Neumann entropy and the $\ensuremath{\alpha}$-R\'enyi entropies cannot be regarded as proxies for each other in studies of entanglement dynamics. We compare these bounds with analytic and numerical results on Hamiltonians with varying degrees of locality and find concrete examples that almost saturate the bound for nonlocal dynamics.
Abstract My previous work [arXiv: 1902.00977 ] studied the dynamics of Rényi entanglement entropy R α in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that R α with Rényi index α > 1 grows no faster than ‘diffusively’ (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-1/2 systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension d ≥ 2 .
We prove an area law for the entanglement entropy in gapped one-dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on matrix product states which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on Rényi entropy for sufficiently large α<1 and implies the ability to approximate the ground state by a matrix product state.
The quantum circuit complexity of an evolving many-body quantum system is believed to exhibit a sustained growth, maintained for timescales much longer than the onset of thermalization. Most previous works have focused on models which violate energy conservation, such as random unitary circuits. Here we study generic, local time-independent Hamiltonian dynamics. We unconditionally prove that for generic local Hamiltonians and typical initial product states at high effective temperature, the robust quantum circuit complexity must grow over a very long period of time, attaining an exponentially large value at late times. Our approach relies on two structural properties that we prove rigorously: (i) generic local Hamiltonians satisfy generalized spectral no-resonance conditions of arbitrary order, and (ii) typical high-temperature product states are effectively supported in exponentially many energy eigenstates. These properties have been widely assumed without proof in prior works. As corollaries of this result, we show the late-time state displays robust volume-law entanglement that is irremovable by polynomial-size circuits, and we establish a no fast-forwarding result for generic local Hamiltonians. We also lower bound the complexity of preparing sufficiently large subsystems with local quantum channels, in contrast with sufficiently small thermalizing regions which we show retain low complexity at late times.