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  • Open Access

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

Daniele Toniolo* and Sougato Bose

  • *Contact author: d.toniolo@ucl.ac.uk; danielet@alumni.ntnu.no
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Phys. Rev. X 15, 031046 – Published 19 August, 2025

DOI: https://doi.org/10.1103/ycdh-z8zf

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.

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Popular Summary

In quantum many-body systems, two key concepts are information propagation—whose speed is bounded by the Lieb-Robinson limit—and the generation of quantum entanglement, which underpins phenomena like thermalization and computational complexity. While both are central to quantum dynamics, their precise relationship has remained unclear. In this work, we rigorously establish a direct connection between the speed at which information spreads through a system and how much entanglement is generated during that process.

We approach this by using the α-Rényi entropy—a generalized measure of information content in a probability distribution—to quantify entanglement in evolving quantum systems. This entropy measure allows us to link the growth of entanglement to the Lieb-Robinson bound, which is a well-understood and experimentally accessible quantity. Crucially, this relationship also informs how difficult it is to simulate a given system using classical methods: Faster information propagation generally implies greater entanglement and reduced simulability. By providing this connection, our framework allows researchers to estimate a system’s computational complexity based on measurable physical quantities.

The broader implication of our findings is a powerful, general tool for understanding quantum dynamics across a wide range of systems. Because our theory applies to all local and quasilocal many-body systems, it extends far beyond prior studies that focused on specific models.

Article Text

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Phys. Rev. X 15, 031046– Published 19 August, 2025
  • Received 16 August 2024
  • Revised 13 November 2024
  • Accepted 17 June 2025
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