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arXiv:2212.07444 · Physical Review A

Bounds on Rényi entropy growth in many-body quantum systems

Zhengyan Darius Shi

Published Apr 8, 2024• 2 citations
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Abstract

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 $A$ 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 $|A|$. For $D$-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 $|A|$ independent for all power-law decaying interactions $V(r)\ensuremath{\sim}{r}^{\ensuremath{-}w}$ with $w>2D+1$. However, for $\ensuremath{\alpha}>1$, the bound is $|A|$ 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 $c$ depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on $k$-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.

DOI
10.1103/physreva.109.042404

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We prove rigorous bounds on the growth of \ensuremathα\ensuremath{\alpha}\ensuremathα-R'enyi entropies S\ensuremathα(t){S}_{\ensuremath{\alpha}}(t)S\ensuremathα​(t) (the von Neumann entropy being the special case \ensuremathα=1\ensuremath{\alpha}=1\ensuremathα=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\ensuremathα\ensuremath′(t)∣|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|∣S\ensuremathα\ensuremath′​(t)∣ (with \ensuremathα\ensuremath≠1\ensuremath{\alpha}\ensuremath{\ne}1\ensuremathα\ensuremath=1) can be exponentially larger than ∣S1\ensuremath′(t)∣|{S}_{1}^{\ensuremath{'}}(t)|∣S1\ensuremath′​(t)∣ as a function of the subsystem size ∣A∣|A|∣A∣. For DDD-dimensional systems with geometric locality, we prove bounds on ∣S\ensuremathα\ensuremath′(t)∣|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|∣S\ensuremathα\ensuremath′​(t)∣ that depend on the decay rate of interactions with distance. When \ensuremathα=1\ensuremath{\alpha}=1\ensuremathα=1, the bound is ∣A∣|A|∣A∣ independent for all power-law decaying interactions V(r)\ensuremath∼r\ensuremath−wV(r)\ensuremath{\sim}{r}^{\ensuremath{-}w}V(r)\ensuremath∼r\ensuremath−w with w>2D+1w>2D+1w>2D+1. However, for \ensuremathα>1\ensuremath{\alpha}>1\ensuremathα>1, the bound is ∣A∣|A|∣A∣ independent only when the interactions are finite-range or decay faster than V(r)\ensuremath∼e\ensuremath−crDV(r)\ensuremath{\sim}{e}^{\ensuremath{-}c\phantom{\rule{0.16em}{0ex}}{r}^{D}}V(r)\ensuremath∼e\ensuremath−crD 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 \ensuremathα\ensuremath{\alpha}\ensuremathα strongly influences the interplay between locality and instantaneous entanglement growth. In other words, the von Neumann entropy and the \ensuremathα\ensuremath{\alpha}\ensuremathα-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.
We show that von Neumann entropy grows algebraically in time for many-body localized systems with long-range interactions. The dephasing time of far-apart localized degrees of freedom is studied first. For a disentangled high-energy initial state, our analysis indicates that the entropy of one half of the system evolves as t1/(p+1){t}^{1/(p+1)}t1/(p+1), where ppp controls the spatial decay of interactions. Numerical simulations of the entropy evolution in the Coulomb and dipolar cases also exhibit an algebraic growth. We find that the saturation value of von Neumann entropy is extensive but small for an ergodic state, and it does not depend on the shape of the interactions. Due to the algebraic entanglement spread, the signature of an extensive entropy should be much easier to observe in experiments with nonlocal interactions than in those with local ones.
Understanding how complex entanglement structures emerge is a central problem in quantum many-body physics. Recent work by Zhang et al. [Phys. Rev. Lett. 137, 020404 (2026)] has considered structured initial states prepared by evolving a product state under a chaotic Hamiltonian for a finite time before quenching to the target Hamiltonian. In this setup, total entanglement entropy growth in many-body localized systems exhibits two distinct regimes, first increasing and then decreasing as the initial entanglement is tuned. In this work, we identify the physical origin of this behavior by analyzing the dynamics of both the Rényi entanglement entropy and the Wehrl-Rényi entropy in the random-field XXZ model, the latter of which characterizes multipartite entanglement. We show that a similar non-monotonic dependence on the initial entanglement also appears in the net growth of the Wehrl-Rényi entropy for product states polarized along the zzz-direction. The first regime is governed by a finite magnetization associated with local integrals of motion, while the second reflects inter-site correlations. In contrast, for product states in the x/yx/yx/y-direction, the entanglement growth exhibits a monotonic decay. Our results provide a more fine-grained picture of how distinct initial-state properties shape entanglement dynamics in many-body localized systems.
Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies which, we conjecture, applies to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system-a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio V_{A}/V, while the Renyi entropy densities depend nonlinearly on V_{A}/V. Surprisingly, Renyi entropies S_{n} for n>1 are convex functions of the subsystem size, with a volume law coefficient that depends on V_{A}/V, and exceeds that of a thermal mixed state at the same energy density. We provide two different arguments to support our results: the first one relies on a many-body version of Berry's formula for chaotic quantum-mechanical systems, and is closely related to the eigenstate thermalization hypothesis. The second argument relies on the assumption that for a fixed energy in a subsystem, all states in its complement allowed by the energy conservation are equally likely. We perform an exact diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions.
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.
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.
The scaling of entanglement with subsystem size encodes key information about phases and criticality, but the von Neumann entropy is costly to access in experiments and simulations, often requiring full state tomography. The second Rényi entropy is readily measured using two-copy protocols and is often used as a proxy for the von Neumann entanglement entropy, where it is assumed to track its asymptotic scaling. Sugino and Korepiny (Int. J. Mod. Phys. B 32, 1850306 (2018)) revealed that in the ground state of some highly constrained spin models, the scaling of the von Neumann and Rényi entropies can differ, varying from power law to logarithmic scaling as a function of the Rényi index. Here, we construct a number-conserving many-body state that demonstrates a Rényi-index-dependent change in the leading entanglement scaling, generalizing previous results to the case of interacting fermions. We introduce a symmetry-aware lower bound on the von Neumann entropy built from charge-resolved Rényi entropies that can provide a protocol for diagnosing anomalous entanglement scaling from experimentally accessible data.