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On propagation of information in quantum many-body systems

Israel Michael Sigal, Jingxuan Zhang

TL;DR

This work establishes a comprehensive framework for bounding the propagation of information in quantum many-body lattice systems, including those with long-range interactions, by proving a maximal velocity bound that enforces a light cone with polynomial tails. It then derives a light-cone approximation for local observables and a weak Lieb–Robinson bound, both formulated in terms of system-specific decay parameters and state-dependent expectations. The authors connect these dynamical bounds to propagation/creation of correlations, quantum messaging constraints, state-control times, and the relationship between spectral gaps and correlation decay, with extensions to macroscopic particle transport. The results reveal a linear light cone in a broad class of systems and provide rigorous limits on information transfer and entanglement generation, offering valuable insights for quantum information processing and many-body dynamics in non-relativistic settings.

Abstract

We prove bounds on the minimal time for quantum messaging, propagation/creation of correlations, and control of states for general lattice quantum many-body systems. The proofs are based on a maximal velocity bound, which states that the many-body evolution stays, up to small leaking probability tails, within a light cone of the support of the initial conditions. This estimate is used to prove the light-cone approximation of dynamics and Lieb-Robinson-type bound, which in turn yield the results above. Our conditions cover long-range interactions. The main results of this paper as well as some key steps of the proofs were first presented in [36].

On propagation of information in quantum many-body systems

TL;DR

This work establishes a comprehensive framework for bounding the propagation of information in quantum many-body lattice systems, including those with long-range interactions, by proving a maximal velocity bound that enforces a light cone with polynomial tails. It then derives a light-cone approximation for local observables and a weak Lieb–Robinson bound, both formulated in terms of system-specific decay parameters and state-dependent expectations. The authors connect these dynamical bounds to propagation/creation of correlations, quantum messaging constraints, state-control times, and the relationship between spectral gaps and correlation decay, with extensions to macroscopic particle transport. The results reveal a linear light cone in a broad class of systems and provide rigorous limits on information transfer and entanglement generation, offering valuable insights for quantum information processing and many-body dynamics in non-relativistic settings.

Abstract

We prove bounds on the minimal time for quantum messaging, propagation/creation of correlations, and control of states for general lattice quantum many-body systems. The proofs are based on a maximal velocity bound, which states that the many-body evolution stays, up to small leaking probability tails, within a light cone of the support of the initial conditions. This estimate is used to prove the light-cone approximation of dynamics and Lieb-Robinson-type bound, which in turn yield the results above. Our conditions cover long-range interactions. The main results of this paper as well as some key steps of the proofs were first presented in [36].
Paper Structure (34 sections, 22 theorems, 235 equations, 6 figures)

This paper contains 34 sections, 22 theorems, 235 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related results
  3. Organization of the paper
  4. Notation
  5. Setup and main results
  6. Maximal velocity bound
  7. Light-cone approximation of evolution \ref{['1.5']}
  8. Lieb-Robinson-type bounds
  9. Propagation/creation of correlations
  10. Constraint on the propagation of quantum signals
  11. Bound on quantum state control
  12. Spectral gap and decay of correlation
  13. Light cone in macroscopic particle transport
  14. Discussions of the results and extensions
  15. Proof of Theorem \ref{['thm1']}
  16. ...and 19 more sections

Key Result

Theorem 2.1

Suppose k-cond holds with some $n\ge1$. Then, for every $c > \kappa$, there exists $C=C(n,\kappa_n,c )>0$ s.th. for all $\eta\ge1,\,X\subset \Lambda$, we have the following estimate for all $\left\lvert t\right\rvert< \eta/c$:

Figures (6)

  • Figure 1: Schematic diagram illustrating $X_\xi$.
  • Figure 2: Schematic diagram illustrating \ref{['3122']}
  • Figure 3: Schematic diagram illustrating \ref{['3132']}.
  • Figure 4: Schematic diagram illustrating the splitting of $H$.
  • Figure 5: Schematic diagram illustrating $Y, Y_\xi, Y_\xi^c$.
  • ...and 1 more figures

Theorems & Definitions (45)

  • Theorem 2.1: MVB for lattice quantum many-body system
  • Theorem 2.2: Light-cone approximation of quantum evolution
  • Theorem 2.3: Weak Lieb-Robinson bound
  • Definition 2.1
  • Theorem 2.4: Propagation/creation of correlation
  • Theorem 2.5
  • Theorem 2.6: Quantum control bound
  • Theorem 2.7: Gap at the ground state implies decay of ground state correlations
  • Theorem 2.8
  • Remark 1
  • ...and 35 more