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On Propagation of Information in Quantum Mechanics and Maximal Velocity Bounds

Israel Michael Sigal, Xiaoxu Wu

TL;DR

This work provides a rigorous, general framework for the propagation of quantum information in zero-density quantum mechanics by establishing a uniform maximal velocity bound with exponential tails (uMVB) for a broad class of dispersion relations. The authors develop an analytic-deformation approach, based on the family $H_\xi = T_\xi H T_\xi^{-1}$ with analytic continuation to a polystrip, to derive light-cone bounds for both density matrices and observables, yielding a quantum-mechanical Lieb-Robinson bound and exponential OTOC decay. The results apply to continuum and lattice settings, allow for matrix-valued and time-dependent Hamiltonians, and extend to multipartite and $N$-particle systems, thereby offering a unified, robust tool for quantifying finite-speed information propagation in fbQM. The approach complements and extends previous scattering-theory and Mourre-type methods, with potential applications in quantum information processing, quantum simulation, and open-system generalizations.

Abstract

We revisit key notions related to the evolution of quantum information in few-body quantum mechanics (fbQM) and, for a wide class of dispersion relations, prove uniform bounds on the maximal speed of propagation of quantum information for states and observables with exponential error bounds. Our results imply, in particular, a fbQM version of the Lieb-Robinson bound, which is known to have wide applications in quantum information sciences. We propose a novel approach to proving maximal speed bounds.

On Propagation of Information in Quantum Mechanics and Maximal Velocity Bounds

TL;DR

This work provides a rigorous, general framework for the propagation of quantum information in zero-density quantum mechanics by establishing a uniform maximal velocity bound with exponential tails (uMVB) for a broad class of dispersion relations. The authors develop an analytic-deformation approach, based on the family with analytic continuation to a polystrip, to derive light-cone bounds for both density matrices and observables, yielding a quantum-mechanical Lieb-Robinson bound and exponential OTOC decay. The results apply to continuum and lattice settings, allow for matrix-valued and time-dependent Hamiltonians, and extend to multipartite and -particle systems, thereby offering a unified, robust tool for quantifying finite-speed information propagation in fbQM. The approach complements and extends previous scattering-theory and Mourre-type methods, with potential applications in quantum information processing, quantum simulation, and open-system generalizations.

Abstract

We revisit key notions related to the evolution of quantum information in few-body quantum mechanics (fbQM) and, for a wide class of dispersion relations, prove uniform bounds on the maximal speed of propagation of quantum information for states and observables with exponential error bounds. Our results imply, in particular, a fbQM version of the Lieb-Robinson bound, which is known to have wide applications in quantum information sciences. We propose a novel approach to proving maximal speed bounds.
Paper Structure (14 sections, 13 theorems, 121 equations, 1 figure)

This paper contains 14 sections, 13 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem and results
  3. Setup
  4. MVB
  5. Localization
  6. Light cone for evolution of states and observables
  7. Proof of Theorem
  8. Localization of observables and proof of Theorem
  9. MVB for evolution of observables: Proof of Theorem
  10. Differentiability and power estimates
  11. $N$-particle dynamics
  12. Acknowledgment
  13. Declarations
  14. Note added in proofs

Key Result

Theorem 1.1

Let Condition (A) hold and let $\mu\in (0,a)$. Then, for any $\mu'\in (0,\mu)$ and for any two disjoint sets $X$ and $Y$ in $\mathbb{R}^n$, we have, where $c'=\frac{\mu c}{\mu'}$, with $c$ given in Eq. c, and $C>0$ is a constant depending on $\frac{\mu}{\mu'}-1,\ \mu$ and $n$.

Figures (1)

  • Figure 1: Light cone diagram of $A$ and $B$

Theorems & Definitions (28)

  • Theorem 1.1: Light cone (maximal propagation velocity) bound
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4: Light cone approximation of Heisenberg evolution
  • Theorem 1.5: Quantum-mechanical Lieb-Robinson bound
  • Corollary 1.6: OTOC estimate
  • proof : Proof of Theorem
  • Proposition 2.1
  • proof
  • Lemma 2.2: Key lemma
  • ...and 18 more