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Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems (2404.15403v1)

Published 23 Apr 2024 in quant-ph, cond-mat.stat-mech, hep-th, math-ph, math.MP, and nlin.CD

Abstract: We prove that the time required for sustained information scrambling in any Hamiltonian quantum system is universally at least logarithmic in the entanglement entropy of scrambled states. This addresses two foundational problems in nonequilibrium quantum dynamics. (1) It sets the earliest possible time for the applicability of equilibrium statistical mechanics in a quantum system coupled to a bath at a finite temperature. (2) It proves a version of the fast scrambling conjecture, originally motivated in models associated with black holes, as a fundamental property of quantum mechanics itself. Our result builds on a refinement of the energy-time uncertainty principle in terms of the infinite temperature spectral form factor in quantum chaos. We generalize this formulation to arbitrary initial states of the bath, including finite temperature states, by mapping Hamiltonian dynamics with any initial state to nonunitary dynamics at infinite temperature. A regularized spectral form factor emerges naturally from this procedure, whose decay is universally constrained by analyticity in complex time. This establishes an exact speed limit on information scrambling by the most general quantum mechanical Hamiltonian, without any restrictions on locality or the nature of interactions.

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Summary

  • The paper derives a logarithmic lower bound on scrambling time based on the entanglement entropy in Hamiltonian systems.
  • It applies a refined energy-time uncertainty principle to relate the spectral form factor with quantum chaos dynamics.
  • The study confirms a generalized fast scrambling conjecture with implications for quantum thermalization and computing.

Universal Speed Limits on Information Scrambling in Quantum Systems

The paper "Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems," authored by Amit Vikram, Laura Shou, and Victor Galitski, addresses pivotal issues in nonequilibrium quantum dynamics by establishing a universal lower bound on the time required for information scrambling in quantum systems. This paper makes significant strides in clarifying the earliest point at which equilibrium statistical mechanics can apply and confirms a generalized version of the fast scrambling conjecture as an inherent characteristic of quantum mechanics.

Overview of Results

The main result of this research proves that the time tst_s needed for sustained scrambling in any Hamiltonian quantum system is at least logarithmic in the entanglement entropy of the scrambled states. Specifically, the authors derive a lower bound for tst_s based on a refined understanding of the energy-time uncertainty principle, which is expressed in terms of the spectral form factor in the context of quantum chaos.

Methodology and Key Concepts

  1. Energy-Time Uncertainty Principle: The authors leverage a refined version of this principle to relate a system’s dynamics to its energy spectrum. This sets a theoretical framework to analyze scrambling times using the spectral form factor.
  2. Spectral Form Factor (SFF): In this work, a regularized form of the SFF is pivotal. It is derived from transformations mapping unitary dynamics with initial state conditions to nonunitary dynamics at infinite temperature. The decay rate of this form factor constrains the scrambling time for general quantum systems.
  3. Analytic Function Boundaries: The decay of time-dependent quantities, notably those related to scrambling, is constrained using the properties of functions analytic within a specific range in the complex plane. This provides a limit on how quickly scrambling processes can evolve in the real-time domain.

Implications and Theoretical Significance

  • Scrambling and Entanglement Generation: The results directly connect the physical process of entanglement generation to the time scales of scrambling, emphasizing a logarithmic relation with the entanglement entropy.
  • Fast Scrambling Conjecture: The paper provides a rigorous basis for a version of the fast scrambling conjecture that applies universally to quantum systems without necessitating assumptions about locality or interaction nature.
  • Thermalization and Quantum Information: This work discusses the link between information spreading in thermal settings (such as black holes) and quantum mechanics' foundational properties, potentially informing theories of quantum gravity.

Future Directions

The paper opens avenues for future research to explore settings where the derived speed limit may achieve or even nearly saturate practical limits in real-world quantum systems. Additionally, there are open questions about the potential for identifying natural systems capable of achieving these theoretical limits and the implications for technologies relying on quantum dynamics, such as quantum computers and simulation devices.

In summary, this research provides a mathematical framework that sets a limit on the dynamics of quantum systems, rooting the theoretical underpinnings of quantum theatrics with experimentally testable implications. This bridges the gap between theoretical constructs and anticipated empirical validation, underscoring the intrinsic link between entanglement and information spreading in quantum systems.

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