Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems

TL;DR

This paper proves a universal lower bound on the time required for information scrambling in quantum systems, establishing a fundamental speed limit linked to entanglement entropy and extending the fast scrambling conjecture.

Contribution

It introduces a universal speed limit on quantum information scrambling applicable to all Hamiltonian systems, regardless of locality or interaction type.

Findings

Scrambling time is at least logarithmic in entanglement entropy.

Established a fundamental speed limit for quantum information dynamics.

Extended the fast scrambling conjecture to general quantum systems.

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.