Exact universal bounds on quantum dynamics and fast scrambling

Abstract

Quantum speed limits such as the Mandelstam-Tamm or Margolus-Levitin bounds offer a quantitative formulation of the energy-time uncertainty principle that constrains dynamics over short times. We show that the spectral form factor, a central quantity in quantum chaos, sets a universal state-independent bound on the quantum dynamics of a complete set of initial states over arbitrarily long times, which is tighter than the corresponding state-independent bounds set by known speed limits. This bound further generalizes naturally to the real-time dynamics of time-dependent or dissipative systems where no energy spectrum exists. We use this result to constrain the scrambling of information in interacting many-body systems. For Hamiltonian systems, we show that the fundamental question of the fastest possible scrambling time -- without any restrictions on the structure of interactions -- maps to a purely mathematical property of the density of states involving the non-negativity of Fourier transforms. We illustrate these bounds in the Sachdev-Ye-Kitaev model, where we show that despite its "maximally chaotic" nature, the sustained scrambling of sufficiently large fermion subsystems via entanglement generation requires an exponentially long time in the subsystem size.

4 pages + references + 11-page supplement, 1+2 figures; v2: added an illustration of the bound in the SYK model, with figures and an expanded supplement. Close to published version

References in corpus (7)

• Thermalization and its mechanism for generic isolated quantum systems
• Many body localization and thermalization in quantum statistical mechanics
• Black holes as mirrors: quantum information in random subsystems
• Chaos, Complexity, and Random Matrices
• Propagation of Correlations in Quantum Lattice Systems
• Quantum Speed Limit for States with a Bounded Energy Spectrum
• Dynamical quantum ergodicity from energy level statistics

Cited by in corpus (9)

• Quantum Dynamics in Krylov Space: Methods and Applications
• Robustness of quantum chaos and anomalous relaxation in open quantum circuits
• Quantum Chaos and Coherence: Random Parametric Quantum Channels
• Theory of Irreversibility in Quantum Many-Body Systems
• Decomposing the Spectral Form Factor
• Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics
• Probing quantum phase transition via quantum speed limit
• Leading and beyond leading-order spectral form factor in chaotic quantum many-body systems across all Dyson symmetry classes
• Swinging small quantum systems out of available values of control parameters