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Exact universal bounds on quantum dynamics and fast scrambling

Amit Vikram, Victor Galitski

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.

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.
Paper Structure (14 sections, 1 theorem, 53 equations, 3 figures)

This paper contains 14 sections, 1 theorem, 53 equations, 3 figures.

Table of Contents

  1. Basis-independent quantum speed limit
  2. Derivation of the inequality Ps(t) >= K(t)
  3. Condition for saturation
  4. Tightness compared to other basis-independent speed limits
  5. Types of scrambling and thermalization
  6. The scrambling of added degrees of freedom to infinite temperature
  7. The scrambling of pure product states to infinite temperature
  8. Thermalization to finite temperature
  9. Entanglement vs operator spreading
  10. Fast Scrambling and Fourier transforms: Formal statement
  11. Example: Slow scrambling via entanglement generation in the SYK model
  12. Defining subsystems in the SYK model
  13. Power-law decay of the SFF
  14. Scrambling in the SYK model: Numerical details

Key Result

Proposition 3

Let $\mathcal{F}$ be the set of functions $\widetilde{\mathcal{N}}: \mathbb{R} \to \mathbb{C}$ normalizable to $\lvert \widetilde{\mathcal{N}}(0)\rvert = 1$ with a non-negative real-valued Fourier transform $\mathcal{N}(E) \geq 0$, and $\mathcal{F}_p \subseteq \mathcal{F}$ be any subset of these fun where $\lambda(x > 0) > 0$ is a given monotonically increasing function, then any quantum system wh

Figures (3)

  • Figure 1: Illustration (log-linear) of scrambling speed limits via Eqs. \ref{['eq:dynamicalinequality']}, \ref{['eq:sffcosbound']}, \ref{['eq:ps_scrambling']} and \ref{['eq:kt_scrambling']} in a realization of the SYK-$4$ model of randomly interacting fermions, for initial fermion Fock states given by Eq. \ref{['eq:SYKinitialstates']} in a subsystem of $N_S = 7$ sites ($D_S = 128$) among $N=10$ total sites ($D=1024$). The power-law decay of $K(t)$ (whose nontrivial effect is evident here up to the first "peak" at $t\approx 6$) leads to exponentially slow sustained scrambling via Eq. \ref{['eq:ts_SYK']} as $N, N_S \to \infty$.
  • Figure 2: Density of states and SFF for the same realization of the SYK model as in other plots; $J=1$, $N=10$ (with $D=2^N = 1024$).
  • Figure 3: Additional numerics (a) demonstrating the high sensitivity of the speed limits such as $P_S(t) \geq K(t)$ required for a many-body system, and (b, c) providing evidence for scrambling to infinite temperature in fermion subsystems of the SYK model. All plots use the same realization of the interaction strengths $J_{jk\ell m}$ (each chosen from a Gaussian distribution as described near Eq. \ref{['eqs:defSYK']}) used in Fig. 1 of the main text.

Theorems & Definitions (1)

  • Proposition 3: Fast scrambling and Fourier transform nonnegativity