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

The gist: The time required for sustained information scrambling in any Hamiltonian quantum system is universally at least logarithmic in the entanglement entropy of scrambled states.

Foundational Problem and Motivation

This work proves 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, addressing two foundational problems (Page 1). It sets the earliest possible time for equilibrium statistical mechanics to apply to a quantum system coupled to a bath at a finite temperature (Page 1). Furthermore, it proves a version of the fast scrambling conjecture as a fundamental property of quantum mechanics itself (Page 1). The setup involves coupling a system HS, initially in state k⟩S, to an external system HE in state ρˆβE, and allowing their mutual interactions to drive time evolution (Page 1). This model is the prototypical setting for the nonequilibrium process of thermalization [1–10] (Page 1).

Methodology and Key Quantities

The approach generalizes a refinement of the energy-time uncertainty principle in terms of the infinite temperature spectral form factor in quantum chaos to arbitrary initial states of the bath, including finite temperature states (Page 1). The core quantity examined is the mean return probability PS(t) (Page 2). This quantity partially measures how much information about the initial state is retained by a basis α⟩S (Page 2). The sustained scrambling time ts is set by the condition PS(t > ts) ≤ Pscr, where Pscr is the scrambled value, typically related to maximal entanglement [19] (Page 2).

Derivation of Universal Bounds

The main result establishes a bound on the sustained scrambling time ts in terms of three quantities: (1) the scrambled return probability Pscr, (2) the information-theoretic fidelity fβ between the thermal state ρˆβE and the infinite temperature maximally mixed state ρˆ0E (Page 2), and (3) an effective partition function Zβ(τ) in a Euclidean time interval [τ1, τ2] of width cβ (Page 2). The derived bound is given by ts ≥ cβ π ln 1/Λeff ln f2βZ2max Pscr! (Page 2). This bound holds for arbitrary Hamiltonians Hˆ without restrictions on locality or the nature of interactions (Page 2).

Quantitative Refinements and Physical Interpretation

The derivation involves mapping the problem to nonunitary dynamics at infinite temperature, leading to a regularized spectral form factor Kβ(t) (Page 3). The constraint PS(t) ≥ Kβ(t) is derived by considering the initial state ρˆk(0) in terms of Kraus operators Kˆr(0), resulting in PS(t) ≥ Kβ(t) (Page 9). This leads to the bound ts ≥ cβ π ln S2,S + 4 ln(fβZmax)2Λeff (Page 8). In the thermodynamic limit, this simplifies to ts ≳ cβ π ln S2,S (Page 8). The paper also shows that for systems constructed with a suitable regularized density of states Neβ(t), the scrambling time is ts ≲ (2cβ/π) ln S2,S for nearly fast scramblers (Page 17).

Implications and Fast Scrambling Conjecture

The results establish statement (a) of the fast scrambling conjecture for this notion of scrambling, generalizing it beyond any specific assumptions of interaction structure (Page 9). The work suggests that systems with a nonvanishing zero temperature entropy are not fast scramblers in large subsystems (Page 9). The paper concludes that the universal bound ts ≥ cβ π ln S2,S completely specifies and establishes statement (a) of the fast scrambling conjecture for this notion of scrambling (Page 9). The analysis also shows that systems constructed in this section nearly saturate the logarithmic-in-entanglement-entropy bound of Eq. (A9) [Eq. (10) in the main text] if the energy eigenbasis is sufficiently random with respect to the computational basis (Page 17).

Supplemental Derivations

The derivation proceeds through several technical steps, including deriving quantitative relations between scrambling in terms of return probabilities and entanglement entropies [supporting Eqs. (4) and (10) of the main text] (Page 7). The proof of Theorem 1 involves a conformal map from a strip to a half-plane to derive the double exponential bound on analytic functions [51] (Page 15). This leads to the final universal result, Eq. (8) of the main text, with the choice of l = √3 − 1 maximizing the bound for ts (Page 16). The construction of nearly fast scramblers in Appendix D demonstrates that double exponential decay rates are achievable for Neβ(t) analytic and bounded in a strip Im t 0 (Page 17).

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Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems Amit Vikram, Laura Shou, and Victor Galitski Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA 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.

How it works

The derivation starts with defining PS(t) as the mean return probability of the computational basis states in HS under evolution by a Hamiltonian Hˆ (Page 2). This quantity partially measures information retention and constrains entanglement generation (Page 2). The sustained scrambling time ts is determined by the condition PS(t > ts) ≤ Pscr, where Pscr is related to maximal entanglement [19] (Page 2).

Derivation of Universal Bounds

The main result shows that ts is bounded in terms of Pscr, fβ, and Zβ(τ) (Page 2). Specifically, for every interval [τ1, τ2], the sustained scrambling time satisfies ts ≥ cβ π ln 1/Λeff ln f2βZ2max Pscr! (Page 2). This bound is exact for any Hamiltonian Hˆ (Page 2).

Quantitative Refinements and Physical Interpretation

The speed limit is expressed in terms of the scrambled Rényi entanglement entropy S2,S, leading to the leading order inequality ts ≳ cβ π ln S2,S (Page 8). The derivation uses a quantitative refinement of an asymptotic double exponential bound on bounded analytic functions (Page 11). This refines the previously undetermined constants in Eq. (21) of the main text to provide concrete numerical bounds for ts that can potentially be tested in experiments (Page 16).

Implications and Fast Scrambling Conjecture

The work specifies statement (a) of the fast scrambling conjecture for this notion of scrambling, generalizing it beyond any specific assumptions of interaction structure (Page 9). The analysis shows that systems constructed in this section nearly saturate the logarithmic-in-entanglement-entropy bound of Eq. (A9) [Eq. (10) in the main text] if the energy eigenbasis is sufficiently random with respect to the computational basis (Page 17).

Supplemental Derivations

The derivation proceeds through mapping unitary dynamics to nonunitary dynamics at infinite temperature, yielding PS(t) ≥ Kβ(t) (Page 9). The proof of Theorem 1 involves a conformal map from a strip to a half-plane to derive the double exponential bound on analytic functions (Page 15). This leads to the final universal result, Eq. (8) of the main text, with the choice of l = √3 − 1 maximizing the bound for ts (Page 16).

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Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems Amit Vikram, Laura Shou, and Victor Galitski Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA 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.

1. Universal Scrambling Time Bound The improved AI system can provide a rigorous, universal speed limit on information scrambling for any Hamiltonian quantum system, proving 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.

2. Applicability to Equilibrium Statistical Mechanics This capability allows the AI to set the earliest possible time for the applicability of equilibrium statistical mechanics in a quantum system coupled to a bath at a finite temperature, providing a concrete constraint on when thermodynamic descriptions become valid.

3. Black Hole Physics Modeling The system can model information spreading within black holes by establishing that a scrambling time of ts ≳ O(β ln N) has been conjectured in sufficiently k-local systems as a fundamental property of quantum mechanics itself, addressing foundational problems in nonequilibrium quantum dynamics.

4. Quantitative Scrambling Metric Analysis The AI can derive quantitative bounds on scrambling based on the mean return probability PS(t), showing that ts ≥ cβ π ln '1/2 ln f2βZmax Pscr !− (Equation 8), allowing for precise calculation of scrambling time given specific system parameters like initial state fidelity and scrambled entropy.

5. Experimental Testability via Entanglement Entropy The system can relate the theoretical bound to experimentally measurable quantities, such as a minimum 2nd Rényi entanglement entropy S2,S required for a state to be considered scrambled, making the bounds testable using local randomized measurements [62].