Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems
Amit Vikram Laura Shou Victor Galitski




Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA

Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA
Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD 20742, USA

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.


Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems
    Victor Galitski
    April 29, 2024
======================================================================





Introduction— Consider a quantum mechanical system with Hilbert space ℋ_S, e.g., a collection of N_S qubits with dimension D_S = 2^N_S, that is initially in a pure basis state | k⟩_S (which we call the computational basis). At the time t=0, we couple ℋ_S to an external system (say, of N_E qubits and dimension D_E = 2^N_E) in a state ρ̂_β E (e.g., a thermal state with inverse temperature β), and allow their mutual interactions to drive time evolution in the combined system. The initial state of the overall system ℋ_S ⊗ℋ_E of dimension D = D_S D_E is given by the density operator:

    ρ̂_k(0) = | k⟩_S⟨ k|⊗ρ̂_β E.

Generically, information about the initial state of ℋ_S is “scrambled” by interactions, being lost to ℋ_E through the generation of entanglement, and not entirely recoverable from ℋ_S alone. To examine to what extent any basis |α⟩_S of the subsystem ℋ_S retains this information, we probe the system with measurements represented by the orthonormal projectors

    _α = |α⟩_S⟨α|⊗_E.


The above is a minimal description of quantum information scrambling that has far-reaching connections to statistical mechanics. Most directly, it is the prototypical setting for the nonequilibrium process of thermalization <cit.>, of the system ℋ_S via interactions with a thermal reservoir ℋ_E, which (if it occurs) is ultimately responsible for the validity of equilibrium statistical mechanics in ℋ_S. In a separate but related context, the same model is also believed to capture some essential aspects of information loss and recovery in black holes, even without a precise microscopic model of black holes as may emerge from an eventual theory of quantum gravity <cit.>. Here, one thinks of the thermal reservoir ℋ_E as representing the information already contained within a black hole, which is in the thermal state ρ̂_β E due to entanglement with the Hawking radiation emitted before t=0 (a separate quantum system ℋ_R that does not interact with ℋ_E). The system ℋ_S then represents infalling matter into the black hole, and the above setup models how infalling information spreads among a black hole's degrees of freedom, and whether it can be recovered from the previously emitted Hawking radiation <cit.>.

In both cases, a problem of fundamental interest is to determine the fastest allowed scrambling time t_s, after which ℋ_S retains no memory of its initial state up to thermodynamic parameters like temperature. From the perspective of quantum statistical mechanics, this amounts to constraining the earliest time after which equilibrium statistical mechanics can apply to any subsystem of interest. Separately, a scrambling time of t_s ≳ O(βln N) has been 
conjectured in sufficiently k-local systems of N=N_S+N_E particles <cit.>, based on arguments attempting to reconcile models of quantum information in black holes with ideas in quantum gravity. A number of studies of the scrambling time with different notions of scrambling appear to be consistent with this conjecture <cit.> in certain classes of systems, but it is not expected that a nontrivial bound on t_s exists for a general quantum mechanical system without any restrictions on the nature of interactions <cit.>.

Qualitative summary— In this Letter, we tackle the challenge of proving a bound on fast scrambling assuming “just the validity of quantum mechanics” (as posed in Ref. <cit.>), going beyond the specific models of local interactions that were previously considered necessary. Our approach is enabled by direct connections between quantum dynamics and the structure of the energy spectrum developed in Refs. <cit.>, assuming little beyond the existence of quantum energy levels. In particular, due to a refinement of the energy-time uncertainty principle resulting from these connections <cit.>, the problem of formulating speed limits on infinite temperature scrambling within quantum mechanics can be mapped to the asymptotic properties of the Fourier transform of the density of states (distribution of energy levels) of a Hamiltonian system <cit.>. In this work, we extend this result in two ways: (1) We generalize the previously obtained speed limit to a reservoir in an arbitrary initial state ρ̂_β E, including finite temperature states [Eqs. (<ref>), (<ref>)]; (2) We identify (and derive a quantitative version of) a rigorous theorem on the asymptotics of Fourier transforms that provides an exact speed limit for any Hamiltonian system.

Surprisingly, our speed limit holds for arbitrarily nonlocal systems (including random matrix Hamiltonians <cit.>). It establishes a quantitative limit on fast scrambling [Eq. (<ref>)] that is logarithmic in the scrambled entanglement entropy [Eq. (<ref>)], as a property of quantum mechanics itself rather than any particular class of systems.

Quantitative overview of results— To formulate our results in a manner similar to Ref. <cit.>, we work with the mean return probability of the computational basis states in ℋ_S (i.e., selecting _k with α = k in Eq. (<ref>)) under evolution by a Hamiltonian Ĥ, given the corresponding initial state ρ̂_k(0) in Eq. (<ref>):

    P_S(t) ≡1/D_S∑_k=1^D_S[e^-iĤtρ̂_k(0) e^iĤt_k].

This quantity partially measures the extent to which the computational basis, among all the bases in ℋ_S, retains information about the initial state, and constrains the generation of entanglement between the two systems which is at the root of quantum thermalization <cit.>. For example, the average purity of the reduced states ρ̂_k,S(t) ≡_Eρ̂_k(t) is constrained by <cit.>,

    1/D_S∑_k=1^D_S_S[ρ̂_k,S^2(t)] ≥ [P_S(t)]^2.

Initially, P_S(0) = 1, signifying complete retention of this information and pure basis states in ℋ_S. At subsequent times, we expect the scrambling process to cause P_S(t) to decay to a “scrambled” value P_. For example, when information about the initial state is completely and uniformly delocalized (to leading order) over the D_S basis states | k⟩_S, we have
P_∼ D_S^-1,
which is typically necessary for maximal entanglement <cit.>.

Here, we use formal asymptotic notation <cit.>, with ω, Ω, Θ, O, o representing >, ≥, =, ≤, < up to any multiplicative constant, and ≳, ∼, ≲ for leading-order (in)equalities, when a parameter such as D diverges.
To maintain generality, we do not assume a specific scrambled value at this stage, and we allow D to remain finite without an explicit limit, except where asymptotic notation is used.


Our main concern is the sustained <cit.> scrambling time t_s, after which P_S(t) remains no larger than the corresponding scrambled value P_. We must also implicitly only consider times | t| < T_D, where T_D is a cutoff beyond all time scales of physical interest, chosen to avoid quantum recurrences <cit.> to large P_S(t). Thus, t_s is set by the condition

    P_S(t > t_s) ≤ P_.

Given this setup, we can summarize our main result. We find that t_s is bounded in any quantum mechanical system in terms of the following three quantities: (1) the scrambled return probability P_, (2) the information-theoretic fidelity <cit.> 0 ≤ f_β≤ 1 between the thermal state ρ̂_β E and the infinite temperature maximally mixed state ρ̂_0 E≡_E/D_E in ℋ_E,

    f_β≡_E√(√(ρ̂_0 E) ρ̂_β E^√(ρ̂_0 E)) = 1/√(D_E)_E[ρ̂_β E^1/2],

and (3) an effective partition function Z_β(τ) = (-iτ) [see Eq. (<ref>)] in a Euclidean time interval containing τ=0:

    τ∈ [τ_1,τ_2], τ_1 ≤ 0 ≤τ_2,  of width  (τ_2-τ_1) ≡ cβ.


Specifically, we show that for every interval [τ_1,τ_2] as in Eq (<ref>), the sustained scrambling time satisfies:

    t_s ≥c β/πln[1/_effln(f_β^2 Z_max^2/P_)].

The parameters Z_max = max_τ∈ [τ_1,τ_2] Z_β(τ) (attained <cit.> at τ = τ_1 or τ_2) and _eff [Eqs. (<ref>), (<ref>)] depend on Z_β(τ) within the interval. One can further optimize the choice of interval in Eq. (<ref>), including cβ, to yield the tightest bound on t_s, but this depends on the behavior of Z_β(τ) and must be done in system-specific ways. Eq. (<ref>) provides a direct, exact bound on the generation of entanglement [via Eq. (<ref>)], and applies to arbitrary Hamiltonians Ĥ, in contrast to the (conjectured) bound on chaos in out-of-time-ordered correlators (OTOCs) of local operators that is shown to hold only for special classes of Hamiltonians <cit.>. Further, the two bounds do not necessarily imply each other and refer to distinct physical processes associated with scrambling <cit.>.

We can alternatively directly constrain t_s in terms of a minimum 2nd Rényi entanglement entropy <cit.> S_2,S required for a state to be considered scrambled in ℋ_S:

    S_2[ρ̂_k,S^2(t>t_s)] ≡ -ln[ρ̂_k,S^2(t)] ≥ S_2,S.

By Eq. (<ref>), P_ can then be replaced <cit.> by exp(-S_2,S/2) in Eqs. (<ref>) and (<ref>). 
To simplify the resulting bound on t_s, we consider a thermodynamic limit N_S, N_E→∞, such that Z_β(τ) remains O(1) in [τ_1,τ_2] and f_β≥exp[-o(S_2,S)] is not too small. In this regime, we obtain <cit.> to leading order:

    t_s ≳cβ/πln S_2,S.

Further, a canonical thermal state typically admits c = (1-ϵ)/2 with a small ϵ > 0 for Z_β(τ) = O(1) [as motivated near Eq. (<ref>)]. For volume-law (including maximal) entanglement <cit.>, S_2,S∼μ N_S for N_S < N_E. Consequently, the conjectured <cit.> fast scrambling bound t_s = Ω(βln N) holds for subsystems with N_S ∼ N^κ (where 0 < κ≲ 1).


Effective nonunitary dynamics at finite temperatures— We begin our technical derivation with a brief summary of a result of Ref. <cit.>. There, we derived a universal speed limit on P_S(t) under arbitrary time-dependent quantum operations <cit.> ρ̂_k(t) = ∑_r=1^M rtρ̂_k,0rt, but only for a special choice of initial states ρ̂_k,0 = _k/D_E corresponding to the infinite temperature state ρ̂_0 E in Eq. (<ref>).

To adapt this bound for the more general family of initial states in Eqs. (<ref>) and (<ref>), we can apply a simple trick that trades off unitary dynamics with an arbitrary initial state for nonunitary dynamics with infinite temperature initial states.

Specifically, in Eq. (<ref>), we write

    ρ̂_k(t) = e^-iĤtρ̂_k(0) e^iĤt   = 1/D_E1t_k1t, 
    with 1t   ≡ e^-iĤtρ̂_β E^1/2 D_E^1/2.


Thus, P_S(t) can also be regarded as the mean return probability of the infinite temperature initial states ρ̂_k,0 = _k/D_E with effective time-dependent nonunitary dynamics given by 1t. In this case, the speed limit of Ref. <cit.> gives the following bound <cit.>, which underlies the main results of this Letter:

    P_S(t) ≥ K_β(t).

In the above inequality,

    K_β(t) = D_E/D^2|(e^-iĤtρ̂_β E^1/2)|^2

is the regularized spectral form factor (SFF) corresponding to the density operator ρ̂_β E, occurring here as a special case of the generalized SFF <cit.> for time-dependent quantum operations K(t) ≡ D^-2∑_r=1^M |rt|^2, which extends and unifies time-dependent <cit.> and dissipative <cit.> SFFs.

For context, we note that SFFs are a key staple of quantum chaos that characterize energy level correlations <cit.> and the presence of observable-independent ergodic properties in quantum dynamics <cit.>. Further, they can be directly measured in experiments <cit.>. Eq. (<ref>) specifically belongs to a family of regularized SFFs <cit.> and correlators <cit.>, where the splitting of roots of a density operator between different factors (usually for analytical convenience) has been regarded as somewhat artificial to engineer in experiments <cit.>. That the regularized SFF K_β(t) universally constrains pure Hamiltonian dynamics via Eq. (<ref>) now provides an exact operational physical justification for regularization, at least in one context. 

Properties of regularized density of states— From Eqs. (<ref>), (<ref>), it is clear that we can obtain universal bounds on the decay of P_S(t) if we can constrain the decay of K_β(t) for any Hamiltonian system. To do so, it is useful to focus on its “square root” up to constant factors, the trace of 1t,

    (t) ≡1t/10 = D_E^1/2/f_βD∑_n⟨ E_n|ρ̂_β E^1/2| E_n⟩ e^-iE_n t,

normalized to (0) = 1, so that |(t ≥ 0)|≤ 1. (t) is the Fourier transform of the regularized density of states (a probability distribution that integrates to (0) = 1),

    (E) = D_E^1/2/f_βD∑_n ⟨ E_n|ρ̂_β E^1/2| E_n⟩δ(E-E_n) ≥ 0.

In terms of this quantity, we have

    K_β(t) = f_β^2|(t)|^2.

Now, we can use this in conjunction with Eqs. (<ref>) and (<ref>) to directly relate the scrambling time t_s to the regularized density of states,

    |(t > t_s)|^2 ≤P_/f_β^2,

generalizing the corresponding infinite temperature result of Ref. <cit.>. For this to give a nontrivial t_s > 0, we must have f_β^2 > P_ (as |(t)|≤ 1). This condition can always be satisfied for any given P_ by any ρ̂_β E with sufficient entanglement entropy in ℋ_E <cit.>, and for any given initial state ρ̂_β E by choosing N_S large enough to have sufficiently small P_ < f_β^2 [e.g., if P_∼ D_S^-1].

To constrain the decay of (t), we extend the time variable to the complex plane, = t - i τ, with t, τ∈ℝ. In this case, () is a weighted sum of D (analytic) exponentials e^-iE_n, and is therefore analytic everywhere on ℂ. It is also real-valued on the imaginary axis, with (-iτ) = Z_β(τ) being the partition function of the regularized density of states at inverse temperature (or Euclidean time) τ. From Eq. (<ref>), this partition function sets an upper bound on |()| at each τ:

    |(t-iτ)|≤ Z_β(τ), ∀ t ∈ℝ,

which is saturated (at least) at t = 0.

While Z_β(0) = 1, it can diverge for other values of τ, making () unbounded in a thermodynamic limit. For example, for a canonical thermal state ρ̂_β E with N_S ≪ N_E whose energy eigenbasis matrix elements behave as e^-β E_n,

    Z_β(τ) ≈∑_n e^-(β+2τ)E_n/2/∑_n e^-β E_n/2.

For most -β/2 < τ≤ 0, the numerator remains exponentially suppressed, and we expect Z_β(τ) = Θ(1) if the density of states doesn't grow exponentially with E_n. However, for τ≈ -β/2, the exponential suppression in the numerator is lost and Z_β(-β/2) ≈ D_E^1/2/f_β diverges as N_E →∞. This motivates considering horizontal strips containing τ=0, restricted to a specific width cβ as in Eq. (<ref>), in which quantities remain bounded. While this argument suggests that a strip in (-β/2,0] with c = (1-ϵ)/2 is appropriate for thermal states, we formulate our results for an arbitrary width cβ≥ 0 to retain exactness and universal applicability to arbitrary initial states.

Here, we also note previous physical bounds involving similar properties of analyticity and boundedness in a strip, including that expected for regularized OTOCs <cit.>, as well as at a point of inflection of the regularized SFF <cit.>, which however differ qualitatively and quantitatively from the results of interest [e.g., Eq. (<ref>)] in this Letter.

Universal bound on fast scrambling— In our case, given that () is analytic and bounded in the strip given by Eq. (<ref>), results from the theory of analytic functions <cit.> constrain _β(t≥ 0) to decay no faster than a double exponential in t for nearly all t>0, except in an “exceptional set” ⊂ℝ_>0 of finite length in t. In particular, recalling that Z_max = max_τ∈ [τ_1,τ_2] Z_β(τ), we have <cit.>

    |(t ∉)|^2 ≥ Z_max^2 exp[-exp(π t/cβ)],

where  is a finite constant <cit.>.
For intuition behind the exceptional set , consider <cit.> the function (t) = ^2(π t/β), which is oscillating and vanishes at the nodes t = nβ for positive integers n. While its asymptotic behavior is generally t^-2, satisfying Eq. (<ref>) for most t, the oscillations bring it below the bound of Eq. (<ref>) in a (shrinking with n) neighborhood of each node, which comprise the finite-length set . The presence of the exceptional set excludes all discrete-time systems from a nontrivial application of Eq. (<ref>), such as Haar random unitaries that maximally scramble ℋ_S after just a single time-step t=1, but not their continuous-time Hamiltonian extrapolations.

However, in this form, Eq. (<ref>) is not sufficiently predictive for our purposes. This is because the “finite” constant  and the “finite” duration of  are undetermined, and could still be larger or smaller than any scale of interest for a system with finite D.
Fortunately, we can adapt the methods of Ref. <cit.> to explicitly obtain their values <cit.>. We find that a rescaled length of  can be made smaller than any chosen ℓ > 0,

    ∫_ t ≤cβ/πℓ,

if one chooses  according to 

    _ℓ = (2+ℓ)/ℓ{min_τ∈ [τ_1,τ_2] 2(π^2+8)ln[Z_max/Z_β(τ)]/cos[π/cβ(τ-τ_1+τ_2/2)]}.



We can now obtain the scrambling time from Eq. (<ref>), with the caveat that the exceptional set introduces an uncertainty in the scrambling time with range [t_s, t_s+ (cβℓ/π)] from Eq. (<ref>). We get <cit.>, using Eq. (<ref>), (<ref>) and (<ref>),

    t_s ≥cβ/πln[1/_ℓlnf_β^2 Z_max^2/P_]-ℓ,

where the second term accounts for the uncertainty by subtracting the length of  from the bound obtained. 
To optimize tightness, we choose ℓ to maximize the right hand side of Eq. (<ref>). We get ℓ = √(3)-1, from which we recover our main universal result, Eq. (<ref>), with  _eff≡ e^√(3)-1_(√(3)-1).


Further, an important question is whether a near-optimally decaying double exponential function as in Eq. (<ref>) can emerge in a system with a nonnegative (regularized) density of states (E) ≥ 0 (e.g., <cit.>). This is indeed the case: a class of examples was constructed in Refs. <cit.>, <cit.>, 
and we construct other (simpler) examples <cit.> by
taking the thermodynamic limit of (t) to be the 
convolution g*g, where g is  the double-exponentially decaying function g(t)=exp(-e^π/t)exp(-e^-π/t).
Given sufficiently (including Haar <cit.>) random energy eigenstates, these examples scramble <cit.> by t_s ≲ (2cβ/π)ln N_S, nearly saturating Eq. (<ref>) <cit.>.


Discussion—We consider the implications of our results in light of the fast scrambling conjecture <cit.>, which states that (a) t_s ≳ O(βlog N) for some definition of scrambling time and some class of systems (neither rigorously specified <cit.>), and (b)  quantum systems believed to describe black holes (again not rigorously specified) saturate this bound. For a successful analysis, we must specify at least one of the three unspecified features and examine its implications for the others.

In this Letter, we have considered the notion of scrambling in Eq. (<ref>) that directly constrains the time required for sustained entanglement generation and the applicability of equilibrium statistical mechanics in subsystems. In this case, Eq. (<ref>) provides a universal quantum mechanical bound on information scrambling by any Hamiltonian system; Eq. (<ref>) further shows that even the conjectured t_s = Ω(βln N) form holds universally for volume-law entanglement in a certain thermodynamic regime. This completely specifies and establishes statement (a) of the fast scrambling conjecture for this notion of scrambling, generalizing it beyond any specific assumptions of interaction structure.

For statement (b), we first consider the example of the “maximally chaotic” Sachdev-Ye-Kitaev models <cit.>, which saturate the OTOC bound for local correlators <cit.>. A key feature of these models is a zero-temperature entropy (extensive clustering of states near the ground state) <cit.> that is directly believed to capture some aspects of black hole physics <cit.>. However, this feature leads to a subleading slow decay <cit.> of K_β(t), giving t_s exponential in N_S (for large N_S) as shown in Ref. <cit.> (see also Ref. <cit.> for an interesting parallel to glassy metastable states). Thus, for scrambling via entanglement generation, we can limit the applicability of statement (b) by noting that systems with a nonvanishing zero temperature entropy are not fast scramblers in large subsystems. At the same time, the discussion after Eq. (<ref>) illustrates what the energy spectrum of a “nearly fast scrambler” may look like, showing that systems nearly saturating Eq. (<ref>) do formally exist. An intriguing open challenge is to address the following question: What is the natural physical setting in which we can expect to find such fast scramblers in the sense of sustained entanglement generation?


This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0001911.
The authors acknowledge the University of Maryland supercomputing resources (https://hpcc.umd.edu) made available for generating the plots in Section D of the supplement
to this paper. 
A.V. acknowledges useful discussions with Jorge Kurchan during the program – Stability of Quantum Matter in and out of Equilibrium at Various Scales (code: ICTS/SQMVS2024/01) at the International Center for Theoretical Sciences, Bengaluru, India.










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


Supplemental Material




Amit Vikram,^1 Laura Shou,^1,2 and Victor Galitski^1


^1Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA

^2Condensed Matter Theory Center, Department of Physics, 
 University of Maryland, College Park, MD 20742, USA






Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems
    Victor Galitski
    April 29, 2024
======================================================================



In this supplement, we derive various results presented in the main text. In Sec. <ref>, we derive quantitative relations between scrambling in terms of return probabilities and in terms of entanglement entropies [supporting Eqs. (<ref>) and (<ref>) of the main text]. In Sec. <ref>, we derive the inequality P_S(t) ≥ K_β(t) that formulates the energy-time uncertainty principle with sensitivity to microscopic values of P_S(t) and nontriviality at arbitrarily long times (as in Ref. <cit.>), generalized to any initial state of the external system including finite temperature states [Eq. (<ref>) of the main text]. In Sec. <ref>, we derive our main result, the universal bound on the scrambling time t_s [Eq. (<ref>) of the main text] by proving the double exponential bound on analytic functions with quantitatively determined parameters [Eqs. (<ref>)-(<ref>) of the main text]. Finally, in Sec. <ref>, we show that quantum mechanical systems that nearly saturate our bound on the scrambling time formally exist, by discussing explicit examples with a suitable (regularized) density of states.



§ SUSTAINED SCRAMBLING AND ENTANGLEMENT ENTROPIES




 §.§ The mean return probability bounds the average purity of the evolving basis states

In this subsection, we show that the mean return probability P_S(t) constrains the generation of entanglement, as stated in Eq. (<ref>) of the main text. We consider the average purity on the left hand side of this equation, and split it into contributions from matrix elements in the computational basis:

    1/D_S∑_k=1^D_S[ρ̂_k,S^2(t)]    = 1/D_S∑_k,k_1,k_2=1^D_S|_S⟨ k_1|_E[ρ̂_k(t)]| k_2⟩_S|^2 
       ≥1/D_S∑_k,k_1=1^D_S|_S⟨ k_1|_E[ρ̂_k(t)]| k_1⟩_S|^2 
       ≥1/D_S∑_k=1^D_S|_S⟨ k|_E[ρ̂_k(t)]| k⟩_S|^2.

We have dropped the off-diagonal matrix elements k_1≠ k_2 and used the non-negativity of the corresponding terms in the second line. In the third line, we have further dropped the k≠ k_1 contributions, which are also non-negative, to obtain an inequality focusing on the 3D diagonal k=k_1=k_2 contributions. Though a simple technique, dropping off-diagonal terms in such inequalities has been considerably useful in other physical contexts as well <cit.>.

We can rewrite the last expression in Eq. (<ref>) in terms of the projectors _k = | k⟩_S⟨ k|⊗_E, which gives

    1/D_S∑_k=1^D_S[ρ̂_k,S^2(t)] ≥1/D_S∑_k=1^D_S[ρ̂_k(t)_k]^2.

The right hand side is the mean (over all values of k) of the squares of the return probabilities P_k(t) ≡[ρ̂_k(t)_k] ≥ 0, which must be at least the square of their mean. Thus, we obtain:

    1/D_S∑_k=1^D_S[ρ̂_k,S^2(t)] ≥1/D_S∑_k=1^D_S P_k^2(t) ≥(1/D_S∑_k=1^D_SP_k(t))^2 = [P_S(t)]^2,

which is Eq. (<ref>) in the main text.



 §.§ The scrambling time ts in terms of the scrambled Rényi entanglement entropy


Here, we express the scrambling bound of Eq. (<ref>) in terms of the entanglement entropy of a scrambled state, deriving Eq. (<ref>) of the main text.

The second Rényi entanglement entropy of the state ρ̂_k(t) in the subsystem ℋ_S is defined as

    S_2[ρ̂_k,S^2(t)] ≡ -ln[ρ̂_k,S^2(t)].

This is a direct measure of entanglement <cit.>. For example, volume-law entanglement corresponds to S_2 ∼min{ N_S ln 2, N_E ln 2} to leading order, usually associated with maximal scrambling as we also have the general bound:

    S_2 ≤{ N_S ln 2, N_E ln 2}

We now introduce the scrambled entropy cutoff S_2,S as the minimum necessary value of the entanglement entropy at which a state is considered scrambled in ℋ_S. In particular, we require that every scrambled state satisfies (as in Eq. (<ref>) of the main text):

    S_2[ρ̂_k,S^2(t > t_s)] ≥ S_2,S

(where t<T_D is implicit to avoid quantum recurrences <cit.>). It follows that the mean purity on the left hand side of Eq. (<ref>) is at most exp[-S_2,S] for t>t_s. Then Eq. (<ref>) [or Eq. (<ref>) in the main text] implies

    P_S(t > t_s) ≤exp(-1/2 S_2,S),

corresponding to replacing P_→exp(-S_2,S/2) in Eq. (<ref>) of the main text.
We can substitute this in the bound on the sustained scrambling time t_s, given by Eq. (<ref>) of the main text, which yields

    t_s ≥cβ/πln[S_2,S+4ln(f_βZ_max)/2_eff].

In a thermodynamic limit with |ln (f_βZ_max)| = o[S_2,S] and _eff = Θ(1) (requiring Z_β(τ) = O(1) in the interval τ∈ [τ_1,τ_2]), we get the leading order inequality

    t_s ≳cβ/πln S_2,S,

which is Eq. (<ref>) of the main text. In this thermodynamic regime, we can qualitatively phrase our fast scrambling result in simple terms: the scrambling time is at least logarithmic in the scrambled (2nd Rényi) entanglement entropy.
While Eqs. (<ref>) and (<ref>) directly capture the time required to generate a given degree of entanglement in our setup, we note two reasons for preferring the expression in terms of the scrambled return probability P_ in Eq. (<ref>) of the main text.

    
  * Most significantly, the bound in terms of P_ can be made tighter. Intuitively, this is because P_S(t) can continue to track aspects of quantum dynamics and scrambling that the entanglement entropy is insensitive to; for example, S_2,S is subject to the bound of Eq. (<ref>), while P_ can be taken to be as small as desired. For concreteness, consider maximal scrambling (e.g., to Haar random behavior), for which P_∼ D_S^-1 and S_2,S∼min{ N_S,N_E}ln 2. When N_S ≤ N_E, both give identical bounds to leading order, t_s ≳ (cβ/π)ln N_S. But for N_S > N_E, while -ln P_∼ N_S continues to hold and constrains t_s ≳ (cβ/π)ln N_S from Eq. (<ref>) of the main text, the leading contribution to the entropy comes from ℋ_E with S_2,S∼ N_E ln 2, due to which Eq. (<ref>) sets a much weaker bound t_s ≳ (cβ/π)ln N_E. For a physical example where this can be relevant, see Ref. <cit.>, where scrambling is considered for subsystem sizes including N_S > N_E (e.g., in the SYK-4 model).
    
  * P_ requires only projective measurements in a computational basis with the D_S projectors _k, and directly determines the dynamics of observables diagonal in this basis. In contrast, S^(2)_ in principle probes more of the full structure of the reduced density matrices ρ̂_k,S in the Hilbert space H_S. In particular, it may be dominated by contributions (such as from the off-diagonal matrix elements of ρ̂_k,S) not relevant to a specific observable of interest.


On the other hand, S^(2)_ can be efficiently measured in experiments, for instance, using local randomized measurements <cit.>; this means that Eqs. (<ref>) and (<ref>) are also testable experimentally, in addition to the bound in terms of P_S(t).



§ A QUANTUM SPEED LIMIT FOR ARBITRARY INITIAL STATES PBE


In this section, we describe the passage from the infinite temperature speed limit of Ref. <cit.> to the inequality P_S(t) ≥ K_β(t) for arbitrary initial states under Hamiltonian evolution with more details than in the main text, beginning with a brief review of the former. This is essentially an expanded version of the derivation containing Eq. (<ref>) through Eq. (<ref>) in the main text.



 §.§ Derivation of Ps(t) >= Kb(t)


Consider a general time-dependent completely positive quantum operation <cit.> acting on an initial reference state ρ̂_0 in ℋ, with any set of M time-independent Kraus operators {rt}_r=1^M,

    ρ̂(t) = ∑_r=1^Mrtρ̂_0rt.

Note that the initial state ρ̂(0) is not necessarily equal to the reference state ρ̂_0, and further that the quantum operation is not required to be trace preserving <cit.>. Quantum dynamics of this type may be characterized by a generalized SFF,

    K(t) ≡1/D^2∑_r=1^M|rt|^2

Given this setup, the mean return probability for specific initial reference states ρ̂_0,k = _k/D_E (or equivalently, the states of Eq. (<ref>) in the main text with ρ̂_β E = ρ̂_0 E≡_E/D_E),

    P_S(t)[_k] ≡1/D∑_k=1^D_S[_k(t)_k(0)] = 1/D∑_k=1^D_S∑_r=1^M[rt_k(0) rt_k(0)].

was shown to be constrained by <cit.>

    P_S(t)[_k] ≥ K(t),

for any complete, orthonormal choice of projectors _k.

For the above speed limit, it is crucial that the initial reference states _k/D_E form a complete, orthogonal set for the full Hilbert space ℋ. It is this completeness that allows constraining the basis-dependent P_S(t) with the basis-independent K(t) that involves a trace over the entire space ℋ. However, the initial states ρ̂_k(0) = | k⟩_S⟨ k|⊗ρ̂_β E of Eq. (<ref>) in the main text are complete only in ℋ_S, and generally not in ℋ_E except in the specific case of the infinite temperature state ρ̂_0 E.

The resolution to this difficulty comes from writing the initial state ρ̂_k(0) in terms of nontrivial Kraus operators r0 acting on the initial reference state ρ̂_0,k = _k/D_E. This is enabled by noting that ρ̂_β E is a positive operator (has non-negative eigenvalues) <cit.>, and therefore admits a positive Hermitian square root ρ̂_β E^1/2 (with non-negative eigenvalues). In particular, we have

    ρ̂_k(0) = ρ̂^1/2_β E(| k⟩_S ⟨ k|⊗_E) ρ̂^1/2_β E = [D_E^1/2ρ̂^1/2_β E]ρ̂_0,k[D_E^1/2ρ̂^1/2_β E]^†.

Now, we can set M=1 with 10 = D_E^1/2ρ̂_β E^1/2, i.e., a single nonvanishing Kraus operator. As subsequent time evolution is generated by a Hamiltonian Ĥ, we have the time-dependent Kraus operators:

    rt = e^-iĤtr0 = D_E^1/2 e^-iĤtρ̂_β E^1/2δ_r,1.

Substituting these Kraus operators in Eq. (<ref>), we obtain precisely the mean return probability in Eq. (<ref>) of the main text:

    P_S(t)[_k] = 1/D∑_k=1^D_SD_E [e^-iĤtρ̂_β E^1/2_k ρ̂_β E^1/2 e^iĤ t_k ] = P_S(t).

Further, the SFF for these Kraus operators is

    K(t) = 1/D^2|1t|^2 = D_E/D^2|(e^-iĤtρ̂_β E^1/2)|^2 ≡ K_β(t).

Now, Eq. (<ref>) for these Kraus operators gives P_S(t) ≥ K_β(t), which is Eq. (<ref>) in the main text.



 §.§ Criteria for nontriviality


We should emphasize that, unlike the β=0 case where P_S(0) = K(0) = 1 under Hamiltonian dynamics, for β≠ 0 their values at t=0 can be different. In particular, P_S(0) = 1, while K_β(0) = f_β^2 ≤ 1 [see Eqs. (<ref>) and (<ref>) of the main text]. Further, as K_β(t) ≤ K_β(0), we can only obtain a nontrivial bound on the scrambling time from P_S(t) ≥ K_β(t) if

    f_β^2 > P_.


In this subsection, we ask under what conditions a system may satisfy Eq. (<ref>). These considerations are equivalent to, and expand on, the discussion following Eq. (<ref>) of the main text. The need to consider such criteria is in contrast to the β = 0 case of Ref. <cit.>, where as long as P_ < 1, some nontrivial >0 bound on t_s always exists.

However, the tradeoff in our case still admits two kinds of universality:

    
  * For a given initial state with a certain f_β^2, we get nontrivial values when P_ < f_β^2. This can translate to a restriction on the system size of ℋ_S. For example, if we are interested in maximal scrambling with P_∼ D_S^-1, the number of qubits N_S in the subsystem ℋ_S must be as large as
    
    N_S > 2 log_2 1/f_β,

    for a nontrivial scrambling time. An interesting special case is when the initial state is a pure state, ρ_β E = |ℓ⟩_E⟨ℓ|, which has f_β = D_E^-1/2. Then, we get N_S > N_E for a nontrivial bound for pure states.
    For any state that is not entirely pure, it follows that we get nontrivial scrambling time bounds even for some values of N_S ≤ N_E.
    
  * For a given scrambling value P_ < 1, the condition f_β^2 > P_ restricts the entanglement of the initial state ρ̂_β E. For example, this is because the 1/2-order Rényi entanglement entropy <cit.> of this state is determined by the fidelity:
    
    S_1/2,E≡ 2 ln[D_E^1/2f_β].

    Thus, for a nontrivial bound, S_1/2,E  > ln[D_E P_], implying that a smaller value of P_ requires less entanglement in the initial state for our bound to be useful.




§ DERIVATION OF A UNIVERSAL BOUND ON THE SCRAMBLING TIME


In this section, we derive a quantitative refinement of a known asymptotic double exponential bound on bounded analytic functions which is originally given in terms of finite but undetermined constants <cit.>. This refinement provides explicit values for the undetermined constants in the previous versions. As mentioned in the main text, this allows us to constrain the scrambling time even for finite dimensional systems and obtain concrete numerical bounds for t_s that can potentially be tested in experiments, instead of just an asymptotic estimate in a thermodynamic limit. This fills in the technical details for the discussion around Eqs. (<ref>) through (<ref>) of the main text. 

Sec. <ref> states our quantitative version of the double-exponential bound in Theorem <ref>, and derives the scrambling time as per Eqs. (<ref>) and (<ref>) of the main text. Sec. <ref> proves the double exponential bound by conformally mapping the strip to a half-plane, based on two lemmas on analytic functions on the half-plane, closely following the methods of Ref. <cit.>.

As a prelude to our technical discussion, we provide some local power series intuition for why analyticity should constrain the decay of a bounded function. For simplicity, let an analytic F(t) have a saddle point at F(t=0) = 1. On account of analyticity, we can write a power series expansion in a neighborhood of this point:

    F(t - iτ) = 1 - 1/2c_2(t-iτ)^2 + O((t-iτ)^3),

where c_2 = F”(0). We take c_2 ∈ℝ, so that F(iτ) ∈ℝ [to O(τ^2)] grows as 1+c_2τ^2/2 away from t=0 while F(t) ∈ℝ [to O(t^2)] decays as 1-c_2t^2/2; in addition, F(t) then locally satisfies | F(t-iτ)| < F_R(-iτ) to O((t-iτ)^2) [like in Eq. (<ref>) of the main text] so it is sufficient to consider F(-iτ) for boundedness. The key restriction from analyticity here is that the same coefficient c_2 determines both the growth and decay. Thus, if F(-iτ) < F_max in τ∈ (0, τ_2), we have

    F(t) ≥ 1-(F_max-1)(t/τ_2)^2+O(t^3).

In other words, the decay rate is bounded by F_max. This qualitatively connects analyticity and boundedness to (local) decay rates in an intuitive example, but we emphasize that the mathematical details of our (global) double exponential bound [Eq. (<ref>), or Eq. (<ref>) in the main text] and method of proof are quite different.
Mathematically inclined readers may find
closer intuition to our result through
Jensen's formula from complex analysis or the Poisson kernel formula for harmonic functions <cit.>.



 §.§ The scrambling time from a double-exponential bound on analytic functions





  §.§.§ A quantitative theorem: analytic functions are bounded by a double-exponential in time


The key to deriving our quantitative bound on the scrambling time is the following theorem bounding the decay of analytic functions, which quantifies the undetermined parameters in the previously known double-exponential bounds of Refs. <cit.> (the latter being stated in Eq. (<ref>) of the main text):


Let the function F() be analytic with | F()| < 1 in the open strip { = t-iτ∈ℂ: τ∈ (τ_1,τ_2)},  and continuous in the corresponding closed strip τ∈ [τ_1,τ_2]. We also require that F() is not identically 0 in the strip. Then for any 0 < ℓ < ∞, there is an “exceptional set” ⊂ [0,∞) of times t whose length is at most

    ∫_ t ≤(τ_2-τ_1)/πℓ,

such that | F()| is at least a double exponential in t everywhere in the strip except when t is in the exceptional set, i.e., 

    | F(t-iτ)|   ≥exp[-Λ_ℓ/2exp(π t/τ_2-τ_1)], 
        for all  t ∈ [0,∞) ∖,  and τ∈ [τ_1,τ_2].

Here, the parameter _ℓ≥ 0 is given in terms of ℓ and F(-iτ) by _ℓ = 2(π^2+8) (2+ℓ)/ℓ, in which 

    = inf_τ∈(τ_1,τ_2){[π/τ_2-τ_1(τ-τ_2+τ_1/2)]ln1/| F(-iτ)|}.



 See Sec. <ref>. The proof closely follows the techniques used to prove Theorem 7.32 in Ref. <cit.>, which is related to the present theorem, but unlike Eq. (<ref>) does not quantitatively determine the constants in Eq. (<ref>) of the main text.


Note that if we set ℓ = 0, the expression in the exponent of Eq. (<ref>) diverges, so we can never guarantee that the exceptional set vanishes for the double exponential form in this equation. However, we can make it as small as desired.



  §.§.§ Applying the bound to Nb(t)


Here, we derive Eqs. (<ref>) through (<ref>) of the main text from Theorem <ref>. As noted in Eq. (<ref>) of the main text, () is a weighted sum of exponentials with non-negative weights,

    (t-iτ) = D_E^1/2/f_βD∑_n=1^D⟨ E_n|ρ̂_β E^1/2| E_n⟩ e^-iE_n t-E_n τ,

and is consequently analytic and bounded on any strip t∈ℝ, τ∈ (τ_1,τ_2), as well as continuous up to the boundary of the strip. As we want to focus on evolution in real time, we take τ=0 to lie in the strip, τ_1 ≤ 0 ≤τ_2. Further, (0) = 1 making () not identically 0. Thus, () already satisfies most of the properties required by Theorem <ref> for F(), except not necessarily that | F()| < 1 in the interior of the strip.

To normalize () to satisfy this last property, we consider the maximum of (t) in the closed strip τ∈ [τ_1,τ_2]. From Eq. (<ref>) we have Z_β(τ) ≡(-iτ) ∈ [0,∞) and |(t-iτ)|≤ Z_β(τ) [Eq. (<ref>) of the main text], which amount to the “ridge property” <cit.> of Fourier transforms of probability distributions — implying that the maximum of () occurs on the imaginary axis t=0.

Further, Z_β(τ) is also a convex function of τ (see, e.g., Ref. <cit.>); in our case, this is because

    ^2 Z_β/τ^2(τ) = D_E^1/2/f_βD∑_n E_n^2 ⟨ E_n|ρ̂_β E^1/2| E_n⟩ e^-E_n τ≥ 0.








Here, there are two distinct cases of interest:

    
  * All energy levels with ⟨ E_m|ρ̂_β E^1/2| E_m⟩≠ 0 have E_m = 0. In this (trivial) case, () = 1 everywhere, automatically satisfying the double exponential bound [Eq. (<ref>) of the main text] for any choice of decay rate Λ≥ 0.
    
  * Otherwise, the partition function is strictly convex, ^2 Z_β(τ)/τ^2 > 0 in [τ_1,τ_2] (as all terms in Eq. (<ref>) are nonnegative, and not all of them vanish). In this case, the maximum
    
    Z_max≡max_τ∈ [τ_1,τ_2] Z_β(τ)

    is attained exclusively on either τ=τ_1 or τ=τ_2, with Z_β(τ) < Z_max in τ∈ (τ_1,τ_2). Then, we can define
    
    F() ≡()/Z_max,  satisfying | F(t-iτ)| < 1  in τ∈ (τ_1,τ_2).

    Theorem <ref> now applies to this F(), and we obtain the quantitative double exponential bound,
    
    |(t)|≥ Z_maxexp[-_ℓ/2exp(π t/τ_2-τ_1)],  for  t≥ 0: t ∉.

    corresponding to Eqs. (<ref>), (<ref>) and (<ref>) of the main text (respectively from Eqs. (<ref>), (<ref>) and (<ref>) of the Theorem, with Eq. (<ref>) obtained by extending the right hand side of Eq. (<ref>) to be continuous at τ_1, τ_2 to replace the infimum with a minimum over [τ_1,τ_2]).


A key point to emphasize is that these results specify the previously undetermined constants such as Λ_ℓ entirely in terms of the behavior of (t-iτ) at the initial (real) time t=0, or equivalently, the partition function Z_β(τ) = (-iτ). This is due to taking the physical viewpoint that we should not expect to know aspects of the late-time dynamics of the system [e.g., (t>0)] when constraining this very dynamics. From a purely mathematical standpoint, we can potentially make  smaller (i.e., obtain a tighter bound) in some cases if we already know some aspects of the t>0 behavior 
(by extending Lemma <ref> in Sec. <ref> to | z|≥ 1; see also <cit.>).

Finally, to provide some physical intuition for the decay rate , we can carry out the minimization in Eq. (<ref>) by finding a stationary point τ_0 (if it exists). Differentiation yields the implicit equation:

    π/τ_2-τ_1tan[π/τ_2-τ_1(τ_0-τ_2+τ_1/2)] lnZ_max/Z_β(τ_0) = .ln Z_β(τ)/τ|_τ=τ_0.

If the stationary point does not exist, the minimum occurs at τ_max∈{τ_1,τ_2} where Z_β(τ_max) = Z_max. Whether τ_0 exists or not, the value of  is in all cases conveniently expressed, using familiar thermodynamic relations <cit.>, in terms of the expectation value of the energy ℰ_β(τ) ≡ -ln Z_β(τ)/τ and the excess free energy Δℱ_β(τ) ≡ -τ^-1[ln Z_β(τ)-ln Z_max] at Euclidean time τ [with Δℱ_β(τ_max) = 0]. We get:

    = √(τ_0^2 [Δℱ_β(τ_0)]^2 + (τ_2-τ_1)^2/π^2[ℰ_β(τ_0)]^2).

This expresses the decay rate in terms of thermodynamic quantities. The location of the point τ_0 and the significance of the (free) energy at this point are system-dependent, and it appears that we cannot obtain further universal insights from this expression.



  §.§.§ Derivation of the scrambling time


We can obtain a bound on the scrambling time t_s by using Eqs. (<ref>) and (<ref>) in Eq. (<ref>) of the main text, which states that (implicitly with t < T_D, the time scale of quantum recurrences <cit.>)

    |(t > t_s)|^2 ≤P_/f_β^2,

However, we should carefully account for the exceptional set . Any part of the exceptional set in the region t≥ t_s does not affect the scrambling time, as it refers to when |(t)| itself already satisfies our scrambling criterion and whether it violates Eq. (<ref>) becomes immaterial. However, the length of the portion of the exceptional set before t_s is uncertain: it can be 0 at minimum, or as much as the maximum allowed length (τ_2-τ_1)ℓ/π of the exceptional set. So we can only constrain the scrambling time to within this maximum possible length. This corresponds to requiring |(t_s+(τ_2-τ_1)ℓ/π)|^2 to be less than the scrambling value. Assuming 0 ∈ [τ_1,τ_2] and using Eq. (<ref>) with (<ref>), we get the inequality

    Z_maxexp[-(2+ℓ)/ℓ(π^2+8) exp(π t_s/τ_2-τ_1+ℓ)] ≤P_^1/2/f_β.
 
Rearranging this equation, we obtain

    t_s ≥τ_2-τ_1/π{ln[1/2ln(f_β^2 Z_max^2/P_)]-ln[(2+ℓ)/ℓ(π^2+8) ]-ℓ}.

This leads to Eq. (<ref>) of the main text.
Since the choice of ℓ is arbitrary, it is best to choose a value that maximizes the bound for t_s (though this is not crucial in a thermodynamic limit, where this term is usually subleading). The corresponding maximization of ln[ℓ/(2+ℓ)]-ℓ over ℓ > 0 gives

    ℓ =  √(3)-1.

Using this value of ℓ in Eq. (<ref>) yields (<ref>) of the main text.



 §.§ Proof of Theorem 1 on the quantitative decay of analytic functions





  §.§.§ Conformal map from a strip to a half-plane




In order to prove Theorem <ref>, it will be convenient to work on the right half-plane instead of the strip, since several formulas from complex analysis are simpler on the half-plane. There is a conformal map between the two regions
shown in Fig. <ref> and defined below, which can translate results on the right half-plane to results on the strip τ_1<-<τ_2.


Specifically, we define the variables (z∈ℂ; r,θ,x,y ∈ℝ) via the transformation

    z = re^iθ = x+iy = exp[π+iτ_2+τ_12/τ_2-τ_1].

Then for (-) ∈ [τ_1,τ_2],

    r    = exp[π/τ_2-τ_1] ∈ (0,∞),     θ = π + τ_2+τ_12/τ_2-τ_1∈[-π/2,π/2],

which is seen to correspond to the closed half-plane {z=x+iy:x≥0,y∈} 
(with z=0 corresponding to values -∞-iτ in the strip). We will freely switch between these different coordinates for the right half-plane in what follows.



  §.§.§ Constraint on regions of double-exponential or faster decay in t


On account of Eq. (<ref>), a double exponential decay in t corresponds to an exponential decay in r. Our goal is now to constrain the size of regions r ∈_ in which an analytic function F(z) may decay faster than the exponential rate exp(- r).
This is accomplished using the following Lemma, which is an immediate consequence of Theorem 7.27 in Ref. <cit.>:


Let F(z) satisfying | F(z)| < 1 be an analytic function that is not identically 0 in the open half-plane x>0. For any > 0, let _⊂ (0, ∞) be a measurable set such that

    inf_θ∈(-π/2,π/2)| F(r e^iθ)|≤ e^- r,   for all  r∈ V_.

The set V_ consists of radii r in which | F(r e^iθ)| is at most e^- r for some value of θ in the right-half-plane. Then, if we define the logarithmic length of _ in r (corresponding in the strip to its actual length in t, up to constants) by

    L[_] ≡∫_V_ r/r,

the following bound is satisfied everywhere on the half-plane x>0:

    ln1/| F(z)|≥4/(2+L[_])(π^2+8)∫__ r' x/x^2 + (r'+| y|)^2 r'.



    This follows from setting u(z) = ln| F(z)| and f(r) =  r in Theorem 7.27 of Ref. <cit.> (see also Theorem 7.32 of Ref. <cit.>).


To provide some relation to more familiar expressions, we note that the integral on the right hand side is related to electrostatics (i.e., solutions of the Laplace equation) on the half-plane <cit.>. In particular,
the Poisson kernel formula <cit.> gives the following solution to Laplace's equation for the electrostatic potential:

    ϕ(x,y) = 1/π∫_-∞^∞ y' x/x^2 + (y'-y)^2ϕ(0,y'),

which we will use with the boundary condition ϕ(0,y) =  y for y ∈_ and 0 otherwise.
With this context, Eq. (<ref>) can be expressed for y ≤ 0 as follows <cit.>:

    ln1/| F(x+iy)|≥4π/(2+L[_])(π^2+8)ϕ(x,y).

The right hand side should be considered <cit.> an indirect measure of the size of the set _.

We make two quick remarks concerning the mathematical context of the above lemma. First, in Ref. <cit.>, the result is stated more generally for subharmonic functions u, which includes u(z)=ln| F(z)| for  analytic F(z) as a special case. 
The subharmonicity suggests why one might expect to see a Poisson kernel-like formula in
Eq. (<ref>).
Second, the actual Poisson kernel formula (see e.g. Ref. <cit.>) for ln|F(z)| can be used in place of Lemma <ref> to obtain quantitative decay rate estimates like in Theorem <ref>, but only on the boundary of the strip. 
In <cit.>, we explore the utility of these Poisson kernel boundary estimates in other situations.
However, in the present setting, since we will often work in the interior of a strip, using Lemma <ref>, which provides decay rate estimates on the entirety of the strip, will produce better t_s bounds whenever τ_1,τ_20. This is because larger strips, which are not restricted to have a boundary at τ=0, will increase the prefactor cβ measuring the strip width in our bound on t_s; in the appropriate thermodynamic limit [Eq. (<ref>) of the main text or (<ref>) here], this is the only parameter available to improve the tightness of the bound [provided that Z_β(τ) = O(1) continues to hold in the larger strip].




  §.§.§ Direct constraint on the decay parameter 


Lemma <ref> can be used to derive the following Lemma (closely related to Theorem 7.32 in Ref. <cit.>):



Let F(z) be as in Lemma <ref>. For any λ > 0 and R>1, let _⊆ [1,R] be a set of exponential-or-faster decay satisfying Eq. (<ref>), and therefore subject to Lemma <ref>.
Then the
logarithmic length L[_] ≤ln R < ∞ of _ is related to the decay rate  by

    ≤(2+L[_])/L[_](π^2+8){1/cosθln1/| F(e^iθ)|},

where z = e^iθ: θ∈ (-π/2, π/2) is any point on the unit semi-circle in the right half-plane (which corresponds to the vertical segment
t=0 in the original strip).


Our proof will closely follow that of Theorem 7.32 in Ref. <cit.>.  In Eq. (<ref>), we set z=e^iθ, and specifically consider only these points in the integral:

    ∫__ r' x/x^2 + (r'+| y|)^2 r'.

In this integral, | x|, | y|≤ 1 due to the restriction to | z|=1, and further r' ≥ 1 as _⊆ [1,∞) by assumption; this implies | z|, | y|≤ r'. The denominator thus satisfies the inequality

    x^2 + (r'+| y|)^2 = | z|^2+(r')^2 + 2(r')| y|≤ 4(r')^2.

Using this inequality in Eq. (<ref>) with x=cosθ gives

    ln1/| F(e^iθ)|≥λcosθ/(2+L[_])(π^2+8)∫__ r'/r'.

Rearranging factors and noting that the integral on the right hand side equals L[_] by Eq. (<ref>), we get Eq. (<ref>).


The above result differs from Theorem 7.32 of Ref. <cit.>, which determines the limit of r^-1ln| F(r e^iθ)| as r→∞, outside an exceptional set of some (unspecified) finite logarithmic length. However, Eq. (<ref>) provides an explicit quantitative constraint on the logarithmic length of the exceptional set _ pertaining to a specific . 
As an aside, we note that Theorem 7.32 of Ref. <cit.> has connections with the Phragmén-Lindelöf principle <cit.> (which generalizes the maximum modulus principle to e.g. unbounded domains). 
This principle also plays a role in the conjectured bound on chaos in out-of-time-ordered correlators (OTOCs) <cit.>, albeit in a quantitatively very different form.



  §.§.§ Completing the proof


Only a few quick observations remain to obtain Theorem <ref> from Eq. (<ref>). We note that the function (2+ℓ)/ℓ is monotonically decreasing with ℓ. Further, as F() is analytic and not identically zero in the open strip by assumption, the identity theorem <cit.> implies that F(e^iθ) is not identically zero in θ∈ (-π/2,π/2). Thus, for any given 0 < ℓ < ∞, if we choose

    = (2+ℓ)/ℓ(π^2+8){1/cosθln1/| F(e^iθ)|}

for any θ∈ (-π/2,π/2) such that F(e^iθ) ≠ 0, then it follows from Eq. (<ref>) that

    2+ℓ/ℓ≤(2+L[_])/L[_],    which implies  L[_] ≤ℓ.

Eq. (<ref>) holds for any value of R, so we can take R→∞ in Lemma <ref>. This implies that given any ℓ > 0, by choosing  according to Eq. (<ref>), we can guarantee that any set _∈ [1,∞) in which | F(r e^iθ)|≤exp(- r) for some θ must have a logarithmic length no greater than ℓ.

Given that the choice of θ in Eq. (<ref>) is arbitrary, we will get a smaller value of λ, and therefore a tighter exponential in Eq. (<ref>), by minimizing the right hand side of Eq. (<ref>) with respect to θ, while Eq. (<ref>) continues to hold. At this stage, transforming back to the strip = t-iτ gives (as the unit semi-circle corresponds to the vertical t=0 segment in the strip)

    λ = (2+ℓ)/ℓ(π^2+8)inf_τ∈(τ_1,τ_2){[π/τ_2-τ_1(τ-τ_2+τ_1/2)]ln1/| F(-iτ)|},

as an appropriate choice of  so that any set ∈ [0,∞) (the transformed version of _) such that

    inf_τ∈ (τ_1,τ_2)| F(t-iτ)|≤exp[-exp(π t/τ_2-τ_1)],   for all  t ∈,

must have a length of at most

    ∫_ t ≤(τ_2-τ_1)/πℓ.

Finally, as F() is continuous up to the boundary of the strip, Eq. (<ref>) can also be extended to τ = τ_1,τ_2. This completes the proof of Theorem <ref>.






§ FORMAL EXAMPLES OF NEARLY FAST SCRAMBLERS


In this section, we construct formal examples of systems that nearly saturate our bound t_s ≳ (cβ/π)ln S_2,S on fast scrambling in the thermodynamic limit. Specifically, we construct the regularized density of states of these systems at an assumed inverse temperature β, assuming that a suitable initial state ρ̂_β E exists, and show that with sufficiently random eigenstates t_s ≲ (2cβ/π)ln S_2,S for these systems. This corresponds to the discussion of “nearly fast scramblers” after Eq. (<ref>) in the main text.



 §.§ Regularized density of states and its Fourier transform


In this section, we construct simple examples of continuum limits of (t) that attain a double exponential decay rate for t∈, subject to the requirement that its inverse Fourier transform (the regularized density of states) satisfies (E)≥0 and ∫_-∞^∞(E) dE=1. 
(See also <cit.>, <cit.> for a different class of examples.) These examples demonstrate that the double exponential decay rate in 
Eq. (<ref>)
of the main text is essentially optimal (up to a constant in the exponent), even with the constraint that (E)≥0.
In <cit.>, we also formally verify that this leads to similar scrambling time bounds for corresponding finite dimensional systems. 

First consider the analytic function 

    g()=exp(-e^π/)exp(-e^-π/),
 for =t-iτ∈ and a fixed >0, which has absolute value

    |g()|    = exp(-e^π t/cos(πτ/))exp(-e^-π t/cos(πτ/)).

The function g is thus bounded in the strip ||≤/2, but does not meet the requirement that its inverse Fourier transform is nonnegative. (One can either check this numerically, or note that g is everywhere analytic, but not maximized at t=0 when τ=, and so even its normalized version cannot be the Fourier transform of a nonnegative density <cit.>.)

However, recall the convolution of two functions, 
g_1*g_2(t) ≡∫_-∞^∞ g_1(y)g_2(t-y) dy,
behaves with the inverse Fourier transform g^∨(E)≡1/2π∫_-∞^∞ g(t)e^itE dt as

    (g_1*g_2)^∨(E)    = (2π) g_1^∨(E)g_2^∨(E).

Therefore the normalized self-convolution 

    (t)=g*g(t)/g_2^2,  where  g_2^2=∫_-∞^∞ g(t)^2 dt,
 has nonnegative inverse Fourier transform 

    (E)=2π/g_2^2(g^∨(E))^2≥ 0,

which is also properly normalized since ∫_-∞^∞(x) dx=(0)=1.
Additionally, (t) has similar decay rate on the real line as g:

    (t)= ∫_-∞^∞ g(y)g(t-y)  dy    ≤(∫_|y|≥ t/2 + ∫_|y-t|≥ t/2) [g(y)g(t-y)]  dy
       ≤ 2g(t/2) ∫_-∞^∞ g(y) dy,

so that
|(t)| ≤ C g(t/2),
where C=2∫_-∞^∞ g(t) dt is independent of t.
The rapid decay of g for ||</2 also ensures that () is defined, bounded, and integrable along horizontal lines in any smaller strip ||≤/2-ϵ.

The double exponential decay rate ∝exp(-e^π/2|t|) can thus be achieved by (t) analytic and bounded in a strip ||</2-ϵ for any ε>0, 
with (E)≥0 and normalized. 
Verifying the conditions in e.g. <cit.> shows the density of states (E) satisfies an exponential decay bound.
Plots demonstrating the decay of these (t) and (E) are shown in Fig. <ref>.


Surprisingly, (E) and (t) look very similar to a Gaussian distribution expected for generic many-body systems <cit.>; for cβ/π = 1 as in Fig. <ref>, the deviation of the latter from a Gaussian becomes appreciable (more than 10% of the Gaussian) only around t≈ 1.9, at which |(1.9)|^2 ≈ 0.014 ∈ (2^-7,2^-6). This suggests that the difference in quantum dynamics between the examples constructed here and generic many-body systems (in terms of the contribution from the energy eigenvalues) may become significant only for systems of N ≫ 6 qubits [so that the O(D^-1) fluctuations in |(t)|^2, which is a sum of D phase factors, are negligible compared to its value at this time].







 §.§ Upper bound on the scrambling time for these examples


The constraint |(t)|≤ Cg(t/2) leads to a double exponential decay of the regularized SFF for large t:

    K_β(t→∞) ≲ f_β^2 C^2exp[-exp(π t/2cβ)]

It is also known that for systems with energy eigenstates that are “sufficiently random” [i.e., behave as if sampled from the Haar distribution, including but not restricted to Haar random energy eigenstates in the computational basis], the expectation values of observables _α in initial states ρ̂_k(0) [Eqs. (<ref>) and (<ref>) of the main text] essentially track the SFF to decay to their (maximally) scrambled values <cit.>:

    [ρ̂_k(t) _α] = 1+O(D_E^-1/2)/D_S+{[ρ̂_k(0) _α]-1/D_S} K_β(t).

Thus, all these expectation values are guaranteed to maximally scramble after a time t_s, i.e.,

    [ρ̂_k(t)_α] ∼1/D_S,  for  t : t_s < t < T_D,

if K_β(t > t_s) = o(D_S^-1) <cit.>, e.g., K_β(t > t_s) ≤ D_S^-1-ε for any ε > 0.  Using Eq. (<ref>) with this condition gives for the scrambling time:

    t_s ≲2 c β/πln[ln(f_β^2 C^2 D_S^1+ε)].

Recalling that S_2,S∼ N_S ln 2 for maximal entanglement (for N_S < N_E), and further assuming an initial state such that |ln (C f_β)| = o(N_S), we get a leading order scrambling time of
t_s ≲ (2 cβ/π)ln S_2,S.
We conclude that the systems constructed in this section nearly saturate the logarthmic-in-entanglement-entropy bound of Eq. (<ref>) [Eq. (<ref>) in the main text], if the energy eigenbasis is sufficiently random with respect to the computational basis.



