Signaling and Scrambling with Strongly Long-Range Interactions

TL;DR

This paper establishes new Lieb-Robinson-type bounds for strongly long-range quantum systems, providing insights into the limits of information signaling and scrambling, which are crucial for understanding quantum dynamics in non-local systems.

Contribution

It introduces two novel bounds on signaling and scrambling times in long-range interacting quantum systems, advancing theoretical understanding of their dynamics.

Findings

Bounds are tight for systems mappable to free-particle Hamiltonians with long-range hopping.

Lower bounds on signaling time are established for all  systems with  < .

Results suggest a pathway to proving the fast scrambling conjecture.

Abstract

Strongly long-range interacting quantum systems---those with interactions decaying as a power-law $1/r^{\alpha}$ in the distance $r$ on a $D$-dimensional lattice for $\alpha\le D$---have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. As a first step towards rectifying this problem, we prove two new Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for $\alpha\le D/2$. Our second bound pertains to generic long-range interacting spin Hamiltonians, and leads to a tight lower bound for the signaling time to extensive subsets of the system for all $\alpha < D$. This result also lower-bounds the scrambling time, and suggests a path towards achieving a tight scrambling bound that can prove the long-standing fast scrambling conjecture.

Figures

Equations

$$H(t)=\sum _{\begin{array}{c}i,j\in \Lambda \\ i<j\end{array}}(J_{ij}(t)c_i^{†}{c}_j+\text{h.c.})+\sum _{i\in \Lambda }{B}_i(t)c_i^{†}{c}_i,$$

$$\Vert \left[c_X(t),c_Y^{†}\right]\Vert \le {\int }_0^{t}d\tau \sqrt{\sum _{i\in \Lambda }{\left|J_{iX}(\tau )\right|}^2}.$$

$$\Vert \left[c_X(t),c_Y^{†}(0)\right]\Vert \le t\times \{\begin{array}{ll}\mathcal{O}\left(1\right)& \alpha >D/2,\\ \mathcal{O}\left(N^{\frac{1}{2}-\alpha /D}\right)& 0\le \alpha \le D/2.\end{array}$$

$$f_{XY}(t)=⟨0|[c_X(t),c_Y^{†}]|0⟩=⟨0|c_X(0)U_I(t)c_Y^{†}|0⟩.$$

$$\frac{d{f}_{XY}}{dt}=-i⟨{\psi }_X|H_I(t)|{\psi }_Y(t)⟩.$$

$$\Vert \frac{d{f}_{XY}}{dt}\Vert$$

$$=\Vert H_I(t)|{\psi }_X⟩\Vert =\sqrt{\sum _{i\in \Lambda }{\left|{\tilde{J}}_{iX}(t)\right|}^2}.$$

$$\left|{⟨0|}_X{⟨0|}^{\otimes N-2}{⟨\psi |}_{Y}U(T){|\psi ⟩}_X{|0⟩}^{\otimes N-2}{|0⟩}_Y\right|=1.$$

$$H(t)=\{\begin{array}{ll}{H}_1=\frac{1}{L^{\alpha }}{\sum }_{i=1}^{N-2}{c}_X^{†}{c}_i+\text{h.c.}& 0<t<\frac{T}{2},\\ H_2=\frac{1}{L^{\alpha }}{\sum }_{i=1}^{N-2}{c}_i^{†}{c}_Y+\text{h.c.}& \frac{T}{2}<t<T,\end{array}$$

$$e^{-i{H}_{1}T/2}|\Psi ⟩=a{|0⟩}^{\otimes N}+b{|0⟩}_X|W⟩{|0⟩}_Y.$$

$$e^{-i{H}_{2}T/2}{e}^{-i{H}_{1}T/2}|\Psi ⟩={|0⟩}_X{|0⟩}^{\otimes N-2}(a{|0⟩}_Y+b{|1⟩}_Y).$$

$$\text{expectationvalue}[c_X^{†}(T),c_Y]\Psi =\frac{1}{2}{\int }_0^{T}d\tau \sqrt{\sum _{i\in \Lambda }{\left|J_{iX}(\tau )\right|}^2}.$$

$$\Vert [A(t),B]\Vert$$

$${\mathcal{J}}^k(X,Y)$$

$$\Vert [A(t),B]\Vert \le 2\Vert A\Vert \Vert B\Vert |X||Y|\left(\frac{e^{\Theta (N^{1-\alpha /D})t}-1}{\Theta (N^{1-\alpha /D})r_{XY}^{\alpha }}\right).$$

$$t_{\text{si}}=\Omega \left(\frac{\log (N)}{N^{1-\alpha /D}}\right).$$

$$\Vert [A(t),B]\Vert \le 2\Vert A\Vert \Vert B\Vert \sum _{i\in X,j\in Y}\frac{e^{\Theta (N^{1-\alpha /D})}t-1}{\Theta (N^{1-\alpha /D})r_{ij}^{\alpha }}.$$

$$t_{\text{sc}}\ge t_{\text{ms}}=\Omega \left(\frac{1}{N^{1-\alpha /D}}\right).$$

$$⟨\psi |[A(t),B]|\psi ⟩=\sin (2t\sum _{j\ne i}{J}_{ij}).$$

Full text

In non-relativistic quantum mechanics, Lieb-Robinson bounds provide a notion of causality Lieb and Robinson (1972), limiting the speed of information propagation (or signaling) to a finite value in lattice systems with short-range interactions. This bounded signaling speed has strong implications for quantum information and condensed matter physics, leading to entanglement area laws Hastings (2007) and the existence of topological order Bravyi et al. (2010). However, it remains an open question whether the signaling speed must be finite if interactions are long-ranged and decay as an inverse power-law $1/r^{\alpha}$ in the inter-particle distance $r$. Such power-law interacting systems arise in experimental platforms for quantum computation and quantum simulation, including Rydberg atoms  Saffman et al. (2010), trapped ions Britton et al. (2012), polar molecules Yan et al. (2013a), defect centers in solids Yao et al. (2012), and atoms trapped along photonic crystals Douglas et al. (2015). The lack of a bounded signaling speed in these systems makes it challenging to understand and predict their dynamics.

For power-law interacting systems with $\alpha$ greater than the lattice dimension $D$, a finite signaling speed has been shown to persist to some intermediate distance and time Gong et al. (2014). At long distances (or times), recent developments show that the signaling speed will diverge at most polynomially in time for $\alpha>2D$ Foss-Feig et al. (2015); Tran et al. (2018), ruling out the exponential divergence suggested by earlier results Hastings and Koma (2006). For $\alpha\leq 2D$, which is the case for most experimental long-range interacting systems, an exponentially growing signaling speed has yet to be ruled out, making the fate of causality far from settled.

In this work, we focus on the regime of strongly long-range interacting systems, where interaction energy per site diverges, thus implying $\alpha\leq D$ Storch et al. (2015). Note that even if one normalizes the interaction strength to make energy extensive (i.e. proportional to the number of lattice sites), these systems are still fundamentally different from those with $\alpha>D$, as energy is in general no longer additive for subsystems Dauxois et al. (2002). To avoid confusion, we will not perform any normalization of interaction strength throughout this paper, as such normalization can always be performed later by a rescaling of time.

Apart from their existence in experimental platforms Islam et al. (2013); Britton et al. (2012); Blatt and Roos (2012); Yan et al. (2013b); Choi et al. (2017), strongly long-range interacting systems have received theoretical interest due to their close relation to fast quantum information scramblers Sachdev and Ye (1993); Gärttner et al. (2017); Kitaev (2015); Maldacena et al. (2016). Two fundamental questions about these systems are (1) what is the shortest time $t_{\text{si}}$ needed to send a signal from one site to a site located an extensive distance away, and (2) what is the shortest time $t_{\text{sc}}$ needed to scramble the information stored in the system 111We will define the quantities $t_{\text{si}}$ and $t_{\text{sc}}$ rigorously later in this Letter?

There have been a number of attempts to answer the above two questions, with limited success. For the first question, Refs. Eisert et al. (2013); Hauke and Tagliacozzo (2013); Eldredge et al. (2017) show that in certain strongly long-range interacting systems with $\alpha\leq D$, information and correlations can spread across the entire system in a finite time, meaning that $t_{\text{si}}$ has no dependence on the number of sites $N$. The Lieb-Robinson-type bound derived in Ref. Storch et al. (2015) does not rule out the possibility of $t_{\text{si}}$ scaling as $\log(N)\,N^{2\alpha/D}/N^{2}$ for $\alpha<D$, which vanishes in the $N\rightarrow\infty$ limit 222While the original bound in Ref. Storch et al. (2015) holds for $n$-body interactions, here we cite the bound on the interaction time as it applies to the specific case of two-body interactions.. However, no protocol that we know of comes close to achieving such fast signaling. For the second question, Ref. Lashkari et al. (2013) shows that the scrambling time can be lower-bounded by $t_{\text{sc}}\gtrsim 1/N$ for $\alpha=0$, but the fastest known scramblers use a time $t_{\text{sc}}\propto\log(N)/N$ to scramble the system Sekino and Susskind (2008).

While the ultimate answers to these two questions remain to be found, we present several advances in this Letter. First, we prove a new bound for systems that can be mapped to free bosons or fermions with $1/r^{\alpha}$ hopping strength, which leads to $t_{\text{si}}\gtrsim N^{\alpha/D}/\sqrt{N}$. Notably, this bound is tight for $\alpha\leq D/2$, as it can be saturated by a new quantum state transfer protocol. Second, we prove that $t_{\text{si}}\gtrsim\log(N)N^{\alpha/D}/N$ for general interacting systems, which improves significantly over the previous bound Storch et al. (2015). Building upon this second result, we also prove a tight bound for many-site signaling (from one site to an extensive part of the system). This many-site signaling bound leads to a scrambling time bound of $t_{\text{sc}}\gtrsim N^{\alpha/D}/N$, which generalizes the result in Ref. Lashkari et al. (2013) to all $\alpha<D$. It also reveals a potential way to obtain a tight scrambling bound that can prove the well-known fast scrambling conjecture Sekino and Susskind (2008), which remains open for general lattice Hamiltonians, despite recent progress Bentsen et al. (2019); Chen and Lucas (2019).

Tight bound for free particles.—We first prove a Lieb-Robinson-type bound for non-interacting bosons/fermions on a lattice. Consider the following free-particle Hamiltonian $H(t)$ defined on a $D$-dimensional lattice $\Lambda$ with $N$ sites:

[TABLE]

where $c_{i}^{\dagger}$ ($c_{i}$) represents the creation (annihilation) operator. The hopping strength $J_{ij}(t)$ and chemical potential $B_{i}(t)$ can depend on time and we do not impose any constraint on them for now. We denote an operator $A$ at time $t$ in the Heisenberg picture as $A(t)=U^{\dagger}(t)AU(t)$, where $U(t)\equiv\mathcal{T}e^{-\frac{i}{\hbar}\int_{0}^{t}H(t^{\prime})\,dt^{\prime}}$ is the time evolution operator ($\hbar=1$). The operator norm of $A$ will be denoted by $\|A\|$.

[TABLE]

We use $[\cdot,\cdot]$ to denote the commutator for bosons and anti-commutator for fermions.

Roughly speaking, the quantity $\|[c_{X}(t),c_{Y}^{\dagger}]\|$ measures the overlap between the support of the operator $c_{X}(t)$ (which expands from site $X$ due to hopping terms) and the site $Y$. As a result, it also quantifies the amount of information that can be sent between $X$ and $Y$ in a given time $t$. Indeed, we define the signaling time $t_{\text{si}}$ as the minimal time required to achieve $\|[c_{X}(t),c_{Y}^{\dagger}]\|>\delta$ for some positive constant $\delta$. Note that we do not expect the chemical potential strength $B_{i}(t)$ to show up in the bound, as on-site Hamiltonian terms do not change the support of $c_{X}(t)$.

If the hopping terms in the Hamiltonian are short-ranged (e.g., nearest-neighbor), one might expect $\|[c_{X}(t),c_{Y}^{\dagger}]\|$ to decay exponentially in the distance $r_{XY}$ between $X$ and $Y$, due to the strong notion of causality that follows from the Lieb-Robinson bound Lieb and Robinson (1972). Additionally, if the hopping strength decays as a power law ($|J_{ij}(t)|\leq 1/r^{\alpha}$) with $\alpha>D$, intuition would suggest that $\|[c_{X}(t),c_{Y}^{\dagger}]\|$ decays algebraically in $r_{XY}$ Hastings and Koma (2006); Gong et al. (2014), indicating a weak notion of causality. However, the right-hand side of Eq. 2 has no dependence on $r_{XY}$. This is because the bound is tailored to strongly long-range hoppings with $\alpha<D$, which makes it loose for shorter-ranged long-range hoppings.

Assuming that $|J_{ij}(t)|\leq 1/r^{\alpha}$, we can simplify Eq. 2 to

[TABLE]

where $N$ is the number of lattice sites and $\mathcal{O}$ is the asymptotic “big-$\mathcal{O}$” notation big . Therefore, for $\alpha\leq D/2$, it takes a time $t_{\text{si}}=\Omega\left(N^{\alpha/D}/\sqrt{N}\right)$ big to signal from site $X$ to site $Y$, independent of the distance between $X$ and $Y$.

In the next section, we show that for $\alpha\leq D/2$, the bound in Eq. 2 can be saturated (up to a factor of 2) by engineered free-particle Hamiltonians. This leads to the conclusion that causality can completely vanish—in the sense that signals can be sent arbitrarily fast given large enough $N$—for a strongly long-range hopping system with $\alpha<D/2$. It remains an open question whether such a statement can be generalized to systems with $D/2\leq\alpha<D$ for either free or interacting particles.

Proof of Theorem 1.—Let us first go into the interaction picture of $\sum_{i}B_{i}(t)c_{i}^{\dagger}c_{i}$ to eliminate the on-site terms from the Hamiltonian in Eq. 1 333 This imparts a time-dependent phase $e^{i\phi_{jk}(t)}$ onto the hopping term $J_{jk}(t)$ for some $\phi_{jk}(t)\in[0,2\pi)$ and $j\neq k$. Since this phase does not change the value of $|J_{jk}(t)|$, it does not affect the bound.. We now have a pure hopping Hamiltonian $H_{I}(t)=\sum_{ij}\tilde{J}_{ij}(t)c_{i}^{\dagger}c_{j}$ with $|\tilde{J}_{ij}(t)|\equiv|J_{ij}(t)|$. Because $H_{I}(t)$ is a quadratic Hamiltonian, $c_{X}(t)$ is a time-dependent linear combination of annihilation operators on every site, and we can write $[c_{X}(t),c^{\dagger}_{Y}]\equiv f_{XY}(t)\mathbbm{1}$, where $f_{XY}(t)$ is a number and $\mathbbm{1}$ represents the identity operator. Given that $U_{I}(t)\ket{0}=\ket{0}$, where $U_{I}(t)$ is the time-evolution operator corresponding to $H_{I}(t)$, and $c_{X}(t)\ket{0}=0$, we have

[TABLE]

For convenience, we define the (normalized) states $\ket{\psi_{X}}\equiv c^{\dagger}_{X}(0)\ket{0}$ and $\ket{\psi_{Y}(t)}\equiv U_{I}(t)c_{Y}^{\dagger}\ket{0}$. Taking the time derivative of Eq. 4 gives

[TABLE]

By the Cauchy-Schwarz inequality,

[TABLE]

The last equality follows from $\ket{\psi_{X}}$ being a single excitation localized on site $X$ and $H_{I}(t)$ consisting only of hopping terms $\tilde{J}_{ij}(t)c_{i}^{\dagger}c_{j}$. Applying the fundamental theorem of calculus yields the bound on $f_{XY}(t)$ and hence Theorem 1.

Saturating the free-particle bound.—We now show that the bound in Theorem 1 can be saturated by engineered Hamiltonians that can also be used to perform fast quantum state transfer. In particular, the protocol presented here has a state transfer time of $T=\mathcal{O}\left(N^{\alpha/D}/\sqrt{N}\right)$, which—for $\alpha\leq D/2$—improves over the fastest-known state transfer protocol using long-range interactions Eldredge et al. (2017).

Our setup for the state transfer task is depicted in Fig. 1. We initialize a lattice with $N$ sites in a tensor product of unoccupied states $\ket{0}$ and some unknown normalized bosonic/fermionic state $\ket{\psi}=a\ket{0}+b\ket{1}$ on a single site $X$. The goal of state transfer is to move $\ket{\psi}$ to the target site $Y$ after the system time-evolves by a $\ket{\psi}$-independent (but possibly time-dependent) Hamiltonian $H(t)$ Nikolopoulos and Jex (2014); Epstein and Whaley (2017).

The unitary time-evolution operator $U(T)$ can be said to implement state transfer in time $T$ if it satisfies the following condition:

[TABLE]

We refer to the left-hand side of Eq. 8 as the fidelity of the state transfer, which can be bounded directly by a Lieb-Robinson-type bound on $H(t)$ such as Eq. 2 Epstein and Whaley (2017).

We label the sites that are not $X$ or $Y$ by $1,\dots,N-2$ and denote the furthest distance between any pair of sites by $L=\mathcal{O}\left(N^{1/D}\right)$. Our state transfer protocol is given by the following piece-wise time-independent Hamiltonian:

[TABLE]

where $T=\pi L^{\alpha}/\sqrt{N-2}$ is the total time for the protocol. Note that while $H(t)$ satisfies the constraint $|J_{ij}(t)|\leq 1/r_{ij}^{\alpha}$ assumed in Eq. 3, the corresponding $J_{ij}(t)$ terms do not actually vary with the distances between sites.

Evolving the initial state $\ket{\Psi}\equiv\ket{\psi}_{X}\ket{0}^{\otimes N-2}\ket{0}_{Y}$ by $H_{1}$ for time $T/2$ yields the intermediate state

[TABLE]

Here, $\ket{W}=\frac{1}{\sqrt{N-2}}\sum_{i=1}^{N-2}c_{i}^{\dagger}\ket{0}^{\otimes N-2}$ is the W state over the $N-2$ remaining sites. Further evolving the state by $H_{2}$ for time $T/2$ yields the final state:

[TABLE]

Thus we have achieved perfect quantum state transfer in time $T=\mathcal{O}\left(N^{\alpha/D}/\sqrt{N}\right)$. Note that the distance between $X$ and $Y$ on the lattice does not appear in the state transfer time. Setting $b=1$ in the above protocol leads to

[TABLE]

Thus, the bound in Eq. 2 is saturated up to a factor of 2.

It should be pointed out that, for $\alpha>D/2$, the above protocol requires a time that increases with $N$, which is slower than for the previous result in Ref. Eldredge et al. (2017). While that protocol has a state transfer time that is constant for $\alpha\leq D$, it uses an engineered Hamiltonian with interactions, and therefore cannot be applied to systems of non-interacting particles. In general, allowing interactions may increase the rate of information propagation, and proving a Lieb-Robinson-type bound in these situations requires a different approach.

Improved bound for general interacting systems.—We now derive bounds on the signaling time that extend beyond free-particle Hamiltonians. Without loss of generality, we study a generic interacting spin Hamiltonian $H(t)=\sum_{i<j}h_{ij}(t)$ where $\|h_{ij}(t)\|\leq 1/r_{ij}^{\alpha}$ and on-site interactions have been eliminated by going into an interaction picture. We will bound the quantity $\|[A(t),B]\|$, where $A$ and $B$ are arbitrary operators supported on sets of sites $X$ and $Y$ respectively, using the following Lieb-Robinson series Hastings and Koma (2006):

[TABLE]

Here, $|X|$ stands for the number of sites $X$ acts on. Each term in Eq. 14 represents a sequence of $k$ directed hops in the lattice that originates at site $X$ and ends at site $Y$. For distinct sites $i$ and $j$, $J_{ij}=1/r_{ij}^{\alpha}$ represents a directed hop from $i$ to $j$. We set $J_{ii}=\sum_{j\neq i}J_{ij}$ 444 The strength of the on-site hop $J_{ii}$ is defined this way for technical reasons that we explain in the Supplemental Material..

Since $J_{ij}$ decays slowly in $r_{ij}$ for $\alpha<D$, our improved bound on $\|[A(t),B]\|$ requires bounding each term in Eq. 14 using new techniques beyond those in previous efforts Hastings and Koma (2006); Storch et al. (2015) and detailed in the Supplemental Material SM . The result (assuming $\alpha<D$) is:

[TABLE]

The factor $\Theta(N^{1-\alpha/D})$ big comes from the total interaction energy per site given by $J_{ii}$.

We consider now the case of signaling between subsystems $X$ and $Y$ of a system $\Lambda$ with $|X|,|Y|=\mathcal{O}\left(1\right)$. We formally define $t_{\text{si}}$—the signaling time from $X$ to $Y$—as the smallest time $t$ such that there exist unit-norm operators $A$ and $B$ supported on $X$ and $Y$ respectively and a constant $\delta=\Theta(1)$ such that $\|[A(t),B]\|>\delta$ Lashkari et al. (2013). If we further assume that $X$ and $Y$ are separated by an extensive distance $r_{XY}=\Omega\left(N^{1/D}\right)$, then the following lower bound holds for the signaling time:

[TABLE]

While we do not know of any examples that saturate this bound, it is the tightest-known signaling time bound for strongly long-range interacting systems. Indeed, the bound is close to being saturated in the limit of $\alpha\rightarrow D^{-}$, as the state transfer protocol in Ref. Eldredge et al. (2017) shows that $t_{\text{si}}=\mathcal{O}\left(\log(N)\right)$ can be achieved at $\alpha=D$. Unfortunately, generalizing our bound in Eq. 15 to the case of $\alpha=D$ leads to $t_{\text{si}}=\Omega\left(1\right)$ SM , which is not saturated by Ref. Eldredge et al. (2017).

Many-site signaling and scrambling bounds.—Of recent interest in the fields of theoretical high-energy and condensed matter physics is the phenomenon of quantum information scrambling Lashkari et al. (2013); Roberts and Swingle (2016); Sekino and Susskind (2008); Hayden and Preskill (2007); Xu and Swingle (2018). Previous work on scrambling in power-law interacting systems has focused primarily on numerical analysis Pappalardi et al. (2018); Chen and Zhou (2018), whereas general mathematical results are lacking. Only in all-to-all interacting systems (which can be treated as the limit $\alpha=0$) have Lieb-Robinson-type bounds been used to bound the scrambling time Lashkari et al. (2013). Using the bound derived in Eq. 15, we can prove a scrambling time bound for systems with $0<\alpha<D$, a regime for which no better result is known.

To derive a bound on the scrambling time, we first derive a bound on the many-site signaling time. We define the many-site signaling time $t_{\text{ms}}$ to be the smallest $t$ required to signal from $X$ to a $Y$ that has extensive size. Lieb-Robinson-type bounds such as Eq. 15 naturally limit the time for many-site signaling. However, a direct application of Eq. 15 to many-site signaling leads to a loose bound. Instead, a more refined technique that sums over all sites within the subsets $X$ and $Y$ yields a tighter bound Gong et al. (2017):

[TABLE]

This bound reduces to Eq. 15 when $|X|,|Y|=1$.

A system on a lattice $\Lambda$ is said to scramble in a time $t_{\text{sc}}$ if any information initially contained in a finite-sized subsystem $S\subset\Lambda$ is no longer recoverable from measurements on $S$ alone tsc . That information is not lost, however, but can be recovered from the complement $\bar{S}=\Lambda\setminus S$ of $S$ Hayden and Preskill (2007); Cotler et al. (2018). As a result, scrambling implies the ability to signal from $S$ to $\bar{S}$ Lashkari et al. (2013). Thus, $t_{\text{sc}}$ is lower bounded by the time it takes to signal from a subset $S$ with size $\Theta(1)$ to its complement with size $\Theta(N)$, which corresponds to the many-site signaling time.

Using Eq. 17, we obtain the following scrambling time bound for $0\leq\alpha<D$:

[TABLE]

Note that this bound differs from Eq. 16 by a $\log(N)$ factor. Additionally, although the bound on $t_{\text{si}}$ in Eq. 16 may allow further tightening, the bound on $t_{\text{ms}}$ in Eq. 18 cannot be generically improved for $0\leq\alpha<D$. To see this, we consider a long-range Ising Hamiltonian $H=\sum_{i\neq j}J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}$, with $J_{ij}=1/r_{ij}^{\alpha}$. For simplicity, we consider the subset $S$ to be a single site indexed by $i$ and construct operators $A=\sigma_{i}^{+}$ and $B=\bigotimes_{j\neq i}\sigma_{j}^{+}$ that are supported on $S$ and $\bar{S}$ respectively. We can analytically calculate the expectation value of $[A(t),B]$ in an initial state $\ket{\psi}=\frac{1}{\sqrt{2}}[\ket{0}^{\otimes N}+\ket{1}^{\otimes N}]$ Gong et al. (2017):

[TABLE]

Using $J_{ij}=1/r_{ij}^{\alpha}$, we find that the signaling time of this protocol is $t=\mathcal{O}\left(N^{\alpha/D-1}\right)$ for $0\leq\alpha<D$, which saturates the many-site signaling time bound in Eq. 18 555A different protocol for many-site signaling was given in Eisert et al. (2013). That result yields a many-site signaling time of $t_{\text{ms}}=\mathcal{O}\left(N^{\alpha/D-1/2}\right)$, which does not saturate the bound in Eq. 18.. This does not, however, imply that the corresponding scrambling time bound is tight. In fact, the fastest known scramblers in long-range interacting systems with $\alpha=0$ have a scrambling time $t_{\text{sc}}=\mathcal{O}\left(\log(N)/N\right)$ Lashkari et al. (2013), which is a factor of $\log(N)$ away from saturating the bound. This suggests that future improvements on the scrambling time bound may be possible.

Conclusions and Outlook.—In this Letter, we make several advances in bounding the signaling and scrambling times in strongly long-range interacting systems. Our results suggest a number of possible future directions. One is to find the optimal signaling time bound for general strongly long-range interacting systems. Previously, this has been an outstanding challenge; we now know of a free-particle bound that is tight for $\alpha\in[0,D/2]$ and a general bound that is nearly tight as $\alpha\rightarrow D^{-}$. The search for the optimal bound for $\alpha\in[0,D]$ has thus been narrowed down significantly.

Additionally, our bound for signaling to an extensive numbers of sites hints at a strategy for achieving a better scrambling bound. In particular, the protocol that saturates our many-site signaling bound relies on an initial entangled state, whereas the definition of scrambling assumes that the system begins in a product state tsc . It may be possible to improve the scrambling time bound by explicitly restricting our attention, when bounding $t_{\text{ms}}$, to initial product states. Obtaining an extra $\log(N)$ factor in $t_{\text{sc}}$ by such restriction would prove the main part of the fast-scrambling conjecture in Ref. Sekino and Susskind (2008).

Finally, we expect that the improved Lieb-Robinson-type bounds derived in this work may lead to a better understanding of the spreading of correlations Luitz and Bar Lev (2019), the growth of entanglement entropy Gong et al. (2017), and thermalization timescales Nandkishore and Huse (2015) in strongly long-range interacting systems.

Acknowledgments.— We thank B. Swingle, M. Foss-Feig, J. Garrison, Z. Eldredge, and A. Deshpande for helpful discussions. MCT, AYG, and AVG acknowledge funding by the DoE ASCR Quantum Testbed Pathfinder program, DoE BES Materials and Chemical Sciences Research for Quantum Information Science program, NSF Ideas Lab on Quantum Computing, ARO MURI, AFOSR, ARL CDQI, and NSF PFC at JQI. This work was supported in part by the Army Research Office (MURI award number W911NF-16-1-0349), the Canadian Institute for Advanced Research, the National Science Foundation (under Grant Nos. CCF-1813814 and PHY-1748958), the Heising-Simons Foundation, and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams and Quantum Testbed Pathfinder (award number DE-SC0019040) programs. ZXG acknowledges funding from NSF RAISE-TAQS program. AYG is supported by the NSF Graduate Research Fellowship Program under Grant No. DGE-1322106.