# Optimal form of light cones for bosonic transport in long-range systems

Tan Van Vu<EMAIL_ADDRESS>Analytical quantum complexity RIKEN Hakubi
Research Team, RIKEN Center for Quantum Computing (RQC), 2-1 Hirosawa, Wako,
Saitama 351-0198, Japan  Tomotaka Kuwahara<EMAIL_ADDRESS>Analytical quantum complexity RIKEN Hakubi Research Team, RIKEN Center for
Quantum Computing (RQC), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan  PRESTO,
Japan Science and Technology (JST), Kawaguchi, Saitama 332-0012, Japan  Keiji
Saito<EMAIL_ADDRESS>Department of Physics, Kyoto
University, Kyoto 606-8502, Japan

###### Abstract

Understanding the ultimate rate at which information propagates is a pivotal
issue in nonequilibrium physics. Nevertheless, the task of elucidating the
propagation speed inherent in quantum bosonic systems presents challenges due
to the unbounded nature of their interactions. In this Letter, we tackle the
problem of particle transport in long-range bosonic systems through the lens
of both quantum speed limits and the Lieb-Robinson bound. Employing a unified
approach based on optimal transport theory, we rigorously prove that the
minimum time required for macroscopic particle transport is always bounded by
the distance between the source and target regions, while retaining its
significance even in the thermodynamic limit. Furthermore, we derive an upper
bound for the probability of observing a specific number of bosons inside the
target region, thereby providing additional insights into the dynamics of
particle transport. Our results hold true for arbitrary initial states under
both long-range hopping and long-range interactions, thus resolving an open
problem of particle transport in generic bosonic systems.

Introduction.—The investigation of the velocity at which information
propagates is a central topic in quantum mechanics. In relativistic theory, a
fundamental constraint, known as information causality, rigorously prohibits
information transfer outside the light cone. Remarkably, Lieb and Robinson [1]
discovered similar restrictions in nonrelativistic quantum theory.
Specifically, they demonstrated the existence of an effective light cone,
outside of which information propagation undergoes an exponential decay with
distance. The Lieb-Robinson bound, as commonly referred to, has proven to be a
powerful tool for the analysis of quantum many-body systems, with diverse
applications across various fields [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14].

For short-range interacting spin systems, the Lieb-Robinson bound is
characterized by a finite velocity, implying that the information propagation
is effectively limited within the linear light cone that grows with a finite
speed. In scenarios involving long-range interactions, however, the existence
of linear light cones becomes significantly intricate, as such interactions
have the capability to instantaneously transmit information to distant places
[15]. Here, long-range interaction means that the interaction strength between
distinct sites obeys a power-law decay as $1/d^{\alpha}$ with distance $d$.
Intuitively, the shape of the light cone is dependent on the exponent $\alpha$
and may no longer be linear. Given the ubiquity of long-range systems in
nature [16] and their fundamental interest, it is of great importance to
elucidate the shape of the light cone with respect to the power decay
$\alpha$. Thus far, a comprehensive characterization of the shape of the light
cone has been established for long-range interacting quantum spin and
fermionic systems [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In Ref. [20], the
condition for the linear light cone was proven to be $\alpha>2D+1$, where $D$
denotes the spatial dimension. Furthermore, for the entire regime of $\alpha$,
the optimal form of the light cone has been identified as $\tau\gtrsim
d^{\min(1,\alpha-2D)}$ [24].

Figure 1: (a) Schematic of long-range bosonic systems, wherein bosons are
transported from region $X$ to a separate region $Y$ with a distance of
$d_{XY}$. (b) Diagram of our first main result on the minimum operational time
$\tau$ for particle transport. The gray area denotes the prohibited region of
the exponent.

As another important class, quantum bosonic systems (e.g., the paradigmatic
Bose-Hubbard model [27]) hold significant importance in the study of cold
atoms in experimental setups. Nevertheless, the theoretical investigation for
bosonic systems has persisted due to the unbounded nature of interactions in
such systems. Classifying the effective light cone within the Bose-Hubbard
model has been substantiated through both experimental observations and
numerical simulations using ultracold gases confined in optical lattices [28,
29, 30, 31]. While the speeds of bosonic transport and information propagation
have been studied for bosonic systems with short-range hoppings [32, 33, 34,
35, 36, 37], the exploration of long-range cases still remains in an early
stage of development. With regard to macroscopic bosonic transport, Faupin and
coworkers [38] have recently demonstrated the finite velocity of macroscopic
particle transport by showing the existence of the linear light cone for
$\alpha\geq D+3$. However, we are still far from the complete classification
of the effective light cone for particle transport. In particular, it is a
critical problem to identify an optimal form of the light cone in bosonic
systems, as has been studied for quantum spin and fermion systems.

In this Letter, we resolve the above problem for macroscopic bosonic transport
in generic systems with long-range hopping and long-range interactions,
spanning the entire regime of $\alpha>D$. By developing a unified approach
based on optimal transport theory, we demonstrate the finite speed of
macroscopic bosonic transport through the lens of quantum speed limits [39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58].
Specifically, we rigorously prove that the minimum time required for the
transport of a macroscopic number of bosons is always bounded by the distance
$d$ as $\tau\gtrsim d^{\min(1,\alpha-D)}$ [cf. Eq. (5); see Fig. 1 for
illustration]. The exponent in this bound is optimal, as it can be achieved
through a constructive state-transfer protocol. Additionally, we derive a
limit for the probability of achieving a specific number of boson transport
into a target region [cf. Eq. (6)]. We emphasize that all our results
unconditionally hold for arbitrary initial states including pure and mixed
states. Notably, our proofs are significantly simple, necessitating only a few
fundamental properties of the Wasserstein distance [59]. Our findings
establish the complete classification for bosonic transport with long-range
interactions.

Setup.—We consider a generic model of bosons on an arbitrary $D$-dimensional
lattice, wherein bosons can hop between arbitrary sites and interact with each
other. The system Hamiltonian is time dependent (i.e., an external control
protocol can be applied to the system) and can be expressed in the following
form:

$H_{t}\coloneqq-\sum_{i\neq
j\in\Lambda}J_{ij}(t)\frac{}{}\hat{b}_{i}^{\dagger}\hat{b}_{j}+\sum_{Z\subseteq\Lambda}h_{Z}(t).$
(1)

Here, $\Lambda$ denotes the set of all the sites in the lattice,
$\hat{b}_{i}^{\dagger}$ and $\hat{b}_{i}$ are the bosonic creation and
annihilation operators for site $i$, respectively, and $h_{Z}(t)$ is an
arbitrary function of $\\{\hat{n}_{i}\\}_{i\in Z}$, where
$\hat{n}_{i}\coloneqq\hat{b}_{i}^{\dagger}\hat{b}_{i}$ is the number operator.
Note that $h_{Z}(t)$ does not need to be local (i.e., $Z$ can be arbitrarily
large). In other words, both long-range hopping and long-range interactions
are allowed in our setup. The hopping terms $J_{ij}(t)$ are upper bounded by a
power law in the Euclidean distance, i.e., $|J_{ij}(t)|\leq
J/\|i-j\|^{\alpha}$, where $\|\cdot\|$ denotes the Euclidean norm and the
power decay $\alpha$ satisfies $\alpha>D$. Examples include the standard Bose-
Hubbard model, specified by $J_{ij}(t)=0$ for any non-neighboring sites $i$
and $j$ and
$\sum_{Z\subseteq\Lambda}h_{Z}(t)=(U/2)\sum_{i}\hat{n}_{i}(\hat{n}_{i}-1)-\mu\sum_{i}\hat{n}_{i}$,
where $U$ and $\mu$ are real constants. Let $\varrho_{t}$ be the system’s
density matrix at time $t$. Then, its time evolution is described by the von
Neumann equation ($\hbar=1$ for simplicity):

$\dot{\varrho}_{t}=-i[H_{t},\varrho_{t}].$ (2)

It can be easily shown that the total number of particles in this system is
conserved, denoted hereafter as $\mathcal{N}$.

We introduce some notations that will be used in the Letter. For an arbitrary
set $X$ of lattice sites, $X_{c}\coloneqq\Lambda\setminus X$ is the set of
sites that are not present in $X$, and $|X|$ denotes its cardinality. The
distance between any two disjoint sets $X$ and $Y$ is defined as the minimum
distance between sites belonging to each of them:

$d_{XY}\coloneqq\min_{i\in X,\,j\in Y}\|i-j\|.$ (3)

The number operator of local bosons occupied inside region $X$ is given by
$\hat{n}_{X}\coloneqq\sum_{i\in X}\hat{n}_{i}$, and the corresponding
concentration operator is $\overline{n}_{X}\coloneqq\hat{n}_{X}/\mathcal{N}$.
Let $\Pi_{\vec{N}}=\outerproduct{\vec{N}}{\vec{N}}$ be the projection onto the
Mott state $\ket{\vec{N}}$ (i.e.,
$\hat{n}_{i}\ket{\vec{N}}=n_{i}\ket{\vec{N}}$ for
$\vec{N}=[n_{i}]_{i\in\Lambda}$). Then, the projection $P_{\hat{n}_{X}\leq
N_{0}}$ is defined as

$P_{\hat{n}_{X}\leq
N_{0}}\coloneqq\sum_{\vec{N}:\matrixelement{\vec{N}}{\hat{n}_{X}}{\vec{N}}\leq
N_{0}}\Pi_{\vec{N}}.$ (4)

The expectation of an observable $A$ with respect to state $\varrho$ is
denoted as $\expectationvalue{A}_{\varrho}\coloneqq\tr{A\varrho}$.

Main results.—Given the aforementioned setup, we are now ready to explain the
results, whose proofs are postponed to the end of the Letter. Our first main
result is the following statement, which describes a fundamental limit on the
operational time required for a macroscopic transport of bosons in long-range
systems.

######  Theorem 1.

Consider a situation where a fraction $\mu\in(0,1]$ of all bosons is
transported from region $X$ to a distant region $Y$. Then, the operational
time $\tau$ required for this macroscopic bosonic transport is lower bounded
by the distance between the two regions as

$\tau\geq\kappa_{1}^{\varepsilon}d_{XY}^{\min(1,\alpha-D-\varepsilon)}.$ (5)

Here, $0<\varepsilon<\alpha-D$ is an arbitrary number and
$\kappa_{1}^{\varepsilon}>0$ is a system-independent constant.

Some remarks on Theorem 1 are in order. (i) First, macroscopic transport here
means that an average number $\mu\mathcal{N}$ of bosons is transported from
$X$ to $Y$, where $\mu$ is a $O(1)$ constant. The result holds for arbitrary
initial states, including both pure and mixed states, and for the entire range
of the power decay $\alpha>D$. Notably, this type of time constraint cannot be
deduced by the standard quantum speed limits [39, 41].

(ii) Second, bound (5) can be physically interpreted as follows. For the case
$\alpha>D+1$, the bound reads $\tau\geq O(d_{XY})$, implying that bosonic
transport always takes time at least proportional to the distance.
Surprisingly, this implication is the same as in the case of short-range
hopping [56], indicating that long-range hopping and long-range interactions
do not enhance the speed of macroscopic bosonic transport [60]. On the other
hand, in the case $D<\alpha\leq D+1$, the minimum operational time is bounded
in terms of the power decay $\alpha$ as $\tau\geq O(d_{XY}^{\alpha-D})$,
indicating the effect of long-range hopping on speeding up bosonic transport.

(iii) Last, we discuss the optimality of the bound (5) and show that it is
optimal. The optimality here is interpreted in the sense that the power
coefficient associated with the distance term in the bound is optimal. In
other words, we need only show that there exist transport protocols such that
bosonic transport can be accomplished within time
$\tau=O(d_{XY}^{\min(1,\alpha-D)})$. To this end, it is sufficient to consider
the case of a single particle (i.e., $\mathcal{N}=1$). In this case, it was
shown that one could construct a rapid state-transfer protocol such that the
particle can be transferred between two sites of distance $d$ within time
$\tau=O(d)$ for $\alpha>D+1$ and $\tau=O(d^{\alpha-D})$ for $D<\alpha\leq D+1$
[61] (see Theorem 11 therein). Therefore, it can be concluded that the bound
(5) is optimal.

Theorem 1 concerns the minimum time required to transport an average number of
bosons between two regions. In the sequel, we focus on the probability of
finding a number of bosons occupied inside the target region within a finite
time. Our second main result, which is stated below, establishes an upper
bound on this probability in terms of the operational time and the distance.

######  Theorem 2.

Assume that the initial boson number outside of region $X$ does not exceed
$N_{0}$, i.e., $\expectationvalue{P_{\hat{n}_{X_{c}}\leq
N_{0}}}_{\varrho_{0}}=1$. Then, the probability of finding bosons inside a
disjoint region $Y$ is upper bounded by the operational time $\tau$ and the
distance between the two regions as

$\expectationvalue{P_{\hat{n}_{Y}\geq N_{0}+\Delta
N_{0}}}_{\varrho_{\tau}}\leq\frac{\kappa_{2}^{\varepsilon}\mathcal{N}}{\Delta
N_{0}}\frac{\tau}{d_{XY}^{\min(1,\alpha-D-\varepsilon)}}.$ (6)

Here, $0<\varepsilon<\alpha-D$ is an arbitrary number and
$\kappa_{2}^{\varepsilon}>0$ is a system-independent constant.

This result holds under a generic setting for both initial states and system
dynamics, same as in Theorem 1. It can be interpreted as follows. Initially,
there are at most $N_{0}$ bosons outside of $X$; therefore, in order to
observe more than $N_{0}$ bosons inside $Y$, a certain number of bosons must
be transferred from $X$ to $Y$. Bound (6) indicates that the probability of
finding at least $N_{0}+\Delta N_{0}$ inside $Y$ is at most linear in time
$\tau$ and inversely proportional to boson number $\Delta N_{0}$ and distance
$d_{XY}$. While the inequality (6) is valid for any boson number $\Delta
N_{0}$, it is particularly relevant and meaningful to discuss macroscopic
transport [i.e., $\Delta N_{0}/\mathcal{N}\eqqcolon\mu$ is a $O(1)$ constant].
In this case, we obtain

$\expectationvalue{P_{\overline{n}_{Y}\geq\xi}}_{\varrho_{\tau}}\leq\kappa_{2}^{\varepsilon}\mu^{-1}\tau
d_{XY}^{-\min(1,\alpha-D-\varepsilon)},$ (7)

where $\xi\coloneqq(N_{0}+\Delta N_{0})/\mathcal{N}>\mu$. In the case
$\alpha\geq D+3$, a similar probability constraint has been derived for pure
states in the absence of long-range interactions [38] [see Eq. (7) therein],
remaining an open question for the case of $D<\alpha<D+3$. Our result thus
completely resolves this open problem of bosonic transport for generic long-
range systems.

Proof of Theorem 1.—The key ingredient in our derivation is using the
Wasserstein distance, which is initially developed in the optimal transport
theory and plays crucial roles in various disciplines, such as statistics and
machine learning [62], computer vision [63], linguistics [64], molecular
biology [65], and stochastic thermodynamics [66, 67, 68, 69]. First, we
introduce the discrete $L^{1}$-Wasserstein distance between two
$|\Lambda|$-dimensional distributions ${\bf\it x}$ and ${\bf\it y}$. Suppose
that we have a transport plan that redistributes ${\bf\it x}$ to ${\bf\it y}$
by sending an amount of $\pi_{ij}\geq 0$ from $x_{j}$ to $y_{i}$ with a cost
of $c_{ij}\geq 0$ per unit mass for all ordered pairs
$\expectationvalue{i,j}$. Here, the coupling
$\pi=[\pi_{ij}]\in\mathbb{R}_{\geq 0}^{N\times N}$ is a joint probability
distribution of ${\bf\it x}$ and ${\bf\it y}$ such that
$\sum_{j}\pi_{ij}=y_{i}$ and $\sum_{j}\pi_{ji}=x_{i}$. Each coupling $\pi$
thus defines an admissible transport plan. Evidently, there is an infinite
number of such transport plans. The Wasserstein distance is then defined as
the minimum transport cost for all feasible plans, given by

$\mathcal{W}({\bf\it x},{\bf\it y})\coloneqq\min_{\pi\in\mathcal{C}({\bf\it
x},{\bf\it y})}\sum_{i,j}c_{ij}\pi_{ij}.$ (8)

Here, $\mathcal{C}({\bf\it x},{\bf\it y})$ denotes the set of couplings $\pi$.
Next, the cost matrix $[c_{ij}]$ needs to be specified. We consider the
following cost matrix:

$c_{ij}=\|i-j\|^{\alpha_{\varepsilon}},$ (9)

where $0<\varepsilon<\alpha-D$ is an arbitrary number and
$\alpha_{\varepsilon}\coloneqq\min(1,\alpha-D-\varepsilon)$ is defined for
convenience. Since $a^{z}+b^{z}\geq(a+b)^{z}$ for any $a,b\geq 0$ and $0\leq
z\leq 1$, we can verify that

$c_{ij}+c_{jk}\geq c_{ik}$ (10)

for any $i,j,k\in\Lambda$. Consequently, the Wasserstein distance defined
using this cost matrix satisfies the triangle inequality. As can be observed,
this distance includes spatial information of the lattice and can be the order
of the system size.

Before proceeding further, we explain some useful notations that will be used
later. For any site $i$, we denote by $i[r]$ the ball of radius $r\geq 0$,
which contains all sites $j$ such that $\|i-j\|<r$. Using this definition, the
set $\Lambda\setminus\\{i\\}$ can be decomposed as

$\Lambda\setminus\\{i\\}=\cup_{\ell=1}^{\infty}(i[\ell+1]\setminus i[\ell]).$
(11)

In addition, we define $\gamma$ as a constant such that the following
inequality holds for any $r\geq 0$:

$|i[r+1]\setminus i[r]|\leq\gamma r^{D-1}.$ (12)

We are now ready to prove Theorem 1. Consider the vector of boson
concentrations occupied at each site,
$x_{i}(t)\coloneqq\tr{\overline{n}_{i}\varrho_{t}}$. Evidently,
$\sum_{i}x_{i}(t)=1$ for any $t$. Using the relation
$[\hat{b}_{i},\hat{n}_{i}]=\hat{b}_{i}$, we can show that the time evolution
of $x_{i}(t)$ is governed by the equation:

$\dot{x}_{i}(t)=\sum_{j(\neq
i)}2J_{ij}(t)\mathcal{N}^{-1}\imaginary\quantity[\tr{\hat{b}_{j}^{\dagger}\hat{b}_{i}\varrho_{t}}]\eqqcolon\sum_{j(\neq
i)}\phi_{ij}(t).$ (13)

Take the time discretization of Eq. (13) with time interval $\delta t=\tau/K$.
Then, for each $k\in[0,K-1]$ and $t=k\delta t$, we have

$x_{i}(t+\delta t)=x_{i}(t)+\sum_{j(\neq i)}\phi_{ij}(t)\delta t.$ (14)

Equation (14) indicates that we can transform ${{\bf\it x}}_{t}$ into
${{\bf\it x}}_{t+\delta t}$ by transporting an amount of $|\phi_{ij}(t)|\delta
t$ between $i$ and $j$ with a cost of $c_{ij}|\phi_{ij}(t)|\delta t$. This
transport plan takes the total cost of

$\frac{1}{2}\sum_{i\neq j}c_{ij}|\phi_{ij}(t)|\delta t\eqqcolon\Phi_{t}\delta
t,$ (15)

which should be larger than or equal to $\mathcal{W}({{\bf\it x}}_{t},{{\bf\it
x}}_{t+\delta t})$. Taking the sum of Eq. (15) from $k=0$ to $k=K-1$ and
applying the triangle inequality for $\mathcal{W}$ yield
$\sum_{k=0}^{K-1}\Phi_{t}\delta t\geq\mathcal{W}({\bf\it x}_{0},{\bf\it
x}_{\tau})$. By taking the $\delta t\to 0$ limit, we immediately obtain

$\int_{0}^{\tau}\Phi_{t}\differential{t}\geq\mathcal{W}({{\bf\it
x}}_{0},{{\bf\it x}}_{\tau}).$ (16)

Since a fraction $\mu$ of bosons must be transported from $X$ to $Y$, we have
$\sum_{i\in Y}\sum_{j\in X}\pi_{ij}\geq\mu$. Therefore,

$\displaystyle\mathcal{W}({{\bf\it x}}_{0},{{\bf\it
x}}_{\tau})\geq\quantity(\min_{i\in Y,j\in X}c_{ij})\sum_{i\in Y,j\in
X}\pi_{ij}\geq\mu d_{XY}^{\alpha_{\varepsilon}}.$ (17)

On the other hand, by applying the Cauchy-Schwarz inequality, $\phi_{ij}(t)$
can be upper bounded as

$|\phi_{ij}(t)|\leq|J_{ij}(t)|[{x}_{i}(t)+{x}_{j}(t)].$ (18)

Consequently, the cost term $\Phi_{t}$ is bounded from above as follows:

$\displaystyle\Phi_{t}$ $\displaystyle\leq\sum_{i}{x}_{i}(t)\sum_{j(\neq
i)}|J_{ij}(t)|\|i-j\|^{\alpha_{\varepsilon}}$
$\displaystyle\leq\sum_{i}{x}_{i}(t)\sum_{\ell=1}^{\infty}\sum_{j\in(i[\ell+1]\setminus
i[\ell])}\frac{J}{\|i-j\|^{\alpha-\alpha_{\varepsilon}}}$ $\displaystyle\leq
J\gamma\sum_{i}{x}_{i}(t)\sum_{\ell=1}^{\infty}\frac{1}{\ell^{\alpha-\alpha_{\varepsilon}-D+1}}$
$\displaystyle=J\gamma\zeta(\alpha-\alpha_{\varepsilon}-D+1).$ (19)

Here, we use the facts that $\|i-j\|\geq\ell$ for any $j\in(i[\ell+1]\setminus
i[\ell])$ and $|i[\ell+1]\setminus i[\ell]|\leq\gamma\ell^{D-1}$ in the third
line, and $\zeta(\cdot)$ denotes the Riemann zeta function. Define
$\kappa_{1}^{\varepsilon}\coloneqq[J\gamma\zeta(\alpha-\alpha_{\varepsilon}-D+1)]^{-1}\mu$.
Note that $\kappa_{1}^{\varepsilon}$ is system-independent and finite because
$\alpha-\alpha_{\varepsilon}-D+1\geq 1+\varepsilon$. Combining Eqs. (16),
(17), and (19) yields

$\tau\geq\kappa_{1}^{\varepsilon}d_{XY}^{\alpha_{\varepsilon}},$ (20)

which completes the proof.

Proof of Theorem 2.—The proof strategy is similar to that of Theorem 1. The
different point is that we consider the probability distribution of boson
numbers at all sites instead of the vector of average boson concentrations.
Define the probability
$p_{\vec{N}}(t)\coloneqq\tr\\{\Pi_{\vec{N}}\varrho_{t}\\}$. Since the total
boson number is invariant, we need only consider feasible states
$\ket{\vec{N}}$ that satisfy
$\tr\\{\hat{n}_{\Lambda}\Pi_{\vec{N}}\\}=\mathcal{N}$. The time evolution of
the probability distribution ${\bf\it p}_{t}=[p_{\vec{N}}(t)]$ can be derived
from the von Neumann equation as

$\displaystyle\dot{p}_{\vec{N}}(t)$
$\displaystyle=-i\tr{\Pi_{\vec{N}}[H_{t},\varrho_{t}]}$
$\displaystyle=\sum_{i\neq
j}{-iJ_{ij}(t)}\sqrt{n_{i}n_{j}^{\prime}}\quantity(\matrixelement{\vec{N}}{\varrho_{t}}{\vec{N}^{\prime}}-\matrixelement{\vec{N}}{\varrho_{t}}{\vec{N}^{\prime}})$
$\displaystyle\eqqcolon\sum_{i\neq j}\varphi_{\vec{N}\vec{N}^{\prime}}(t),$
(21)

where $\vec{N}^{\prime}$ is obtained from $\vec{N}$ by setting
$n_{i}^{\prime}=n_{i}-1$, $n_{j}^{\prime}=n_{j}+1$, and $n_{k}^{\prime}=n_{k}$
for all $k\neq i,j$. Note that
$\varphi_{\vec{N}\vec{N}^{\prime}}(t)\in\mathbb{R}$. Two states $\vec{N}$ and
$\vec{M}$ are said to be neighboring if there exist $i\neq j$ such that
$|n_{i}-m_{i}|=|n_{j}-m_{j}|=1$ and $n_{k}=m_{k}$ for any $k\neq i,j$. For
such neighboring states, the transport cost is defined as
$c_{\vec{N}\vec{N}^{\prime}}=\|i-j\|^{\alpha_{\varepsilon}}$. For arbitrary
states $\vec{N}$ and $\vec{M}$, the cost is defined as the shortest-path cost
over all possible paths connecting the two states:

$c_{\vec{M}\vec{N}}=\min\sum_{k=1}^{K}c_{\vec{N}_{k}\vec{N}_{k-1}},$ (22)

where $\vec{N}_{0}=\vec{N}$, $\vec{N}_{K}=\vec{M}$, and $\vec{N}_{k-1}$ and
$\vec{N}_{k}$ are neighboring states for all $1\leq k\leq K$. Obviously,
$c_{\vec{N}\vec{N}}=0$. Using the defined shortest-path costs, we consider the
following Wasserstein distance:

$\mathcal{W}({\bf\it p},{\bf\it q})\coloneqq\min_{\pi\in\mathcal{C}({\bf\it
p},{\bf\it q})}\sum_{\vec{M},\vec{N}}c_{\vec{M}\vec{N}}\pi_{\vec{M}\vec{N}}.$
(23)

Following the same strategy as in Eqs. (14)–(16), we can similarly prove that

$\frac{1}{2}\int_{0}^{\tau}\sum_{\vec{N}}\sum_{i\neq
j}c_{\vec{N}\vec{N}^{\prime}}|\varphi_{\vec{N}\vec{N}^{\prime}}(t)|\differential{t}\geq\mathcal{W}({\bf\it
p}_{0},{\bf\it p}_{\tau}).$ (24)

Applying the Cauchy-Schwarz inequality, we obtain

$|\varphi_{\vec{N}\vec{N}^{\prime}}(t)|\leq{|J_{ij}(t)|}\quantity[n_{i}p_{\vec{N}}(t)+n_{j}^{\prime}p_{\vec{N}^{\prime}}(t)].$
(25)

Consequently, the left-hand side of Eq. (24) can be upper bounded as

$\displaystyle\frac{1}{2}\sum_{\vec{N}}\sum_{i\neq
j}c_{\vec{N}\vec{N}^{\prime}}|\varphi_{\vec{N}\vec{N}^{\prime}}(t)|$
$\displaystyle\leq\sum_{\vec{N}}p_{\vec{N}}(t)\sum_{i}n_{i}\sum_{j(\neq
i)}\frac{J}{\|i-j\|^{\alpha-\alpha_{\varepsilon}}}$
$\displaystyle\leq\sum_{\vec{N}}p_{\vec{N}}(t)\mathcal{N}J\gamma\zeta(\alpha-\alpha_{\varepsilon}-D+1)$
$\displaystyle=\mathcal{N}J\gamma\zeta(\alpha-\alpha_{\varepsilon}-D+1).$ (26)

Let $\mathcal{S}_{0}=\\{\vec{N}\,|\,\sum_{i\in X_{c}}n_{i}\leq N_{0}\\}$ and
$\mathcal{S}_{\tau}=\\{\vec{N}\,|\,\sum_{i\in Y}n_{i}\geq N_{0}+\Delta
N_{0}\\}$. To reach state $\vec{M}\in\mathcal{S}_{\tau}$ from state
$\vec{N}\in\mathcal{S}_{0}$ by hopping bosons between sites, an amount of
$\Delta N_{0}$ bosons must be transferred from sites of $X$ to sites of $Y$.
Let $\vec{N}_{i_{0}}\to\vec{N}_{i_{1}}\to\dots\to\vec{N}_{i_{L}}$ be a
sequence of states that transfers a boson from site $i_{0}\in X$ to site
$i_{L}\in Y$ by iteratively hopping bosons between sites $i_{l-1}$ and $i_{l}$
for $1\leq l\leq L$. Then, by applying the triangle inequality, we obtain that
the cost of transferring one boson is lower bounded as

$\sum_{l=1}^{L}c_{\vec{N}_{i_{l}}\vec{N}_{i_{l-1}}}=\sum_{l=1}^{L}\|i_{l}-i_{l-1}\|^{\alpha_{\varepsilon}}\geq\|i_{L}-i_{0}\|^{\alpha_{\varepsilon}}\geq
d_{XY}^{\alpha_{\varepsilon}}.$ (27)

Therefore, we immediately obtain $c_{\vec{M}\vec{N}}\geq\Delta
N_{0}d_{XY}^{\alpha_{\varepsilon}}$ for any $\vec{M}\in\mathcal{S}_{\tau}$ and
$\vec{N}\in\mathcal{S}_{0}$. Since
$\expectationvalue{P_{\hat{n}_{X_{c}}>N_{0}}}_{\varrho_{0}}=0$,
$p_{\vec{N}}(0)=0$ for any $\vec{N}\notin\mathcal{S}_{0}$. Combining this with
$p_{\vec{N}}(0)=\sum_{\vec{M}}\pi_{\vec{M}\vec{N}}$ immediately derives
$\pi_{\vec{M}\vec{N}}=0$ for any $\vec{N}\notin\mathcal{S}_{0}$ and $\vec{M}$.
Using these facts, the Wasserstein distance can be lower bounded as follows:

$\displaystyle\mathcal{W}({\bf\it p}_{0},{\bf\it p}_{\tau})$
$\displaystyle=\min_{\pi}\sum_{\vec{M},\vec{N}}c_{\vec{M}\vec{N}}\pi_{\vec{M}\vec{N}}$
$\displaystyle\geq\min_{\pi}\sum_{\vec{M}\in\mathcal{S}_{\tau},\vec{N}\in\mathcal{S}_{0}}c_{\vec{M}\vec{N}}\pi_{\vec{M}\vec{N}}$
$\displaystyle\geq\Delta
N_{0}d_{XY}^{\alpha_{\varepsilon}}\min_{\pi}\sum_{\vec{M}\in\mathcal{S}_{\tau}}\sum_{\vec{N}\in\mathcal{S}_{0}}\pi_{\vec{M}\vec{N}}$
$\displaystyle=\Delta
N_{0}d_{XY}^{\alpha_{\varepsilon}}\min_{\pi}\sum_{\vec{M}\in\mathcal{S}_{\tau}}p_{\vec{M}}(\tau)$
$\displaystyle=\Delta
N_{0}d_{XY}^{\alpha_{\varepsilon}}\expectationvalue{P_{\hat{n}_{Y}\geq
N_{0}+\Delta N_{0}}}_{\varrho_{\tau}}.$ (28)

Here, we use the relation
$\sum_{\vec{N}\in\mathcal{S}_{0}}\pi_{\vec{M}\vec{N}}=\sum_{\vec{N}}\pi_{\vec{M}\vec{N}}=p_{\vec{M}}(\tau)$
in the fourth line. Combining Eqs. (24), (26), and (28) yields

$\expectationvalue{P_{\hat{n}_{Y}\geq N_{0}+\Delta
N_{0}}}_{\varrho_{\tau}}\leq\frac{\mathcal{N}J\gamma\zeta(\alpha-\alpha_{\varepsilon}-D+1)\tau}{\Delta
N_{0}d_{XY}^{\alpha_{\varepsilon}}},$ (29)

which completes the proof by setting $\kappa_{2}^{\varepsilon}\coloneqq
J\gamma\zeta(\alpha-\alpha_{\varepsilon}-D+1)$.

Conclusions.—Using optimal transport theory, we elucidated fundamental limits
on the speed of particle transport in long-range bosonic systems from both
perspectives of quantum speed limits and the Lieb-Robinson bound. The results
hold for the entire range of power decay and arbitrary initial states under
generic long-range characteristics. Our findings completely resolve the
critical open problem of the speed of macroscopic particle transport. It is
worth noting that the speed of bosonic transport may not be finite in the
presence of higher-order hopping terms (i.e., terms like
$\hat{b}_{i}\hat{b}_{j}\hat{b}_{k}^{\dagger}\hat{b}_{l}^{\dagger}$), such as
interaction-induced tunneling terms [70] (see the Supplemental Material [71]
for a discussion). The exploration of the speed of information propagation is
left as future work.

###### Acknowledgements.

T.V.V. was supported by JSPS KAKENHI Grant Number JP23K13032. T.K.
acknowledges Hakubi projects of RIKEN and was supported by JSPS KAKENHI Grant
Number 18K13475 and JST, PRESTO Grant Number JPMJPR2116. K.S. was supported by
JSPS KAKENHI Grant Numbers JP23H01099, JP19H05603, and JP19H05791.

## References

  * Lieb and Robinson [1972] E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972).
  * Hastings and Wen [2005] M. B. Hastings and X.-G. Wen, Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance, Phys. Rev. B 72, 045141 (2005).
  * Bravyi  _et al._ [2006] S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb-Robinson bounds and the generation of correlations and topological quantum order, Phys. Rev. Lett. 97, 050401 (2006).
  * Hastings and Koma [2006] M. B. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265, 781 (2006).
  * Nachtergaele and Sims [2006] B. Nachtergaele and R. Sims, Lieb-Robinson bounds and the exponential clustering theorem, Commun. Math. Phys. 265, 119 (2006).
  * Osborne [2006] T. J. Osborne, Efficient approximation of the dynamics of one-dimensional quantum spin systems, Phys. Rev. Lett. 97, 157202 (2006).
  * Hastings [2007] M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech.: Theory Exp. 2007, P08024.
  * Van Acoleyen  _et al._ [2013] K. Van Acoleyen, M. Mariën, and F. Verstraete, Entanglement rates and area laws, Phys. Rev. Lett. 111, 170501 (2013).
  * Roberts and Swingle [2016] D. A. Roberts and B. Swingle, Lieb-Robinson bound and the butterfly effect in quantum field theories, Phys. Rev. Lett. 117, 091602 (2016).
  * Iyoda  _et al._ [2017] E. Iyoda, K. Kaneko, and T. Sagawa, Fluctuation theorem for many-body pure quantum states, Phys. Rev. Lett. 119, 100601 (2017).
  * Anshu  _et al._ [2021] A. Anshu, S. Arunachalam, T. Kuwahara, and M. Soleimanifar, Sample-efficient learning of interacting quantum systems, Nat. Phys. 17, 931 (2021).
  * Alhambra and Cirac [2021] A. M. Alhambra and J. I. Cirac, Locally accurate tensor networks for thermal states and time evolution, PRX Quantum 2, 040331 (2021).
  * Kuwahara and Saito [2022] T. Kuwahara and K. Saito, Exponential clustering of bipartite quantum entanglement at arbitrary temperatures, Phys. Rev. X 12, 021022 (2022).
  * Chen  _et al._ [2023] C.-F. Chen, A. Lucas, and C. Yin, Speed limits and locality in many-body quantum dynamics, arXiv:2303.07386 (2023).
  * Eisert  _et al._ [2013] J. Eisert, M. van den Worm, S. R. Manmana, and M. Kastner, Breakdown of quasilocality in long-range quantum lattice models, Phys. Rev. Lett. 111, 260401 (2013).
  * Saffman  _et al._ [2010] M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  * Foss-Feig  _et al._ [2015] M. Foss-Feig, Z.-X. Gong, C. W. Clark, and A. V. Gorshkov, Nearly linear light cones in long-range interacting quantum systems, Phys. Rev. Lett. 114, 157201 (2015).
  * Matsuta  _et al._ [2016] T. Matsuta, T. Koma, and S. Nakamura, Improving the Lieb-Robinson bound for long-range interactions, Ann. Henri Poincaré 18, 519 (2016).
  * Chen and Lucas [2019] C.-F. Chen and A. Lucas, Finite speed of quantum scrambling with long range interactions, Phys. Rev. Lett. 123, 250605 (2019).
  * Kuwahara and Saito [2020] T. Kuwahara and K. Saito, Strictly linear light cones in long-range interacting systems of arbitrary dimensions, Phys. Rev. X 10, 031010 (2020).
  * Else  _et al._ [2020] D. V. Else, F. Machado, C. Nayak, and N. Y. Yao, Improved Lieb-Robinson bound for many-body Hamiltonians with power-law interactions, Phys. Rev. A 101, 022333 (2020).
  * Tran  _et al._ [2021a] M. C. Tran, A. Y. Guo, A. Deshpande, A. Lucas, and A. V. Gorshkov, Optimal state transfer and entanglement generation in power-law interacting systems, Phys. Rev. X 11, 031016 (2021a).
  * Kuwahara and Saito [2021a] T. Kuwahara and K. Saito, Absence of fast scrambling in thermodynamically stable long-range interacting systems, Phys. Rev. Lett. 126, 030604 (2021a).
  * Tran  _et al._ [2021b] M. C. Tran, A. Y. Guo, C. L. Baldwin, A. Ehrenberg, A. V. Gorshkov, and A. Lucas, Lieb-Robinson light cone for power-law interactions, Phys. Rev. Lett. 127, 160401 (2021b).
  * Chen and Lucas [2021] C.-F. Chen and A. Lucas, Optimal Frobenius light cone in spin chains with power-law interactions, Phys. Rev. A 104, 062420 (2021).
  * Gong  _et al._ [2023] Z. Gong, T. Guaita, and J. I. Cirac, Long-range free fermions: Lieb-Robinson bound, clustering properties, and topological phases, Phys. Rev. Lett. 130, 070401 (2023).
  * Bloch  _et al._ [2008] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  * Läuchli and Kollath [2008] A. M. Läuchli and C. Kollath, Spreading of correlations and entanglement after a quench in the one-dimensional Bose-Hubbard model, J. Stat. Mech.: Theory Exp. 2008, P05018.
  * Carleo  _et al._ [2014] G. Carleo, F. Becca, L. Sanchez-Palencia, S. Sorella, and M. Fabrizio, Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids, Phys. Rev. A 89, 031602 (2014).
  * Cheneau  _et al._ [2012] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of correlations in a quantum many-body system, Nature 481, 484 (2012).
  * Takasu  _et al._ [2020] Y. Takasu, T. Yagami, H. Asaka, Y. Fukushima, K. Nagao, S. Goto, I. Danshita, and Y. Takahashi, Energy redistribution and spatiotemporal evolution of correlations after a sudden quench of the Bose-Hubbard model, Science Advances 6 (2020).
  * Schuch  _et al._ [2011] N. Schuch, S. K. Harrison, T. J. Osborne, and J. Eisert, Information propagation for interacting-particle systems, Phys. Rev. A 84, 032309 (2011).
  * Wang and Hazzard [2020] Z. Wang and K. R. Hazzard, Tightening the Lieb-Robinson bound in locally interacting systems, PRX Quantum 1, 010303 (2020).
  * Kuwahara and Saito [2021b] T. Kuwahara and K. Saito, Lieb-Robinson bound and almost-linear light cone in interacting boson systems, Phys. Rev. Lett. 127, 070403 (2021b).
  * Yin and Lucas [2022] C. Yin and A. Lucas, Finite speed of quantum information in models of interacting bosons at finite density, Phys. Rev. X 12, 021039 (2022).
  * Faupin  _et al._ [2022a] J. Faupin, M. Lemm, and I. M. Sigal, On Lieb-Robinson bounds for the Bose-Hubbard model, Communications in Mathematical Physics (2022a).
  * Kuwahara  _et al._ [2022] T. Kuwahara, T. Van Vu, and K. Saito, Optimal light cone and digital quantum simulation of interacting bosons, arXiv:2206.14736 (2022).
  * Faupin  _et al._ [2022b] J. Faupin, M. Lemm, and I. M. Sigal, Maximal speed for macroscopic particle transport in the Bose-Hubbard model, Phys. Rev. Lett. 128, 150602 (2022b).
  * Mandelstam and Tamm [1945] L. Mandelstam and I. Tamm, The uncertainty relation between energy and time in non-relativistic quantum mechanics, J. Phys. USSR 9, 249 (1945).
  * Uhlmann [1992] A. Uhlmann, An energy dispersion estimate, Phys. Lett. A 161, 329 (1992).
  * Margolus and Levitin [1998] N. Margolus and L. B. Levitin, The maximum speed of dynamical evolution, Physica D 120, 188 (1998).
  * del Campo  _et al._ [2013] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Quantum speed limits in open system dynamics, Phys. Rev. Lett. 110, 050403 (2013).
  * Deffner and Lutz [2013] S. Deffner and E. Lutz, Quantum speed limit for non-Markovian dynamics, Phys. Rev. Lett. 111, 010402 (2013).
  * Taddei  _et al._ [2013] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Quantum speed limit for physical processes, Phys. Rev. Lett. 110, 050402 (2013).
  * Pires  _et al._ [2016] D. P. Pires, M. Cianciaruso, L. C. Céleri, G. Adesso, and D. O. Soares-Pinto, Generalized geometric quantum speed limits, Phys. Rev. X 6, 021031 (2016).
  * Mondal  _et al._ [2016] D. Mondal, C. Datta, and S. Sazim, Quantum coherence sets the quantum speed limit for mixed states, Phys. Lett. A 380, 689 (2016).
  * Deffner [2017] S. Deffner, Geometric quantum speed limits: a case for Wigner phase space, New J. Phys. 19, 103018 (2017).
  * Shanahan  _et al._ [2018] B. Shanahan, A. Chenu, N. Margolus, and A. del Campo, Quantum speed limits across the quantum-to-classical transition, Phys. Rev. Lett. 120, 070401 (2018).
  * Campaioli  _et al._ [2018] F. Campaioli, F. A. Pollock, F. C. Binder, and K. Modi, Tightening quantum speed limits for almost all states, Phys. Rev. Lett. 120, 060409 (2018).
  * Funo  _et al._ [2019] K. Funo, N. Shiraishi, and K. Saito, Speed limit for open quantum systems, New J. Phys. 21, 013006 (2019).
  * García-Pintos and del Campo [2019] L. P. García-Pintos and A. del Campo, Quantum speed limits under continuous quantum measurements, New J. Phys. 21, 033012 (2019).
  * Sun  _et al._ [2021] S. Sun, Y. Peng, X. Hu, and Y. Zheng, Quantum speed limit quantified by the changing rate of phase, Phys. Rev. Lett. 127, 100404 (2021).
  * Van Vu and Hasegawa [2021] T. Van Vu and Y. Hasegawa, Geometrical bounds of the irreversibility in Markovian systems, Phys. Rev. Lett. 126, 010601 (2021).
  * Shiraishi and Saito [2021] N. Shiraishi and K. Saito, Speed limit for open systems coupled to general environments, Phys. Rev. Res. 3, 023074 (2021).
  * Hamazaki [2022] R. Hamazaki, Speed limits for macroscopic transitions, PRX Quantum 3, 020319 (2022).
  * Van Vu and Saito [2023a] T. Van Vu and K. Saito, Topological speed limit, Phys. Rev. Lett. 130, 010402 (2023a).
  * Hasegawa [2023] Y. Hasegawa, Unifying speed limit, thermodynamic uncertainty relation and Heisenberg principle via bulk-boundary correspondence, Nat. Commun. 14 (2023).
  * Deffner and Campbell [2017] S. Deffner and S. Campbell, Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control, J. Phys. A 50, 453001 (2017).
  * Villani [2008] C. Villani,  _Optimal Transport: Old and New_ (Springer, Berlin, Heidelberg, 2008).
  * [60] Note that it may not be the case when considering the speed of information propagation.
  * Tran  _et al._ [2020] M. C. Tran, C.-F. Chen, A. Ehrenberg, A. Y. Guo, A. Deshpande, Y. Hong, Z.-X. Gong, A. V. Gorshkov, and A. Lucas, Hierarchy of linear light cones with long-range interactions, Phys. Rev. X 10, 031009 (2020).
  * Kolouri  _et al._ [2017] S. Kolouri, S. R. Park, M. Thorpe, D. Slepcev, and G. K. Rohde, Optimal mass transport: Signal processing and machine-learning applications, IEEE Signal Process. Mag. 34, 43 (2017).
  * Haker  _et al._ [2004] S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, Optimal mass transport for registration and warping, Int. J. Comput. Vision 60, 225 (2004).
  * Huang  _et al._ [2016] G. Huang, C. Guo, M. J. Kusner, Y. Sun, F. Sha, and K. Q. Weinberger, Supervised word mover’s distance, in  _Advances in Neural Information Processing Systems_ , Vol. 29 (2016).
  * Schiebinger  _et al._ [2019] G. Schiebinger, J. Shu, M. Tabaka, B. Cleary, V. Subramanian, A. Solomon, J. Gould, S. Liu, S. Lin, P. Berube, _et al._ , Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming, Cell 176, 928 (2019).
  * Aurell  _et al._ [2011] E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Optimal protocols and optimal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011).
  * Nakazato and Ito [2021] M. Nakazato and S. Ito, Geometrical aspects of entropy production in stochastic thermodynamics based on wasserstein distance, Phys. Rev. Res. 3, 043093 (2021).
  * Dechant [2022] A. Dechant, Minimum entropy production, detailed balance and Wasserstein distance for continuous-time Markov processes, J. Phys. A 55, 094001 (2022).
  * Van Vu and Saito [2023b] T. Van Vu and K. Saito, Thermodynamic unification of optimal transport: Thermodynamic uncertainty relation, minimum dissipation, and thermodynamic speed limits, Phys. Rev. X 13, 011013 (2023b).
  * Sowiński  _et al._ [2012] T. Sowiński, O. Dutta, P. Hauke, L. Tagliacozzo, and M. Lewenstein, Dipolar molecules in optical lattices, Phys. Rev. Lett. 108, 115301 (2012).
  * [71] See Supplemental Material for a discussion on the speed of bosonic transport in quantum bosonic systems with interaction-induced tunneling terms.