Macroscopic particle transport in dissipative long-range bosonic systems

Domain: physics
Published: 2026-03-21
Editorial outcome: minor_revision

Abstract
Dissipation in quantum many-body systems provides a more general and experimentally realistic perspective on particle transport than closed quantum systems. In this work, we determine the maximal speed of macroscopic particle transport in dissipative bosonic systems featuring both long-range hopping and long-range interactions. By developing a generalized optimal transport theory for open quantum systems, we rigorously establish the relationship between the minimum transport time and the source-target distance, and investigate the maximal transportable distance of bosons. We demonstrate that optimal transport exhibits a fundamental distinction depending on whether the system experiences one-body loss or multi-body loss. Moreover, we present the minimal transport time and the maximal transport distance for systems with both gain and loss. We observe that even an arbitrarily small gain rate enables transport over long distances if the lattice gas is dilute. Importantly, we generally reveal that the emergence of decoherence-free subspaces facilitates the long-distance and perfect transport process. Additionally, we derive an upper bound for the probability of transporting a given number of particles during a fixed period in the presence of particle loss. Possible experimental protocols for observing our theoretical predictions are also discussed.

Introduction
The maximal velocity of information propagation is a central topic in quantum mechanics. Information causality, a fundamental constraint in relativistic quantum mechanics, gives rise to an upper bound for information propagation, known as a light cone. Remarkably, Lieb and Robinson pointed out that quantum information propagation in locally interacting quantum spin systems on lattices also exhibits an effective light cone structure [1, 2]. This result, named the Lieb-Robinson (LR) bound, demonstrates locality even in non-relativistic quantum many-body systems and has undergone substantial improvements and extensive extensions since its discovery. The LR bound has been extended to general interacting quantum spin systems and fermionic systems [3–10] and interacting bosonic systems [11–15]. Recently, it has been further generalized to open quantum systems [16–20]. The LR bound has now become a powerful tool in condensed matter physics [21–26], quantum information science [27–32], high energy physics [33–35], etc.

Despite the great success in applying the Lieb-Robinson bound, its extension to bosonic systems remains limited due to the fact that the norm of the bosonic operators is unbounded. Constant progress has been made in understanding the maximal speed for information propagation in bosonic systems with short-range hopping by restricting the initial boson number [13, 15] to one-dimensional systems [12] or to suitable initial states [14]. However, for general unbounded operators in high-dimensional systems, the traditional arguments yield only unsatisfactory results on the maximal information-propagation speed [11]. On the other hand, some studies focus on the speed limit for macroscopic particle transport, which quantifies the minimum time required for a given number of particles to propagate to a reachable regime. Recent results have elucidated the maximal speed of bosonic macroscopic particle transport, from the Bose-Hubbard model [13, 36–38] to more general systems with both long-range hopping and long-range interactions [39, 40].

Nevertheless, the existing findings are limited to closed quantum systems. Due to intrinsic chemical reactions and inelastic collisions between particles [41–57], photoassociation [58–62] or interactions with external environments [63–70], these systems are usually not strictly closed and inevitably suffer from dissipation, such as dephasing and particle loss, which is common in atomic, molecular, and optical systems. The dissipation causes the breakdown of the optimal transport theory in closed quantum systems since the traditional optimal transport theory only applies to the transport between two balanced particle distributions with equal total particle numbers. However, particle loss disrupts the balance between the two distributions. While previous studies have investigated particle transport in Markovian open quantum systems [71–73], they are primarily restricted to few-body regimes. In contrast, the maximal transport speed in general many-body open quantum systems remains largely unexplored.

In this work, we develop the macroscopic particle transport theory for generic bosonic systems with long-range interaction and long-range hopping in the presence of particle loss. Inspired by the methods in Ref. [39], we construct the optimal transport theory applicable to open quantum systems and demonstrate the relationship between the transport time and the distance between the source and target regions for the Lindbladian dynamics. Specifically, for local particle loss, we show that the optimal transport time for the case of one-body loss in Eq. (4) is even longer than that with multi-body loss in Eq. (8), which can be attributed to the existence of decoherence-free subspaces [74] of the systems. The emergence of decoherence-free subspaces inherently supports long-distance and perfect (i.e., 100%) transport (see Result 2 below).

Furthermore, we analyze Lindbladian dynamics with both gain and loss and demonstrate that particle gain can effectively reduce transport time, as shown in Eq. (10). We compare the maximal transport distance between the case with only one-body loss and the case with both gain and loss, as given in Eqs. (7) and (14). Our results indicate that particle gain extends the transport distance. In particular, if the lattice gas is dilute, even an infinitesimally small gain rate can enable arbitrarily long transport distances over the lattice. We also derive an upper bound for the probability of transporting a given number of bosons between regions in a fixed period, as shown in Eq. (17). We conclude by discussing potential experimental protocols to test the predicted transport bounds and highlighting open questions that arise from our study.

Results
A. Local particle loss

We consider a quantum many-body system on a D-dimensional lattice with Λ standing for all sites. For an arbitrary subset Z ⊆ Λ, the number of sites in Z is denoted by |Z|, while the system size is represented by |Λ|. The bosonic Hamiltonian can be written as

H := \sum_{i\neq j \in \Lambda} J_{ij}(t)b_i^\dagger b_j + \sum_{Z \subseteq \Lambda} h_Z(t), (1)

where b_i^\dagger and b_i are the bosonic creation and annihilation operators at the site i ∈ Λ, respectively. Without any assumptions on boson-boson interactions in Eq. (1), h_Z(t) is an arbitrary function of the boson number operators {n_i}_{i \in \Lambda}, where n_i := b_i^\dagger b_i. The hopping amplitudes J_{ij}(t) are symmetric and upper bounded by a power law in the Euclidean distance, i.e., |J_{ij}(t)| \leq J/\|i-j\|^\alpha for a positive constant J and the power decay α satisfies α > D. In other words, both the long-range hopping and long-range interaction can be considered.

Let ρ_t be the density matrix of the system. In the presence of dissipation, its dynamics follows the Lindblad equation:

\frac{d}{dt} \rho_t = i[\rho_t, H] + \sum_{i} \frac{\gamma}{2} (2L_i \rho_t L_i^\dagger - L_i^\dagger L_i \rho_t - \rho_t L_i^\dagger L_i), (2)

where we assume the Born-Markov approximation, which is valid in the weak-coupling regime. Here, γ represents the jump rate, and L_i denotes the Lindblad operator. In this study, we focus on the local Lindbladian, where the Lindblad operator L_i only acts on one site. Specifically, we consider two types of Lindblad operators: one characterizing purely particle loss and the other incorporating both particle loss and gain [75]. The corresponding transport process is illustrated in Fig. 1. Figure 1(a) depicts the case of one-body loss or multi-body loss, while Fig. 1(b) presents the scenario where particle loss and gain coexist.

Consider a scenario where a fraction μ ∈ (0, 1] of all bosons is transported from region X to a distant region Y, which in the open quantum systems is defined as

x_Y(\tau) - x_{X^c}(\tau) \geq \mu. (3)

Here x_Y(\tau) is the total particle number in region Y at time τ with the Hamiltonian H and x_{X^c}(\tau) is the total particle number in region X^c := Λ \ X at time τ when the Hamiltonian is switched off throughout the process, with N being the initial total boson number on the lattice and X \ Y representing the set of sites in region X that is not included in region Y. We define the transport in this way because the change in particle number within region Y can arise from three sources: particle exchange with the environment, transport from region X^c \ Y, and transport from region X. Our definition effectively eliminates contributions from the first two, isolating the net transfer from X to Y. Accordingly, the inequality (3) indicates that μN particles are effectively transported from X to Y within time τ, taking into account both dissipation and hopping processes on the lattice. The distance between any two disjoint sets X and Y is defined as the minimum distance between their respective sites, i.e., d_{XY} := min_{i \in X, j \in Y} \|i-j\|. The lattice satisfies |i[r] \ i[r-1]| \leq φr^{D-1}, where φ is an O(1) constant determined by the lattice structure and i[r] denotes the ball of radius r centered at site i. For the above setting, we develop the macroscopic particle transport theory for long-range open systems by introducing a unified approach for particle transport. The detailed proofs are outlined in the Methods section IV.

Result 1. For the local one-body-loss case, i.e., L_i = b_i, the transport time τ satisfies

\tau e^{-\gamma\tau} \geq \kappa_{\epsilon} d_{XY}^{\alpha_{\epsilon}}, (4)

where \kappa_{\epsilon} := \mu/(J\phi\zeta(\alpha - \alpha_{\epsilon} - D + 1)) and \zeta(x) represents the Riemann-zeta function. Here, \alpha_{\epsilon} := min(1, \alpha - D - \epsilon) with 0 < \epsilon < \alpha - D being an arbitrary number.

Compared with the extant result in closed quantum systems [39]

\tau \geq \kappa_{\epsilon} d_{XY}^{\alpha_{\epsilon}}, (5)

we find that the transport time τ is suppressed by an exponential decay, as the particle number decreases exponentially over time. This fact follows directly from the equation

\frac{d\langle\hat{N}\rangle}{dt} = \frac{\gamma}{2} \sum_i \langle b_i^\dagger[\hat{N}, b_i] + [b_i^\dagger, \hat{N}]b_i \rangle = -\gamma\langle\hat{N}\rangle,

which yields the exponential decay N_t = Ne^{-\gamma t}. Here, \hat{N} denotes the total particle number operator, and N_t = \langle\hat{N}\rangle := tr(\hat{N}\rho_t) is the expected total particle number at time t.

From Eq. (4), we observe that there exists a maximal particle number that can be transported from region X to region Y. The upper bound for μ takes the form of

\mu \leq min \left( \frac{J\phi\zeta(\alpha - \alpha_{\epsilon} - D + 1)}{e^{\gamma\tau} d_{XY}^{\alpha_{\epsilon}}} N, 1 \right) (6)

by optimizing the transport time τ in Eq. (4). If the one-body loss rate is significant, it is more likely that the amount μN of bosons can never be transported from region X to region Y even in the infinite-time limit. Furthermore, we consider another transport limit of the bosons, defined as the farthest distance d_l for which one boson can be transported. In the case of minimal transport (i.e., μ = 1/N), the transport limit is given by

d_l = \left( \frac{NJ\phi\zeta(\alpha - \alpha_{\epsilon} - D + 1)}{e^{\gamma\tau}} \right)^{1/\alpha_{\epsilon}}. (7)

This result indicates that if N bosons are initially confined at the origin, their transport remains restricted within a region of radius d_l. This constraint, introduced by particle loss, highlights a form of locality and provides an upper bound on the effective trapping range in experiments.

Result 2. For the local multi-body-loss case, i.e., L_i = b_i^n with n > 1, n ∈ Z, the transport time τ satisfies

\tau \geq \kappa_{\epsilon} d_{XY}^{\alpha_{\epsilon}}. (8)

Notably, despite the presence of particle loss, this lower bound remains identical to that of the closed system [cf. Eq. (5)]. The difference between the one-body-loss case and the multi-body-loss case stems from the presence of decoherence-free subspaces [74]. For one-body loss, it is evident that the decoherence-free subspace consists only of the vacuum state. However, for multi-body loss, the decoherence-free subspace is given by the projection P = \sum_n |n\rangle\langle n|, where |n\rangle is the Mott state with no more than (n - 1) particles on each site. If the initial state is dilute and the dynamics remain confined to the subspace Ran P, particle transport proceeds without dissipation. As a result, the optimal transport under multi-body loss coincides with that of a closed quantum system. This explains why Eq. (8) is independent of the exponent n. A particularly natural example exhibiting the decoherence-free subspace is a system that contains at most (n - 1) particles. However, the particle number is strictly limited in this example. For quantum many-body systems with N ≥ n, the equality in Eq. (8) can still be asymptotically saturated by the following Hamiltonian:

H = \sum_{i\neq j} J_{ij}(b_i^\dagger b_j + b_j^\dagger b_i) + U \sum_i \hat{n}_i(\hat{n}_i - 1), (9)

where the limit U → ∞ imposes the hard-core boson constraint, corresponding to the Mott-insulating phase in which only up to a single occupancy per site is allowed [76]. In this case, if the initial state lies within the decoherence-free subspace, the quantum state remains confined to the subspace due to the suppression of multi-particle occupation. Consequently, the transport process effectively mimics that of a closed quantum system. Therefore, this nontrivial decoherence-free subspace enables perfect transport over arbitrary distances without limitations on μ and d_l. For instance, particles initialized in a Mott state can be perfectly transported using the protocol proposed in Ref. [39]. In contrast, for one-body loss, no nontrivial decoherence-free subspace exists, making the equality in Eq. (8) unattainable; instead, a tighter bound is provided in Eq. (4) of Result 1. Experimentally, multi-body loss typically arises from internal chemical reactions or photoassociation.

B. Local particle loss and gain

Thus far, we have only considered local particle loss. In what follows, we further investigate particle transport behavior in the presence of both particle loss and gain.

Result 3. For the Lindbladian with both local one-body loss and one-body gain, where the Lindblad operator takes the form of L_i := \sqrt{\gamma_1/\gamma}b_i + \sqrt{\gamma_2/\gamma}b_i^\dagger with \gamma_2 < \gamma_1, the transport time satisfies

\tau e^{-\Delta\gamma\tau} + \frac{\gamma_2}{\Delta\gamma} \left( \frac{1 - e^{-\Delta\gamma\tau}}{\Delta\gamma} N^{-1} - \tau e^{-\Delta\gamma\tau} \right) \geq \kappa_{\epsilon} d_{XY}^{\alpha_{\epsilon}}, (10)

where \Delta\gamma := \gamma_1 - \gamma_2, and N := N/|\Lambda| is the initial particle density on the lattice. Here, \gamma_1 and \gamma_2 represent the loss and gain rates, respectively.

We note that Result 1 is a special case of Result 3 with \gamma_1 = \gamma and \gamma_2 = 0. Notably, the left-hand side of Eq. (10) does not vanish in the long-time limit, making it fundamentally different from the one-body-loss case. This indicates that gain intrinsically changes the particle transport process.

From Eq. (10), the maximal fraction of particles that can be transported from region X to region Y is given by

\mu \leq min \left( \frac{J\phi\zeta(\alpha - \alpha_{\epsilon} - D + 1)}{d_{XY}^{\alpha_{\epsilon}}} B(\tau_c), 1 \right), (11)

where

B(t) := \begin{cases} e^{-\Delta\gamma t} + \frac{\gamma_2 N^{-1}}{(\Delta\gamma)^2 t} & \text{if } N > \frac{\gamma_2}{\Delta\gamma}, \ \frac{(\Delta\gamma)^2}{(\gamma_2 N^{-1} - \Delta\gamma)^2} & \text{if } N \leq \frac{\gamma_2}{\Delta\gamma}, \end{cases} (12)

and \tau_c := (\Delta\gamma - \gamma_2/N)^{-1}. We examine two limiting cases of this result. In the dense limit (N → ∞), B(\tau_c) asymptotically approaches 1/(e^{\Delta\gamma}), indicating an enhancement of the upper bound on μ compared to Eq. (6). In contrast, in the dilute limit (N → 0), the upper bound diverges as B(\tau_c) = \gamma_2 N^{-1}/(\Delta\gamma)^2 → ∞. This divergence suggests that an arbitrary number of particles can be transported from X to Y when the lattice is sufficiently large. The underlying reason for this behavior can be explained as follows. The dark state |ψ⟩, which satisfies L_i |ψ⟩ = 0, takes the product form of squeezed states as

|ψ⟩ = \bigotimes_i \exp \left( \frac{r}{2} (b_i^2 - b_i^{\dagger 2}) \right) |0⟩, (13)

where r = arctanh(\sqrt{\gamma_2/\gamma_1}). This state is also the unique steady state of the Lindbladian when the Hamiltonian is switched off. One can explicitly show that the average particle number per site in |ψ⟩ is given by \langle\psi|\hat{n}_i|\psi\rangle = \gamma_2/\Delta\gamma := \bar{n}, which coincides with the critical particle density in Eq. (12).

Importantly, the mechanism enabling long-distance transport here differs from that in Result 2. There, the decoherence-free subspace comprises an extensive set of states, allowing the dynamics to remain confined within it. In contrast, the present decoherence-free subspace consists solely of the single dark state |ψ⟩, preventing the system from evolving fully within the subspace. Nevertheless, when the initial particle density is below \bar{n}, the gain process drives the system toward |ψ⟩, making the transport effectively immune to losses and thereby enabling nearly perfect transport.

Similarly, we figure out the transport limit d_l

d_l = (NJ\phi\zeta(\alpha - \alpha_{\epsilon} - D + 1)B(\tau_c))^{1/\alpha_{\epsilon}}. (14)

In the dilute limit, the distance d_l can be rewritten as

d_l = \left( \frac{\gamma_2 |\Lambda| J\phi\zeta(\alpha - \alpha_{\epsilon} - D + 1)}{(\Delta\gamma)^2} \right)^{1/\alpha_{\epsilon}}. (15)

Therefore, the transport limit d_l can be comparable with the system size, allowing particles to reach any site in the lattice—even for arbitrarily small gain rates \gamma_2—provided the initial average density is below \bar{n}. To be specific, in a three-dimensional lattice, if we take \gamma_2/\Delta\gamma \sim O(|\Lambda|^{-1/2}) and J/\Delta\gamma \sim O(1), then the transport limit satisfies d_l \sim O(|\Lambda|^{1/2}), which is larger than the system length O(|\Lambda|^{1/3}) in the thermodynamic limit. Combining the results above, we conclude that the particle gain plays a nontrivial role in the dilute limit, more than simply compensating for the particle loss.

In addition, our results also hold for local Lindbladian dynamics

\frac{d\rho_t}{dt} = i[\rho_t, H] + \sum_{i \in \Lambda} \frac{\gamma_{\alpha}}{2} (2L_{\alpha, i} \rho_t L_{\alpha, i}^\dagger - L_{\alpha, i}^\dagger L_{\alpha, i} \rho_t - \rho_t L_{\alpha, i}^\dagger L_{\alpha, i}), \alpha = 1, 2, (16)

where L_{1, i} := b_i and L_{2, i} := b_i^\dagger. The Lindbladian can be realized by coupling the system to two baths with different chemical potentials: one higher and one lower than the system.

C. Transport Probability

Result 4. Assume that the initial boson number outside region X is N_{X^c}(0). In the presence of local n-body loss, the probability of finding N_{X^c}(0) + μ bosons in a disjoint region Y in a given time τ is upper bounded by

P_{x_Y(\tau) \geq x_{X^c}(0) + \mu} \leq \frac{J\phi\zeta(\alpha - \alpha_{\epsilon} - D + 1)\tau}{\mu d_{XY}^{\alpha_{\epsilon}}}, (17)

where the ratios x_{X^c}(0) and μ are of the order O(1).

It is worth noting that Result 2 can be interpreted as a corollary of Result 4 by following the same procedure in Ref. [39]. To be specific, by choosing an arbitrary constant μ' ∈ (0, μ) and applying Markov's inequality, one can show that

P_{x_Y(\tau) \geq x_{X^c}(0) + \mu} \geq \frac{x_Y(\tau) - x_{X^c}(0) - \mu'}{1 - x_{X^c}(0) - \mu'}. (18)

When the particle transport satisfies x_Y(\tau) \geq x_{X^c}(0) + \mu, we obtain

\tau \geq \kappa_{\epsilon} \frac{\mu'(\mu - \mu')}{\mu[1 - x_{X^c}(0) - \mu']} d_{XY}^{\min(1, \alpha - D - \epsilon)}, (19)

by combining Eqs. (17) and (18). Since the factor μ'(μ - μ')/{μ[1 - x_{X^c}(0) - μ']} ≤ 1 can be regarded as an O(1) constant, the upper bound of the probability in Eq. (17) shows fundamentally the same limit in particle transport as Result 2 on macroscopic transport. In comparison with the result for closed quantum systems [39], we conclude that dissipation does not generally enhance transport speed.

Methods
Before going into detailed proofs, we briefly explain the optimal transport theory, which concerns the optimal way to transport distributions. The main concept in the theory is the Wasserstein distance [80], which is defined as the minimum transport cost over all feasible plans that redistribute a given distribution to a desired one. Specifically, the Wasserstein distance between two finite-dimensional distributions x and y is defined as

W(x, y) := \min_{\pi} \sum_{m, n} \pi_{mn} c_{mn}, (20)

where π denotes a joint probability distribution of x and y satisfying \sum_n \pi_{mn} = x_m and \sum_m \pi_{mn} = y_n. The quantity π_{mn} can be interpreted as the amount of probability transported from x_n to y_m. The cost function c_{mn} is positive and symmetric as c_{mn} = c_{nm}, which satisfies c_{mn} + c_{np} \geq c_{mp} for any m, n, p. These properties guarantee that the Wasserstein distance also satisfies the triangle inequality. Notably, the Kantorovich-Rubinstein duality [39, 80] derives another variational form of the Wasserstein distance,

W(x, y) = \max_w w^\top(y - x), (21)

where the maximum is over all vectors w satisfying |w_m - w_n| \leq c_{mn}.

The Wasserstein distance defined in Eq. (20) applies to balanced distributions x and y. That is, ∥x∥_1 and ∥y∥_1 must be identical, where ∥x∥_1 := \sum_n x_n. However, in the presence of particle loss, the particle number decays, and we may encounter cases where the 1-norms of x and y differ. Hence, the conventional distance cannot be straightforwardly applied in general. To address this issue, we define a generalized Wasserstein distance between two vectors x and y satisfying ∥x∥_1 \geq ∥y∥_1 as

\tilde{W}(x, y) := \min_{x \succeq x' \succeq 0, y' \succeq y, \|x'\|_1 = \|y'\|_1} W(x', y'), (22)

where x ⪰ x' indicates that x_n \geq x'_n for all n. We can further prove that the generalized Wasserstein distance satisfies the triangle inequality (the proof is provided in Section II of Supplemental Information).

A. Outline of the proof for Result 1

Here we consider the case of local one-body loss (i.e., L_i = b_i). The dynamics can be rewritten as

\frac{d}{dt} \rho_t = -i[H, \rho_t] + \sum_i \frac{\gamma}{2} (2b_i \rho_t b_i^\dagger - b_i^\dagger b_i \rho_t - \rho_t b_i^\dagger b_i). (23)

By defining the particle ratio in site i at time t as x_i(t) := tr(b_i^\dagger b_i \rho_t)/N, the time evolution of x_i(t) can be derived from Eq. (23) as

\dot{x}_i(t) = \frac{1}{N} \sum_{j, j \neq i} 2J_{ij}(t) \text{Im}[tr(b_j^\dagger b_i \rho_t)] - \gamma x_i(t). (24)

By solving this equation, we obtain

x_i(\tau) = x_i(0)e^{-\gamma\tau} + \int_0^\tau dt \sum_{j, j \neq i} \tilde{\phi}_{ij}(t), (25)

with the definition

\tilde{\phi}_{ij}(t) := 2J_{ij}(t)e^{\gamma(t - \tau)} \text{Im}[tr(b_j^\dagger b_i \rho_t)] / N. (26)

Here we can see that the particle current originates from the coherence in the density matrix, which is intrinsically different from the transport mechanism in the classical rate equation, such as the reaction-diffusion equation. From Eq. (25), we can see the dynamics of the particle number at site i can be separated into two parts. The first term on the right-hand side of Eq. (25) represents the particle decay due to the one-body loss since the total number also exponentially decays with time:

\sum_i x_i(t) = \sum_i x_i(0)e^{-\gamma t} = e^{-\gamma t}. (27)

The second term on the right-hand side of Eq. (25) is the current flow from all the sites other than i. Therefore, \tilde{\phi}_{ij} represents the current from the site j to the site i at time t.

Since we consider the case where a fraction μ of the total bosons must be transported from X to Y within the time period τ, the following relation should hold:

x_Y(\tau) - x_{X^c}(\tau) \geq \mu, (28)

where X^c := Λ \ X and x_{X^c}(\tau) = e^{-\gamma\tau}x_{X^c}(0) denotes the density at time τ in region X^c in the presence of one-body loss by switching off the Hamiltonian. We consider the Wasserstein distance between the initial distribution e^{-\gamma\tau}x_0 and the final distribution x_\tau, with the cost function is defined as c_{ij} = \|i-j\|^{\alpha_{\epsilon}}. Any feasible coupling π should satisfy \sum_j \pi_{ij} = e^{-\gamma\tau}x_j(0) and \sum_i \pi_{ij} = x_i(\tau). Accordingly, we obtain

\mu \leq x_Y(\tau) - e^{-\gamma\tau}x_{X^c}(0)

= \sum_{i \in Y, j \in \Lambda} \pi_{ij} - \sum_{i \in \Lambda, j \in X^c} \pi_{ij}

= \sum_{i \in Y, j \in X} \pi_{ij} - \sum_{i \in Y^c, j \in X^c} \pi_{ij} \leq \sum_{i \in Y, j \in X} \pi_{ij}. (29)

Therefore, the Wasserstein distance can be lower bounded by

W(e^{-\gamma\tau}x_0, x_\tau) \geq \min_{\pi} \sum_{i \in Y, j \in X} c_{ij} \pi_{ij} \geq \mu d_{XY}^{\alpha_{\epsilon}}. (30)

On the other hand, applying the Kantorovich-Rubinstein duality yields

W(e^{-\gamma\tau}x_0, x_\tau) = \max_w w^\top (x_\tau - e^{-\gamma\tau}x_0)

= \max_w \int_0^\tau dt \sum_{i \neq j} \tilde{\phi}_{ij}(t)(w_i - w_j)

\leq \frac{1}{2} \sum_{i \neq j} \int_0^\tau dt c_{ij} |\tilde{\phi}_{ij}(t)|, (31)

where we use the relations \tilde{\phi}_{ij}(t) = -\tilde{\phi}_{ji}(t) and |w_i - w_j| \leq c_{ij}. Meanwhile, the current \tilde{\phi}_{ij} can be upper bounded by

|\tilde{\phi}_{ij}(t)| = J_{ij}(t) |tr[(b_j^\dagger b_i - b_i^\dagger b_j)\rho_t]| e^{\gamma(t - \tau)} / N

\leq J_{ij}(t) [x_i(t) + x_j(t)] e^{\gamma(t - \tau)}. (32)

Consequently, the Wasserstein distance can be upper bounded as

W(e^{-\gamma\tau}x_0, x_\tau) \leq \sum_{i \neq j} \int_0^\tau dt J_{ij} x_i(t) e^{\gamma(t - \tau)}

\leq \sum_i \int_0^\tau dt \sum_{l=1}^\infty J \phi x_i(t) l^{\alpha - \alpha_{\epsilon} - D + 1} e^{\gamma(t - \tau)}

= \tau e^{-\gamma\tau} J \phi \zeta(\alpha - \alpha_{\epsilon} - D + 1), (33)

where we use Eq. (27) to obtain the last inequality. Therefore, by combining Eq. (33) with Eq. (30), we obtain the lower-bound condition for the transport time τ as

\tau e^{-\gamma\tau} \geq \kappa_{\epsilon} d_{XY}^{\alpha_{\epsilon}}. (34)

This completes the proof.

B. Outline of the proof for Result 2

For the case of local multi-body loss (i.e., L_i = b_i^n, n > 1), an analogy of the analysis in the previous section fails since we cannot exactly solve the dynamics of x_i(t) anymore. Nevertheless, with the help of the generalized Wasserstein distance (22), we can resolve the problem. In the presence of particle loss, the dynamics of x_i(t) can be written in a general form of

\dot{x}_i(t) = -d_i(t) + \sum_{j(\neq i)} \phi_{ij}(t). (35)

Here, d_i(t) := -(\gamma/2N)tr(\hat{d}_i \rho_t) represents the local loss rate of bosons at site i, \phi_{ij}(t) := 2N^{-1}J_{ij}(t) \text{Im}[tr(b_j^\dagger b_i \rho_t)] is the particle current exchanged between sites due to hopping, and \hat{d}_i := 2(b_i^\dagger)^n \hat{n}_i b_i^n - \hat{n}_i (b_i^\dagger)^n b_i^n - (b_i^\dagger)^n b_i^n \hat{n}_i. Crucially, we can show that the loss rate d_i(t) is nonnegative for any i as follows:

d_i(t) = \frac{\gamma}{N} \sum_{\vec{n}} \Theta(n_i - n) tr(\Pi_{\vec{n}} \rho_t) \frac{n_i!}{(n_i - n)!} (n_i - k) \geq 0, (36)

where \Pi_{\vec{n}} := |\vec{n}\rangle\langle\vec{n}|$ is the projection operator onto the state with particle distribution {n_i}. Define the generalized Wasserstein distance W(x_0, x_\tau) as W(x_0, x_\tau) := \min_{x_0 \succeq x' \succeq 0, x'' \succeq x_\tau, \|x'\|_1 = \|x''\|_1} W(x', x''). Applying the Kantorovich-Rubinstein duality, the distance W(x_0, x_\tau) can be upper bounded as follows:

W(x_0, x_\tau) = \max_{w} w^\top(x'' - x') \leq \frac{1}{2} \sum_{i \neq j} \int_0^\tau dt c_{ij} |\phi_{ij}(t)|, (37)

where the first equality is obtained by substituting the Kantorovich-Rubinstein duality (21) into the generalized Wasserstein distance (22) and the second line is achieved by assigning x' = x_0 and x'' = x_\tau. Similarly, its lower bound can be established as

W(x_0, x_\tau) \geq \mu d_{XY}^{\alpha_{\epsilon}}, (38)

where the last equality is obtained using the following fact:

\mu \leq x_Y(\tau) - x_{X^c}(0) \leq x''_Y - x'_{X^c} = \sum_{i \in Y, j \in X} \pi_{ij} - \sum_{i \in Y^c, j \in X^c} \pi_{ij} \leq \sum_{i \in Y, j \in X} \pi_{ij}. (39)

Following the same procedure as in Eq. (33) and noting that ∥x_t∥_1 \leq 1, we can show that

\frac{1}{2} \sum_{i \neq j} \int_0^\tau dt c_{ij} |\phi_{ij}(t)| \leq \tau J \phi \zeta(\alpha - \alpha_{\epsilon} - D + 1). (40)

Combining Eqs. (37), (38), and (40) yields the desired result.

C. Outline of the proof for Result 3

Under the Lindbladian dynamics (16), the dynamics of x_i is given by

\dot{x}_i(t) = \frac{1}{N} \sum_{j, j \neq i} 2J_{ij}(t) \text{Im}[tr(b_j^\dagger b_i \rho_t)] - \gamma_1 x_i(t) + \gamma_2 x_i(t) + \frac{\gamma_2}{N}. (41)

Here, \gamma_1 and \gamma_2 represent the loss and gain rates, respectively. By defining

\tilde{x}_i(t) := x_i e^{(\gamma_1 - \gamma_2)t} - \frac{N^{-1}(\gamma_2 - \gamma_1)}{\gamma_1 - \gamma_2} e^{(\gamma_1 - \gamma_2)t}, (42)

we obtain the evolution of \tilde{x}_i as

\frac{d}{dt} \tilde{x}_i(t) = \frac{1}{N} \sum_{j, j \neq i} 2J_{ij}(t) \text{Im}[tr(b_j^\dagger b_i \rho_t)] e^{(\gamma_1 - \gamma_2)t}. (43)

Therefore, the solution to x_i takes the form of

x_i(\tau) = x_i(0)e^{-\Delta\gamma\tau} + \frac{\gamma_2}{N\Delta\gamma}(1 - e^{-\Delta\gamma\tau}) + \int_0^\tau dt \frac{1}{N} \sum_{j, j \neq i} 2J_{ij}(t) \text{Im}[tr(b_j^\dagger b_i \rho_t)] e^{\Delta\gamma(t - \tau)}, (44)

where \Delta\gamma := \gamma_1 - \gamma_2. The first and second terms on the right-hand side represent the joint effect of the particle gain and loss, while the third term represents the particle transport. The total number evolves with time as

\sum_i x_i(\tau) = \sum_i x_i(0)e^{-\Delta\gamma\tau} + \gamma_2 N^{-1}(1 - e^{-\Delta\gamma\tau}) / \Delta\gamma = e^{-\Delta\gamma\tau} + \frac{\gamma_2 N^{-1}}{\Delta\gamma}(1 - e^{-\Delta\gamma\tau}). (45)

Therefore, we can define an effective particle current for transport

\tilde{\phi}_{ij}(t) := 2J_{ij}(t) \text{Im}[tr(b_j^\dagger b_i \rho_t)] e^{\Delta\gamma(t - \tau)} / N. (46)

Similar to the analysis in Sec. IV A, we here consider the Wasserstein distance between the distribution y_\tau and the final distribution x_\tau with the cost function defined as c_{ij} = \|i-j\|^{\alpha_{\epsilon}}. Here the distribution y_\tau satisfies

y_i(\tau) = y_i(0)e^{-\Delta\gamma\tau} + \frac{\gamma_2}{N\Delta\gamma}(1 - e^{-\Delta\gamma\tau}). (47)

We note that y_\tau represents the evolution of the particle distribution in the absence of hopping. If a fraction μ of the total particles is transported from X to Y in a time period τ, we have the following relation:

x_Y(\tau) - y_{X^c}(\tau) \geq \mu. (48)

Then a feasible coupling π should satisfy \sum_j \pi_{ij} = y_j(\tau) and \sum_i \pi_{ij} = x_i(\tau). Accordingly, we have

\mu \leq x_Y(\tau) - y_{X^c}(\tau) \leq \sum_{i \in Y, j \in X} \pi_{ij}. (49)

Therefore, the Wasserstein distance can be similarly lower bounded by

W(y_\tau, x_\tau) \geq \min_{\pi} \sum_{i \in Y, j \in X} c_{ij} \pi_{ij} \geq \mu d_{XY}^{\alpha_{\epsilon}}. (50)

To obtain the upper bound, we still apply the Kantorovich-Rubinstein duality and obtain

W(y_\tau, x_\tau) \leq \frac{1}{2} \sum_{i \neq j} \int_0^\tau dt c_{ij} |\tilde{\phi}_{ij}(t)|, (51)

where the analysis is the same as the one in Eq. (31). The current can also be upper bounded by

|\tilde{\phi}_{ij}(t)| = J_{ij}(t) |tr[(b_j^\dagger b_i - b_i^\dagger b_j)\rho_t]| e^{\Delta\gamma(t - \tau)} / N \leq J_{ij}(t) (x_i(t) + x_j(t)) e^{\Delta\gamma(t - \tau)}. (52)

Hence, we obtain an upper bound for the summation of the cost function, given by

W(y_\tau, x_\tau) \leq \sum_{i \neq j} \int_0^\tau dt J_{ij} l^{\alpha - \alpha_{\epsilon} - D + 1} e^{\Delta\gamma(t - \tau)} = A(\tau) J \phi \zeta(\alpha - \alpha_{\epsilon} - D + 1), (53)

where

A(\tau) := \tau e^{-\Delta\gamma\tau} + \frac{\gamma_2}{\Delta\gamma} \left( \frac{1 - e^{-\Delta\gamma\tau}}{\Delta\gamma} N^{-1} - \tau e^{-\Delta\gamma\tau} \right). (54)

Here we use Eq. (45) in the second inequality of Eq. (53). By combining Eq. (53) with Eq. (50), we obtain the final result

\tau e^{-\Delta\gamma\tau} + \frac{\gamma_2}{\Delta\gamma} \left( \frac{1 - e^{-\Delta\gamma\tau}}{\Delta\gamma} N^{-1} - \tau e^{-\Delta\gamma\tau} \right) \geq \kappa_{\epsilon} d_{XY}^{\alpha_{\epsilon}}, (55)

which completes the proof.

D. Outline of the proof for Result 4

Under the Lindbladian dynamics (2), the evolution equation of the probability distribution p_{\vec{n}}(t) := \langle\vec{n}|\rho_t|\vec{n}\rangle is generally shown as

\dot{p}_{\vec{n}}(t) = +i \sum_{i \neq j} J_{ij}(t) \sqrt{n_i n'_j} (\langle\vec{n}|\rho_t|\vec{n}'\rangle - \langle\vec{n}'|\rho_t|\vec{n}\rangle) + \gamma \sum_i \frac{(n_i + 1)!}{n_i!} p_{\vec{n}_i^+} - \gamma \sum_i \frac{n_i!}{n_i!} p_{\vec{n}}, (56)

where the state \vec{n}_i^+ is the state with the particle distribution {n_i + 1} for i ∈ Λ and n_j = n_j for j ≠ i. We note that the state \vec{n} with the particle distribution {n_i} satisfies n_i = n_i. The particle distribution {n'_i} of the state \vec{n}' satisfies n'_i = n_i - 1 and n'_j = n_j + 1 for i, j ∈ Λ. For a general case with arbitrary n-body loss, the traditional Wasserstein distance breaks down since the quantum jump process changes the support of the probability distribution. However, since the particle loss is local, we only consider two types of neighboring states. The first type is for any two vectors \vec{n} and \vec{m} that satisfy n_i - m_i = -(n_j - m_j) = ±1 for some i, j, and n_k = m_k for all k ≠ i, j. For these states, we define the cost as c_{\vec{m}\vec{n}} = \|i-j\|^{\alpha_{\epsilon}}. For the second type of states, we define two vectors \vec{n} and \vec{m} = \vec{n}_i^- that satisfy n_i - m_i = n for some i, and n_k = m_k for all k ≠ i. For these states, we define the cost as c_{\vec{m}\vec{n}} = 0. For general vectors \vec{n} and \vec{m} that satisfy ∥\vec{m}∥_1 \leq ∥\vec{n}∥_1, we can define the cost between them as the shortest path cost over all possible paths connecting the two states as follows:

c_{\vec{m}\vec{n}} = \min \sum_{k=1}^K c_{\vec{n}_k \vec{n}_{k-1}}, (57)

where \vec{n}_0 = \vec{n}, \vec{n}_K = \vec{m}, and \vec{n}_k and \vec{n}_{k-1} are neighboring states for all 1 ≤ k ≤ K. For general states \vec{m} and \vec{n} satisfying ∥\vec{m}∥_1 > ∥\vec{n}∥_1, we define the cost c_{\vec{m}\vec{n}} = \Omega + 1 where \Omega := \max_{\|\vec{m}\|_1 \leq \|\vec{n}\|_1} c_{\vec{m}\vec{n}}. One can show that the cost we defined satisfies the triangular inequality. The time evolution can be expressed as

\dot{p}_{\vec{n}}(t) = \sum_{i \neq j} \phi_{\vec{n}\vec{n}'}(t) + \sum_i [\varphi_{\vec{n}\vec{n}^+}(t) + \varphi_{\vec{n}\vec{n}^-}(t)], (58)

where

\phi_{\vec{n}\vec{n}'} := iJ \sqrt{n_i n'_j} (\langle\vec{n}|\rho_t|\vec{n}'\rangle - \langle\vec{n}'|\rho_t|\vec{n}\rangle), (59)

which holds even for asymmetric c_{ij}, we bound the Wasserstein distance W(p_0, p_\tau) as below:

W(p_0, p_\tau) \leq \max_{f_{\vec{m}} - f_{\vec{n}} \leq c_{\vec{m}\vec{n}}} f^\top(p_0 - p_\tau) \leq \int_0^\tau dt \sum_{\vec{n}, \vec{n}'} (f_{\vec{n}} - f_{\vec{n}'}) \phi_{\vec{n}\vec{n}'} + \sum_i (f_{\vec{n}} - f_{\vec{n}_i^+}) \varphi_{\vec{n}\vec{n}_i^+}, (61)

where the state \vec{n}_i^+ is the state with the particle distribution {n_i + 1} for i ∈ Λ and n_j = n_j for j ≠ i. Since |f_{\vec{n}} - f_{\vec{n}'}| \leq c_{\vec{n}\vec{n}'} and f_{\vec{n}} - f_{\vec{n}^+} \leq 0, we subsequently obtain

W(p_0, p_\tau) \leq \frac{1}{2} \int_0^\tau dt \sum_{\vec{n}, \vec{n}'} c_{\vec{n}\vec{n}'} |\phi_{\vec{n}\vec{n}'}(t)|. (62)

Since the current |\phi_{\vec{n}\vec{n}'}(t)| is the same as the one in closed quantum systems and has no relation to the dissipation, we can obtain W(p_0, p_\tau) \leq N\tau J \phi \zeta(\alpha - \alpha_{\epsilon} - D + 1). (63)

On the other hand, the Wasserstein distance can be rewritten as

W(p_0, p_\tau) = \min_{\pi} \sum_{\vec{m}, \vec{n}} c_{\vec{m}\vec{n}} \pi_{\vec{m}\vec{n}}, (64)

where the joint probability distribution π_{\vec{m}\vec{n}} satisfies \sum_{\vec{m}} \pi_{\vec{m}\vec{n}} = p_{\vec{n}}(\tau) and \sum_{\vec{n}} \pi_{\vec{m}\vec{n}} = p_{\vec{m}}(0). Since the cost is nonzero if and only if the transport process, rather than particle loss, takes place, the analysis is identical to that for closed quantum systems, and we can obtain

W(p_0, p_\tau) \geq \mu N d_{XY}^{\alpha_{\epsilon}} P_{x_Y(\tau) \geq x_{X^c}(0) + \mu}. (65)

Combining Eqs. (63) and (65), we achieve the same upper bound on the probability distribution as in closed quantum systems:

P_{x_Y(\tau) \geq x_{X^c}(0) + \mu} \leq \frac{J \phi \zeta(\alpha - \alpha_{\epsilon} - D + 1) \tau}{\mu d_{XY}^{\alpha_{\epsilon}}}, (66)

which completes the proof.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors present an extension of optimal transport theory to open quantum systems with long-range hopping and interactions. They highlight the importance of decoherence-free subspaces for transport over long distances. The results are derived analytically and framed within a general open-system formalism, aiming to extend known bounds on transport velocities to dissipative settings.\nWhile technically competent, the manuscript suffers from a lack of conceptual advances or striking results, making it unsuitable for a high-impact venue\nMajor issues\n- In the case of driven-dissipative systems, the authors mention the existence of decoherence-free subspaces but fail to relate them to the relatively longer distances over which transport can occur. In particular, it remains unexplained why, despite the existence of decoherence-free subspaces comprised of extensive states, transport in general remains limited to finite distances.\n- The authors emphasize that they treat quantum systems; however, the results equally apply to bosons and fermions, and the described transport appears to be entirely classical. Hence, the question arises to what extent these results are useful in any quantum applications. One could speculate that, due to the exponential loss of coherence in the presence of loss, the quantum mechanical description provided here offers no additional information compared to the equivalent classical statements derived from rate equations. Along these lines, it would be desirable, if possible applications were highlighted in the manuscript.\n- The discussion section concludes with a long, unstructured list of platforms and techniques that realize individual parts of the model under consideration. Instead, the authors should focus on those candidates that can (potentially) realize all parts of the model, including the required preparation and measurements. In the current form, the reader is left to wonder whether such an experiment is at all feasible. The paper does not convincingly bridge theory and experiment.\nMinor issues\n- In the first sentence of the abstract, '... provides a more general [...] perspective on particle transport.', it is not mentioned what perspective the present work is to be compared with.\n- The introduction should specify to which systems their treatment applies. In particular, it is incorrect to claim that all quantum many-body systems inevitably lose particles. It appears the authors exclusively refer to AMO systems, where this is typically the case, but disregard solid-state materials and platforms.\n- In Eq. (1), the index notation in h_Z is not accurately defined.\nThe paper is well written and accessible to a broad audience of physicists; it provides a useful generalization of optimal transport theory to open systems. However, in light of the major issues raised above, the manuscript does not meet the necessary criteria to warrant publication in a high-impact journal. In particular, the relevance of these results in light of the inevitable loss of coherence is insufficiently justified. In fact, it could be argued that similar results are obtained from the related classical rate equations. This lack of broader impact weakens its suitability for a multidisciplinary journal like Nature Communications. Furthermore, the manuscript offers limited conceptual progress, deriving only intuitively anticipated results.", "reviewer_2": "Report on the manuscript “Macroscopic Particle\nTransport in Dissipative Long-Range\nBosonic\nSystems” by Li, Shang, Kuwahara, and Vu\nSeptember 13, 2025\nThis paper considers macroscopic particle transport bounds in open quantum\nsystem driven by long-range Bose-Hubbard type Hamiltonians interacting with\nthe environment with (i) local 1-body loss, (ii) n-body loss, and (iii) simultane-\nous loss and gain (but with no net gain). Constraints on minimal time needed\nto transport a macroscopic fraction of particles from a region X to a disjoint\nregion Y are obtained in each of the cases (i)–(iii), constituting Results 1–3 in\nthe paper. Result 4 is a related upper bound on transport probability\nPriorworks, includingsomebytheauthorsofthearticleunderreview, have\ninvestigated the same problem for closed particle systems. Specifically, the\noptimal transport method, which is the main mechanism of the proof in the\npresent paper, has been used in Ref. [40] by the last two authors together with\nK. Saito to investigate this problem for the same type of Hamiltonian without\nparticledissipation. Thus,theobjectiveofthepaperunderreviewis,essentially,\nto extend the method of Ref. [40] to systems with dissipative interaction with\nthe environment.\nTechnically, though the proofs here appear to follow closely the correspond-\ning ones in Ref. [40], the results in the present paper are clearly not implied by\nprior work such as Ref. [40], since suitable modification of the Wasserstein dis-\ntance is needed for unbalanced distributions arising due to particle dissipation.\nIn some sense, the main technical advance of the present paper is proposing\nsuch a modification (see eq. (20)), and then establishing appropriate lower and\nupper bounds for it, which in turn leads to the desired transport bound.\nConceptually,thepaperpointstopossibleparadigmaticdifferencesintrans-\nportpropertiesbetween(A)opensystemswith1-bodyandmulti-bodyloss,and\n(B) closed and open systems in the dilute limit. Possible experimental setup to\nverify these differences are also indicated. These findings do not appear to be\npredicated by any prior works and could be of interest to practitioners.\n(Remarks to the Author)\nIbelievetheresultsofthepaperunderreviewcouldbeofgeneralinteresttothe\nreadearship of Nature Communications. The results are novel, clearly stated,\nand appear to be fundamentally correct. The proofs are based on established\nmethod and are easy to follow. Overall, I think the paper is well-written and\nscientifically sound. There are, however, a number of confusion in the analysis\nand possible improvement in the presentation, as I will explain further below.\nI would recommend the paper for publication in Nature Communications,\nprovided the author addresses the remarks listed below.\nIwouldstartwithsomemajorpointsofconfusion,mainlyregardingtherelation\namong Results 1–3 and their proofs, and then proceed to include some further\nminor issues. I wish to emphasize that the remarks below do not necessarily\nunderminethevalidityoftheresults,andagainitisoverallanicepaper,worthy\nof a NC publication after revision to ensure that all confusions are eliminated.\n1. MymainconfusionisabouttheuseofthegeneralizedWassersteindistance\nW(cid:102)(x,y), as defined in eq. (20). As mentioned earlier, this is perhaps the\nmost important technical advance in this paper, and therefore deserves\nthe most careful elucidation.\nOn the one hand, it is pointed out in the paragraph preceding (20) that\ntheclassicalWassersteindistanceW(x,y)requiresthedistributionstobe\nbalanced, i.e., ∥x∥ = ∥y∥ . On the other hand, it appears that only in\n1 1\nthe proof of Result 2 (Sect. IV.B) is W(cid:102)(x,y) actually used, whereas both\nSects. IV.A and C use W(x,y), albeit the distributions involved do not\nappear to be balanced; see, e.g., eqs. (28) and (47).\nItseemsnaturalthatthetreatmentfortheproofsinSects.IV.A–Cshould\nbeunifiedusingW(cid:102)(x,y). Ifthisisnotthecase,theauthorsshouldinclude\na justification for the difference in treatment.\n2. Another major point of confusion is the relation between Results 1–2. A\nhighlight of the paper, as pointed out in the abstract, is “a fundamental\ndistinction depending on whether the system experiences one-body loss\nor multi-body loss”. This distinction is attributed to “the existence of\ndecoherence-free subspaces” in the n-body loss scenario.\nAssuch,itshouldbepointedoutwhereexactlyintheanalysisinSect.IV.B\ndiddecoherence-freesubspacesplayarolethatleadstothequantitatively\ndifferent bound from the 1-body scenario.\nSpecifically,itispointedoutprecedingeq.(37)thatthisestimateisderived\nfrom “same procedure as in eq. (31) ”. But which aspect of the n-body\nloss, with n > 1, prevents us from getting the extra e−γτ factor as in\n(Remarks to the Author)\neq. (31)? This is crucial for the readers to understand the mechanism\nbehind the paradigmatic difference.\n3. A related issue to the point above is that, upon inspection, the inequality\n(8) appears to be independent of the loss parameter n in the Lindblad\noperator. Ifeq.(8)iscorrectasitstands,thenatleastashortjustification\nshould be included and, ideally, some explanation should be added in\nappropriate place in Sect. IV.B that explains this n-independence.\n4. Result 4 is very condensed as it stands. It is neither motivated from nor\ncompared with the preceding results, and no interpretation is given for\nthe estimate (17). As such, I have several questions about this result:\nFirst, what is the order of magnitude of N compared with N? I grant\n0\ncourse still valid). If this is indeed the case, then it would be natural to\nunify the notation with Results 1–3 by dividing N , ∆N , etc., all by N,\n0 0\nand reformulate eq. (17) in terms of the resulting fractions. This should\nlead to a result with comparable notations as in eq. (3).\nSecond,regardingtheproofof(17),theauthorssuggestprecedingeq.(60)\nonp.7(whichisexactly(17))thatthisestimatecanbederived“following\na similar analysis in closed quantum systems”. But in SM I only find a\nproof for closed system (essentially the same as in Ref. [40]), with little\nhintonhowtomodifyitinthepresenceofparticleloss. Icannotruleout\nthe possibility that the closed-system proof extend right away; but even\nif this is indeed the case, it should be indicated clearly in the main text,\nand the reason explained in the relevant places in SM. If this is not the\ncase, then please give a complete proof of (60).\nThird, what is the range of n in Result 4? In view of the distinction\nproposed in Results 1–2, one might expect (17) to be different for n = 1\nand n > 1. Also, estimate (17) again appears to be independent of n.\nPlease justify if this is indeed the case.\n5. ItappearsbothinthestatementandintheproofthatResult1isaspecial\ncase of Result 3 with γ = 0 and γ = γ. This seems natural and could\n2 1\nbe pointed out, e.g., immediately after the statement of Result 1.\n6. Page 1, right column. The second paragraph ends with “Thus, the max-\nimal speed for particle transport in open quantum systems remains elu-\nsive.” Insofar as I am aware, exactly this problem has been studied in\nBreteaux et al (EMS Press, 2022, Rev. Math. Phys. 2024) and Sigal-Wu\n(arXiv:2503.20635), for the case of few-body (i.e., zero-density) Hamilto-\nnian. Of course, these results do not extend readily to the many-body\ncase, so they do not imply any impairment on the novelty of paper under\nreview.\n7. p.2,leftcolumn. Thetop-leftparagraphendswith“Moreover,incorporat-\ning the results of multi-body loss, we demonstrate that the emergence of\n(Remarks to the Author)\ndecoherence-free subspaces inherently supports long-distance and perfect\n(i.e., 100%) transport. ” In addition to the issue about how decoherence-\nfree subspaces affect the proof, I also wonder how exactly does n-body\nloss supports perfect transport. I did not find where in the sequel is this\npoint expounded.\n8. p.2, eq. (3). Here the authors defined by words two macroscopic distri-\nbutions x (τ) and y (τ), as the Hamiltonian is switched on and off.\nY Xc\nHowever, thisnotationisnotusedconsistentlyintheproof. Forexample,\ncompare (26) and (45).\n9. p.3, left column, line 3. Readers could benefit from a quick derivation of\nthe equation for N (and, more generally, eq. (42)).\nt\n10. p.3, the paragraph after Result 2. As mentioned earlier, the difference\ndeserves more careful justification with quantitative analysis.\n11. p.3, Result 3. Here N seem to be time-dependent, and this should be\nincorporated in the notation. Perhaps this would lead to some change in\nestimate (10).\n12. In view of the preceding point, when discussing different thermodynamic\nregime, one should specify the asymptotic conditions as on which time\ninstance of N (presumably, on τ =0).\n13. p.4. At the beginning of the 3rd paragraph of Sect. III, it is written that\n“Furthermore,wealsoexpecttoobservethepossiblelong-rangetransport\nin the experiments enabled by the particle gain.”\nConceptually, I wonder how does one distinguish whether an observed in-\ncrementofparticlesinaregionY iscausedbymovementofparticlesfrom\nadisjointregionX thereto,orbytheparticlegainedfromtheenvironment\ndirectly landing in Y? This should be justified in order to substantiate\nthe claim cited above. Some quantitative analysis would be helpful.\n14. p.5, eq. (27). I don’t understand why the last inequality should hold. It\nwould make sense if the equality were ≤. Similarly for eqs. (36) and (46).\n15. p.5, bottom-right paragraph. Please include a short proof for the claim\ndˆ ≥0.\ni\n16. p.6, left column. Please explain why ∥x ∥ = ∥x˜∥ , as well as the first\n0 1 1\ninequality in (34).\n17. p.7, eq. (53): What’s the definition of N⃗? It is important to specify at\nwhichtimeinstance isN⃗ defined, sincethetotalnumberofparticleisnot\nconserved.\n18. p.7, both eq. (53) and (55) are said to be the evolution equation for p ,\nN⃗\nbutthelasttermslookdifferent. Pleasecheckand,ideally,includeashort\nderivation for the correct one.\n4"}, "author_response": "Response letter to referees\nWe thank both the referees for carefully reading our manuscript and for insightful\ncomments and criticisms. We are especially grateful to Referee 2 for supporting the\npublication of our work in Nature Communications. We also thank Referee 1 for the\nimportant criticism.\nBelow we list the summary of changes to the main text and the point-to-point response to\nthe reviewers’ comments before. In the main text, all the revisions have been highlighted\nin blue. We hope that the revised manuscript, together with our reply below, will meet with\nyour approval.\nSummary of Changes\n1. To address the concern of both Referee 1 and 2, we have written the specific form of\nthe decoherence-free subspace and discussed the role of it in Result 2 in the main text.\n2. To address the concern of Referee 1, we have stressed the long-distance transport in\nthe dilute limit in Result 3 and its relationship to the density of the dark state in the\ndecoherence-free subspace in the main text.\n3. To address the concern of Referee 1, we have compared our work with classical rate\nequations in the main text.\n4. To address the concern of Referee 1, we have restructured the discussion section to\nfurther explain the experimental setups to verify our theoretical results.\n5. To address the concern of Referee 1, we have emphasized that pure particle loss can\nonly take place in AMO systems instead of solid-state platform in the Introduction part\nin the main text.\n6. To address the concern of Referee 2, we have further explained the reason for\nintroducing a new generalized Wasserstein distance in the calculation steps in Results\n1 and 3 in the main text.\n7. To address the concern of Referee 2, we have discussed the reason why Eq. (8) is\nindependent of 𝑛 in Result 2.\n8. To address the concern of Referee 2, we have emphasized the magnitude of 𝑁 /𝑁\n!\nand Δ𝑁 /𝑁 and rewritten Result 4 correspondingly.\n!\n9. To address the concern of Referee 2, we have explained that the proof of Result 1 is\na special case of Result 3 in the statement of Result 3 in the main text.\n10. In response to Referee 2, we have cited relevant references on the transport of few-\nbody open quantum systems and compared our results with them in the Introduction\npart in the main text.\n11. To address the concern of Referee 2, we have added more details of the proof of Eq.\n(60) of the original main text (Eq. (66) in the revised main text) in the section IV.D.\n12. To address the concern of Referee 2, we have further explained how multi-body loss\ncan support perfect and long-distance transport in both Result 2 and Result 3 in the\nmain text.\n13. To address the concern of Referee 2, we have unified the notation of the macroscopic\ndistribution functions throughout the main text.\n14. To address the concern of Referee 2, we have added a brief derivation of 𝑁 in Result\n\"\n1 in the main text.\n15. To address the concern of Referee 2, we have stressed that 𝒩 is the initial particle\ndensity which is independent of time in Result 3 in the main text.\n16. To address the concern of Referee 2, we have further clarified the definition of the\ntransport in open quantum systems below Eq. (3) of the main text.\n17. To address the concern of Referee 2, we have fixed the typo in the last equality of Eqs.\n(27), (36) and (46) of the original main text (Eqs. (29), (39) and (49) in the revised main\ntext).\n18. To address the concern of Referee 2, we have included a brief proof for 𝑑' ≥0 in Eq.\n#\n(36) in Sec. IV.B.\n19. To address the concern of Referee 2, we have further explained the reason of the\nbalanced norms of 𝒙 and 𝒙+ and the first inequality in Eq. (34) of the original main\n!\ntext (Eq. (37) of the revised version).\n20. To address the concern of Referee 2, we have clarified the definition of 𝑁,,⃗ in Eq. (53)\nof the original main text (Eq. (56) of the revised one) in Sec. IV.D.\n21. Several minor changes have been made to correct typos and improve the readability\nof the manuscript.\nReply to Referee 1\nWe appreciate Referee 1 for his/her critical comments.\n“The authors present an extension of optimal transport theory to open quantum\nsystems with long-range hopping and interactions. They highlight the importance of\ndecoherence-free subspaces for transport over long distances. The results are derived\nanalytically and framed within a general open-system formalism, aiming to extend\nknown bounds on transport velocities to dissipative settings.\nWhile technically competent, the manuscript suffers from a lack of conceptual\nadvances or striking results, making it unsuitable for a high-impact venue”\nWe thank the referee for aptly summarizing the main results of our manuscript. Indeed,\nthere are several main contributions from our works: (i) We constructed the first general\noptimal transport theory in open quantum systems, and (ii) we discovered the\nenhancement of transport by decoherence-free subspaces. Especially in the case of both\ngain and loss, we have shown that even an arbitrarily small gain rate can facilitate long-\ndistance transport, which is counterintuitive since the particle gain plays a nontrivial role\nbeyond simply compensating for particle loss. While the contribution (i) is the technical\nadvancement, the contribution (ii) promotes the understanding of particle transport in open\nquantum systems and provides the striking result. We have further stressed the\ncontribution (ii) in the main text and carefully studied the major issues raised by the referee.\nLet us reply to each comment in the following.\n“Major issues\n- In the case of driven-dissipative systems, the authors mention the existence of\ndecoherence-free subspaces but fail to relate them to the relatively longer distances\nover which transport can occur. In particular, it remains unexplained why, despite the\nexistence of decoherence-free subspaces comprised of extensive states, transport in\ngeneral remains limited to finite distances.”\nWe thank the referee for this insightful comment, which helps us clarify our main result. In\nour work, there are two cases where decoherence-free subspaces (DFSs) can emerge:\none is the case with multi-body loss, and the other is the case with both gain and loss.\nWhen the system has local 𝑛-body loss with 𝑛 >1, there exists a DFS where the local\nparticle number is always less than 𝑛. Within such DFS, there exist an extensive number\nof quantum states that can realize non-dissipative transport for an arbitrary distance, where\nthe dissipative part in the Lindbladian does not contribute to the dynamics of density matrix\nand the system can be effectively considered as a closed quantum system.\nCorrespondingly, we have added more discussion to the relationship between the\npresence of DFS and the long-range transport in the part associated with Result 2.\nFor the other case with both particle gain and loss, after careful consideration, we have\nshown that there is only one state in the DFS (namely, the dark state) instead of extensive\nstates, which is exhibited in Eq. (13) in the revised main text. Unlike the case with multi-\nbody loss, the transport process in the driven-dissipative systems cannot be restricted in\nthe DFS. This is why the transport limit 𝑑 shown in Eq. (14) remains a finite value.\n$ \nHowever, we can see that in the dilute limit, the value of the transport limit 𝑑 in Eq. (15)\n$ \nis comparable with the size of the lattice even if we take the thermodynamic limit: |Λ|→∞.\nTherefore, the transported particles in the dilute limit can reach each position of the lattice\nin general and the transport is not limited to a finite distance in the lattice. The physical\nreason is that the particle gain can offset the loss of current into the environment if the\naverage density is lower than the particle-number density of the dark state in the DFS. In\nresponse to this point, we have discussed the long-distance transport of Eq. (15) and its\nrelation to the dark state in the DFS in the part associated with Result 3.\n“- The authors emphasize that they treat quantum systems; however, the results\nequally apply to bosons and fermions, and the described transport appears to be\nentirely classical. Hence, the question arises to what extent these results are useful in\nany quantum applications. One could speculate that, due to the exponential loss of\ncoherence in the presence of loss, the quantum mechanical description provided here\noffers no additional information compared to the equivalent classical statements\nderived from rate equations. Along these lines, it would be desirable, if possible\napplications were highlighted in the manuscript.”\nWe respectfully disagree with this comment of the referee report. Even though our results\nequally apply to bosonic and fermionic systems, our study highly relies on the quantum\nproperties of the Lindbladian or the system Hamiltonian. This point can be seen from Eq.\n(24) in the revised main text, which describes the time evolution of the local particle density:\nHere, the first term originates from the coherent dynamics of the density matrix in the basis\nof particle number, which has no correspondence in the classical rate equations, such as\nthe reaction-diffusion equation. Hence, the transport in the quantum systems cannot be\nsimply represented by a classical rate equation, even though the dissipation part looks like\nthe rate equation in classical chemical reactions. To clarify this point, we have added more\ncomparisons between our work and the rate equation in the main text.\n“- The discussion section concludes with a long, unstructured list of platforms and\ntechniques that realize individual parts of the model under consideration. Instead, the\nauthors should focus on those candidates that can (potentially) realize all parts of the\nmodel, including the required preparation and measurements. In the current form, the\nreader is left to wonder whether such an experiment is at all feasible. The paper does\nnot convincingly bridge theory and experiment.”\nWe thank the referee for this valuable comment. In the revised main text, we have\nintroduced a potential experimental platform—based on Rydberg-dressed neutral atoms\ntrapped in optical lattices or tweezer arrays (T. Macri and T. Pohl, Phys. Rev. A 89, 011402\n(2014), V. Borish et al., Phys. Rev. Lett. 124, 063601 (2020))—which can realize all parts\nof our model, including the required preparation and measurements. The details of feasible\nquantum simulation procedure based on this platform are depicted in Sec. IV. E in the main\ntext.\n“Minor issues\n- In the first sentence of the abstract, '... provides a more general [...] perspective on\nparticle transport.', it is not mentioned what perspective the present work is to be\ncompared with.”\nWe thank the referee for this helpful comment. Here the words “more general and\nexperimentally realistic” are used to compare our present work with the quantum transport\nlimit in closed quantum systems. Due to the common dissipative nature in ultracold atomic\nsystems, we need to consider the dissipative transport theory in general. If we take the\ndissipation rate to be 0, our theory can also reproduce the results in closed quantum\nsystems. To avoid ambiguity, we have revised the expression in the main text.\n“- The introduction should specify to which systems their treatment applies. In particular,\nit is incorrect to claim that all quantum many-body systems inevitably lose particles. It\nappears the authors exclusively refer to AMO systems, where this is typically the case,\nbut disregard solid-state materials and platforms.”\nWe are grateful to the referee for this comment. The Lindbladian dynamics with only\nparticle loss is only common in AMO systems and cannot be seen in condensed matter\nsystems. However, the case with both particle gain and loss can also be found in solid-\nstate materials and platforms. One famous example is the semiconductor microcavity,\nwhich concerns the exciton-polariton condensate in condensed matter systems (see M.\nWouters, et al., PRL 99, 140402 (2007)). To clarify this point, we have revised the\ncorresponding expressions in the main text.\n“- In Eq. (1), the index notation in h_Z is not accurately defined.”\nWe thank the referee for this comment. The index of ℎ has been explained above Eq.\n%\n(1), which represents an arbitrary subset of the lattice Λ. To avoid any misunderstanding,\nwe have moved the explanation of the index to the text below Eq. (1).\n“The paper is well written and accessible to a broad audience of physicists; it provides\na useful generalization of optimal transport theory to open systems. However, in light\nof the major issues raised above, the manuscript does not meet the necessary criteria\nto warrant publication in a high-impact journal. In particular, the relevance of these\nresults in light of the inevitable loss of coherence is insufficiently justified. In fact, it\ncould be argued that similar results are obtained from the related classical rate\nequations. This lack of broader impact weakens its suitability for a multidisciplinary\njournal like Nature Communications. Furthermore, the manuscript offers limited\nconceptual progress, deriving only intuitively anticipated results.”\nWe thank the referee for recognizing our work as a well-written and useful one. However,\nwe respectfully disagree with the referee’s statement that “similar results are obtained from\nthe related classical rate equation” since quantum coherence in the density matrix is\nintrinsic to quantum systems and does not have a classical correspondence. In addition,\nwe have substantially revised the manuscript to fully address the comments and criticism\nraised by the referee, especially by further specifying the relevance between the transport\ntime and the decoherence-free subspace in Result 2, and the arbitrarily long-distance\ntransport in Result 3 favored by even an arbitrarily small gain which is counterintuitive.\nTherefore, we believe that our results show conceptual progress on the optimal transport\ntheory in open quantum systems. We do hope that our revised manuscript will meet the\napproval by the referee for publication of our manuscript in Nature Communications.\nReply to Referee 2\nWe are very grateful to Referee 2 for his/her positive comments and pertinent questions.\nBelow let us reply to each of them.\n“1 Summary\nThis paper considers macroscopic particle transport bounds in open quantum system\ndriven by long-range Bose-Hubbard type Hamiltonians interacting with the\nenvironment with (i) local 1-body loss, (ii) n-body loss, and (iii) simultaneous loss and\ngain (but with no net gain). Constraints on minimal time needed to transport a\nmacroscopic fraction of particles from a region X to a disjoint region Y are obtained in\neach of the cases (i)–(iii), constituting Results 1–3 in the paper. Result 4 is a related\nupper bound on transport probability\nPrior works, including some by the authors of the article under review, have\ninvestigated the same problem for closed particle systems. Specifically, the\noptimal transport method, which is the main mechanism of the proof in the\npresent paper, has been used in Ref. [40] by the last two authors together with K. Saito to\ninvestigate this problem for the same type of Hamiltonian without particle dissipation.\nThus, the objective of the paper under review is, essentially, to extend the method of Ref.\n[40] to systems with dissipative interaction with the environment.\nTechnically, though the proofs here appear to follow closely the corresponding ones in\nRef. [40], the results in the present paper are clearly not implied by prior work such as\nRef. [40], since suitable modification of the Wasserstein distance is needed for\nunbalanced distributions arising due to particle dissipation. In some sense, the main\ntechnical advance of the present paper is proposing such a modification (see eq. (20)),\nand then establishing appropriate lower and upper bounds for it, which in turn leads to\nthe desired transport bound.\nConceptually, the paper points to possible paradigmatic differences in transport\nproperties between (A) open systems with 1-body and multi-body loss, and\n(B) closed and open systems in the dilute limit. Possible experimental setup to\nverify these differences are also indicated. These findings do not appear to be\npredicated by any prior works and could be of interest to practitioners.”\nWe thank the referee very much for aptly summarizing the main result of our manuscript\nand recognizing the strengths of our work.\n“2 Recommendation\nI believe the results of the paper under review could be of general interest to the\nreadearship of Nature Communications. The results are novel, clearly stated, and\nappear to be fundamentally correct. The proofs are based on established method and\nare easy to follow. Overall, I think the paper is well-written and scientifically sound.\nThere are, however, a number of confusion in the analysis and possible improvement\nin the presentation, as I will explain further below. I would recommend the paper for\npublication in Nature Communications, provided the author addresses the remarks\nlisted below. ”\nWe sincerely thank the referee for acknowledging our work as a well-written and\nscientifically sound one. We have carefully studied the questions and comments raised by\nthe referee. Let us reply to each one in the following.\n“3 Questions and comments\nI would start with some major points of confusion, mainly regarding the relation among\nResults 1–3 and their proofs, and then proceed to include some further minor issues.\nI wish to emphasize that the remarks below do not necessarily undermine the validity of\nthe results, and again it is overall a nice paper, worthy of a NC publication after revision\nto ensure that all confusions are eliminated.\n1. My main confusion is about the use of the generalized Wasserstein distance 𝑊6(𝒙,𝒚),\nas defined in eq. (20). As mentioned earlier, this is perhaps the most important\ntechnical advance in this paper, and therefore deserves the most careful elucidation.\nOn the one hand, it is pointed out in the paragraph preceding (20) that the classical\nWasserstein distance 𝑊(𝒙,𝒚) requires the distributions to be balanced, i.e., ;|𝑥|; =\n& \n;|𝑦|; . On the other hand, it appears that only in the proof of Result 2 (Sect. IV.B) is\n& \n𝑊6(𝒙,𝒚) actually used, whereas both Sects. IV.A and C use 𝑊(𝒙,𝒚), albeit the\ndistributions involved do not appear to be balanced; see, e.g., eqs. (28) and (47).\nIt seems natural that the treatment for the proofs in Sects. IV.A–C should be unified using\n𝑊6(𝒙,𝒚). If this is not the case, the authors should include a justification for the\ndifference in treatment.”\nWe thank the referee very much for this insightful comment. The generalized Wasserstein\ndistance 𝑊6(𝒙,𝒚) we introduced can be applied to general cases with 𝑛-body loss.\nHowever, the lower bound for the transport time in Eq. (8) in the main text derived from the\ngeneralized distance 𝑊6(𝒙,𝒚) is only tight for the case 𝑛 >1. The reason is that the multi-\nbody loss gives rise to a decoherence-free subspace (DFS) where the particle number\ndoes not decay during the transport. Therefore, the equality in Eq. (8) can only be taken\nwhen the transport process takes place in the DFS, which shows the same result as the\none in closed quantum systems. For the one-body-loss case, there does not exist a DFS\nsuch that particles can transport without particle loss, and the bound in Eq. (8) can be\nimproved. This is why we used the conventional Wasserstein distance to get a tighter\nrelation between the transport time and the distance for 𝑛 =1. For the case with both one-\nbody loss and one-body gain, the transport process generally cannot stay in the DFS,\nand we need to use the conventional Wasserstein distance to obtain a tighter relation.\nIt is worth noting that the particle distributions at the initial and final times are generally\nunbalanced for the dissipative case. Therefore, we consider the distance in Eq. (28) in the\noriginal main text (Eq. (30) in the revised version) as 𝑊(𝑒'()𝒙 ,𝒙 ) instead of simply\n! ) \n𝑊(𝒙 ,𝒙 ) as in closed quantum systems. Due to the one-body loss, the total particle\n! ) \nnumber at time 𝜏 is given by 𝑁 =𝑁 𝑒'(), indicating that the distributions 𝑒'()𝒙 and 𝒙\n) ! ! ) \nhave the same 1-norm. In other words, the two distributions 𝑒'()𝒙 and 𝒙 are balanced.\n! ) \nA similar analysis can also be applied to Eq. (47) in the original main text (Eq. (50) in the\nrevised version). Since the evolution of the total particle number for the case with both\nparticle gain and loss is shown in Eq. (45), we correspondingly define a distribution 𝒚 in\n) \nEq. (47) such that 𝒚 has the same 1-norm as 𝒙 . To avoid possible confusions, we have\n) ) \nadded more explanations on the Wasserstein distance used in Eqs. (28) and (47) in the\noriginal version (Eqs. (30) and (50) in the revised version) to the main text.\n(Note: Unless otherwise indicated, equation numbers refer to those in the revised version.)\n“2. Another major point of confusion is the relation between Results 1–2. A highlight of\nthe paper, as pointed out in the abstract, is “a fundamental distinction depending on\nwhether the system experiences one-body loss or multi-body loss”. This distinction is\nattributed to “the existence of decoherence-free subspaces” in the n-body loss scenario.\nAs such, it should be pointed out where exactly in the analysis in Sect. IV.B did\ndecoherence-free subspaces play a role that leads to the quantitatively different bound\nfrom the 1-body scenario.\nSpecifically, it is pointed out preceding eq. (37) that this estimate is derived from “same\nprocedure as in eq. (31)”. But which aspect of the n-body loss, with 𝑛 >1, prevents us\nfrom getting the extra 𝑒'() factor as in eq. (31)? This is crucial for the readers to\nunderstand the mechanism behind the paradigmatic difference.”\nWe are grateful to the referee for raising this important question. As mentioned by the\nreferee, the difference arises from the existence of the decoherence-free subspace (DFS).\nFor the case with multi-body loss, there exists a DFS given by 𝑃 =∑ |𝒏⟩⟨𝒏|, where |𝒏⟩\n𝒏 \nis the Mott state with no more than 𝑛−1 particle occupation on each site of the lattice.\nHence, if we consider the Bose-Hubbard model with strong repulsive interaction given by\nEq. (9) in the main text, the double occupancy will be forbidden, and the particles can\ntransport within the DFS without particle loss. More specifically, in the analysis in Sec. IV.B,\nthe dissipative loss 𝑑 (𝑡) in Eq. (36) can become zero in the presence of the DFS, and\n# \nthe dynamics becomes the same as that of closed systems for such cases. However, it is\nnot the case when the one-body loss is considered, where 𝑑 (𝑡) becomes strictly positive\n# \nand there does not exist a DFS such that dissipation is absent. Therefore, the result derived\nin Eq. (8) is not tight for the one-body-loss case. The better estimation for the transport\ntime is given in Eq. (4), where the extra 𝑒'() factor comes from particle loss.\nCorrespondingly, there is no factor 𝑒'() in Eq. (37) in the original main text since the upper\nbound is determined by the transport process within the DFS, where the particle loss is\nabsent. To clarify the role of the DFS, we have further explained the difference between\nthe case with 𝑛 >1 and the case with 𝑛 =1 in Result 2 in the main text.\n“3. A related issue to the point above is that, upon inspection, the inequality (8) appears\nto be independent of the loss parameter n in the Lindblad operator. If eq. (8) is correct\nas it stands, then at least a short justification should be included and, ideally, some\nexplanation should be added in appropriate place in Sect. IV.B that explains this n-\nindependence.”\nWe thank the referee for this comment. As we explained above, if the particle transport\ntakes place in the DFS when 𝑛 >1, there will be no particle loss and hence the relation\nbetween the transport time and the distance has no dependence on the value of 𝑛. We\nhave also clarified the reason for this independence of 𝑛 in Result 2 in the main text.\n“4. Result 4 is very condensed as it stands. It is neither motivated from nor compared\nwith the preceding results, and no interpretation is given for the estimate (17). As such,\nI have several questions about this result:\nFirst, what is the order of magnitude of 𝑁 compared with 𝑁? I grant 𝑁 /𝑁 is 𝑂(1),\n! ! \nbecause otherwise this bound would seem trivial (but of course still valid). If this is\nindeed the case, then it would be natural to unify the notation with Results 1–3 by\ndividing 𝑁 , 𝛥𝑁 , etc., all by 𝑁, and reformulate eq. (17) in terms of the resulting\n! ! \nfractions. This should lead to a result with comparable notations as in eq. (3).”\nWe thank the referee for this helpful comment. It is correct that the magnitude of 𝑁 /𝑁 is\n! \n𝑂(1). For the consistency of the main text, we have reformulated Eq. (17) with the ratios\n𝑁 /𝑁 and 𝛥𝑁 /𝑁, and comparable notations as in Eq. (3).\n! ! \n“Second, regarding the proof of (17), the authors suggest preceding eq. (60) on p.7\n(which is exactly (17)) that this estimate can be derived “following a similar analysis in\nclosed quantum systems”. But in SM I only find a proof for closed system (essentially\nthe same as in Ref. [40]), with little hint on how to modify it in the presence of particle\nloss. I cannot rule out the possibility that the closed-system proof extend right away;\nbut even if this is indeed the case, it should be indicated clearly in the main text,\nand the reason explained in the relevant places in SM. If this is not the case, then please\ngive a complete proof of (60).”\nWe thank the referee for this comment. The proof of Eq. (60) in the original version (Eq.\n(66) in the revised manuscript) from Eq. (59) (Eq. (62) in the revised manuscript) is nearly\nthe same as the analysis in the closed quantum systems. To avoid any confusion, we have\nadded more details on how to prove Eq. (60) from Eq. (59) (in the original version) to the\nmain text and revised the corresponding expressions in the Supplemental Material.\n“Third, what is the range of n in Result 4? In view of the distinction proposed in Results\n1–2, one might expect (17) to be different for 𝑛 =1 and 𝑛 >1. Also, estimate (17)\nagain appears to be independent of n. Please justify if this is indeed the case.”\nWe thank the referee for this question. Equation (17) in Result 4 has no restrictions on the\nvalue of 𝑛. As we explained above, there exists a DFS when 𝑛 >1 and the upper bound\nfor the probability of transporting particles is indeed tight. However, for the case with 𝑛 =\n1, this result is not tight enough yet. It is required to take all the possible trajectories for\nparticle transport into account for the microscopic transport in Result 4, which greatly\nincreases the complexity and difficulty to bound the probability compared with the\nmacroscopic transport. Therefore, we leave the possibly better bound for the probability of\nthe one-body-loss case to the future work since the microscopic transport is not our main\nfocus in this work. In addition, we have also added more interpretation of Eq. (17) to Result\n4 to illustrate that Result 2 is a corollary of Result 4.\n“5. It appears both in the statement and in the proof that Result 1 is a special case of\nResult 3 with 𝛾 =0 and 𝛾 =𝛾. This seems natural and could be pointed out, e.g.,\n+ & \nimmediately after the statement of Result 1.”\nWe are grateful to the referee for this suggestion. We have noted this in the statement of\nResult 3.\n“6. Page 1, right column. The second paragraph ends with “Thus, the maximal speed\nfor particle transport in open quantum systems remains elusive.” Insofar as I am aware,\nexactly this problem has been studied in Breteaux et al (EMS Press, 2022, Rev. Math.\nPhys. 2024) and Sigal-Wu (arXiv:2503.20635), for the case of few-body (i.e., zero-\ndensity) Hamiltonian. Of course, these results do not extend readily to the many-body\ncase, so they do not imply any impairment on the novelty of paper under review. ”\nWe are very grateful to the referee for pointing out these relevant references.\nCorrespondingly, we have also cited these references in the Introduction part and revised\nthe related expressions to clarify that our results should be considered for dissipative\nquantum many-body systems.\n“7. p.2, left column. The top-left paragraph ends with “Moreover, incorporating the\nresults of multi-body loss, we demonstrate that the emergence of decoherence-free\nsubspaces inherently supports long-distance and perfect (i.e., 100%) transport.” In\naddition to the issue about how decoherence-free subspaces affect the proof, I also\nwonder how exactly does n-body loss supports perfect transport. I did not find where\nin the sequel is this point expounded.”\nWe thank the referee for this comment. We have further explained how 𝑛-body loss with\n𝑛 >1 (i.e., there exists a decoherence-free subspace) supports long-distance and perfect\ntransport in Result 2.\n“8. p.2, eq. (3). Here the authors defined by words two macroscopic distributions\n𝑥\n, \n(𝜏) and 𝑦 -!(𝜏), as the Hamiltonian is switched on and off. However, this notation\nis not used consistently in the proof. For example, compare (26) and (45). ”\nWe are grateful to the referee for this comment. Indeed the notation is consistent with each\nother since we have 𝑦 -! (𝜏)=𝑒'()𝑥 -! (0) for the case with one-body loss if the\nHamiltonian is switched off. With this equation, we can see that Eq. (26) and Eq. (45) in\nthe original main text (Eq. (28) and Eq. (48) in the revised version) are consistent. To avoid\nany confusion, we have further explained Eq. (26) (in the original version) in the main text.\n“9. p.3, left column, line 3. Readers could benefit from a quick derivation of the equation\nfor 𝑁 (and, more generally, eq. (42)).”\n\" \nWe thank the referee for this valuable comment. We have added a brief derivation for 𝑁\n\" \nto the main text for the case with one-body loss. Equation (45) (Eq. (42) in the original main\ntext) follows directly from Eq. (44) by summing over all lattice sites 𝑖, where the hopping\nterms vanish since 𝐽 =−𝐽 is antisymmetric.\n#. .# \n“10. p.3, the paragraph after Result 2. As mentioned earlier, the difference deserves\nmore careful justification with quantitative analysis.”\nAs we explained above, we have further explained the difference between the Result 1\nand Result 2 in the main text.\n“11. p.3, Result 3. Here 𝒩 seem to be time-dependent, and this should be\nincorporated in the notation. Perhaps this would lead to some change in estimate (10).”\nWe thank the referee for this question. In Result 3, 𝒩 represents the particle density at\nthe initial time instead of at time 𝑡 and is time independent. To avoid any\nmisunderstanding, we have clarified the definition of 𝒩 in Result 3.\n“12. In view of the preceding point, when discussing different thermodynamic\nregime, one should specify the asymptotic conditions as on which time instance of 𝒩\n(presumably, on 𝜏 =0).”\nAs we explained above, 𝒩 is time independent, which has been clarified in the main text.\nThe thermodynamic limit is thus taken with respect to 𝑁 (the total particle number at the\ninitial time).\n“13. p.4. At the beginning of the 3rd paragraph of Sect. III, it is written that “Furthermore,\nwe also expect to observe the possible long-range transport in the experiments\nenabled by the particle gain.\nConceptually, I wonder how does one distinguish whether an observed increment of\nparticles in a region Y is caused by movement of particles from a disjoint region X\nthereto, or by the particle gained from the environment directly landing in Y?\nThis should be justified in order to substantiate the claim cited above. Some quantitative\nanalysis would be helpful.”\nWe thank the referee for this excellent question. The change in the particle number within\nregion Y generally has two contributions: (i) direct particle gain from the environment and\n(ii) particle flow from the region outside Y. As the referee correctly pointed out, in practice\nit may be difficult to distinguish whether particles arriving in Y originate from (i) or (ii). To\naddress this issue effectively, we define the transported particle number as the difference\nbetween the particle number in Y at time 𝜏 with the Hamiltonian switched on (i.e., the\nactual transport process) and the particle number at time 𝜏 with the Hamiltonian switched\noff under identical environmental conditions. Since the latter captures only the contribution\nfrom (i), this subtraction effectively eliminates the direct environmental gain from our\ndefinition. We have clarified this point more explicitly in the revised manuscript.\n“14. p.5, eq. (27). I don’t understand why the last inequality should hold. It would make\nsense if the equality were ≤. Similarly for eqs. (36) and (46).”\nWe thank the referee for this helpful comment. It is correct that the last equality in Eqs.\n(27), (36), and (46) of the original main text (Eqs. (29), (39), and (49) in the revised main\ntext) should all be ≤. We have fixed these typos in the main text.\n“15. p.5, bottom-right paragraph. Please include a short proof for the claim 𝑑 ≥0.”\n# \nWe thank the referee for this suggestion. We have now included the proof for 𝑑 ≥0 in\n# \nEq. (36) in the main text.\n“16. p.6, left column. Please explain why ||𝒙 || =||+𝒙|| , as well as the first inequality\n! & & \nin (34).”\nWe thank the referee for this good comment. From the definition of 𝒙+, we find that\nHere 𝜙 represents the current flow from the site 𝑗 to the site 𝑖, which is antisymmetric:\n#. \n𝜙 =−𝜙 . Then by taking the summation over the index 𝑖, we obtain ||𝒙 || =||𝒙+|| .\n#. .# ! & & \nIn addition, the first equality in Eq. (34) of the original main text (Eq. (37) in the revised main\ntext) can be obtained from the Kantorovich-Rubinstein duality, which is shown in Eq. (21).\nBy applying the Kantorovich-Rubinstein duality (21) to the Wasserstein distance 𝑊(𝒙,𝒚)\nin the generalized Wasserstein distance in Eq. (22), we have the first equality in Eq. (37)\n(Eq. (34) of the original version). We have further clarified these two points in the main text.\n“17. p.7, eq. (53): What’s the definition of 𝑁,,⃗? It is important to specify at which time\ninstance is 𝑁,,⃗ defined, since the total number of particle is not conserved.”\nWe thank the referee for this question. In the main text, we use 𝑁,,⃗ for two notations:\nparticle number vectors 𝑁,,⃗ and quantum states |𝑁,,⃗⟩. When we use single vector 𝑁,,⃗, it\nrepresents the vector state of particle numbers as 𝑁,,⃗ =𝑛 . On the other hand, for each 𝑁,,⃗,\n# # \nwe define the corresponding Mott states ;𝑁,,⃗Q=∏ |𝑛 ⟩, where |𝑛 ⟩ is the quantum state\n# # # \nwith 𝑛 bosons occupied at the site 𝑖. In this sense, both 𝑁,,⃗ and |𝑁,,⃗⟩ are time\n# \nindependent. What is time-dependent is the probability distribution 𝑝 (𝑡) for each particle\n/00⃗ \nnumber vector 𝑁,,⃗ in a quantum state. To clarify this point, we have further explained the\nmeaning of 𝑁,,⃗ in Eq. (53) of the original version (Eq. (56) in the revised main text) in the\nmain text.\n“18. p.7, both eq. (53) and (55) are said to be the evolution equation for 𝑝 , but the\n/00⃗ \nlast terms look different. Please check and, ideally, include a short derivation for the\ncorrect one.”\nWe thank the referee for this comment. Equations (53) and (55) of the original main text\n(Eqs. (56) and (58) for the revised one) are indeed equivalent to each other by substituting\nthe definition of 𝜑 and 𝜑 under Eq. (56) into Eq. (55) of the original main text (Eqs.\n/00⃗/00⃗# /00⃗/00⃗$ \n\" \" \n(59) and (58) for the revised one). These two quantities represent the probability flow from\nthe states 𝑁,,⃗2 and 𝑁,,⃗' to the state 𝑁,,⃗.\n# # \nIn conclusion, we have tried our best to address all the comments and criticisms raised by\nthe referees and revised the manuscript to reflect our reply to each of them. Now we believe\nthat our revised manuscript clarifies the unclear points pointed out by the referees and\nmeets the acceptance criteria of the journal. We do hope that our revised manuscript,\ntogether with our reply above, will meet with the editor’s approval for publication of our\nmanuscript in Nature Communications.\nSincerely yours,\nHongchao Li1, Cheng Shang2, Tomotaka Kuwahara2,3,4, and Tan Van Vu5\n1Department of Physics, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan\n2Analytical quantum complexity RIKEN Hakubi Research Team, RIKEN Center for\nQuantum Computing (RQC), Wako, Saitama 351-0198, Japan\n3PRESTO, Japan Science and Technology (JST), Kawaguchi, Saitama 332-0012, Japan\n4RIKEN Cluster for Pioneering Research (CPR), Wako, Saitama 351-0198, Japan\n5Center for Gravitational Physics and Quantum Information, Yukawa Institute for\nTheoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-\n8502, Japan\nEmail: lhc@cat.phys.s.u-tokyo.ac.jp"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I was previously critical of this manuscript; however, the authors’ detailed and thoughtful response to my comments and those of the second referee has fully addressed my main concerns. The revisions have significantly strengthened the manuscript, and I am satisfied that the work now meets the standards for publication in Nature Communications. I therefore recommend acceptance of the paper in its present form.", "reviewer_2": "This is a thorough revision, in which most of the comments from my previous report have been sufficiently addressed. The main results and key conceptual novelties have been further expounded, making the work more accessible to a broader readership.\nSome minor remaining suggestions:\n1. Regarding the claim “The emergence of decoherence-free subspaces inherently supports long-distance and perfect (i.e., 100%) transport” on p.2: Some reference to specific subsequent text that substantiates this claim would be helpful. For example, it seems to me that two different mechanisms are referred to in the sequel, one after the statement of result 2 and another after the statement of result 3. Perhaps these statements can be integrated to a separate section and compared with each other.\n2. On this note, regarding the first suggested mechanism on p. 3, the authors wrote “Consequently, the transport process effectively mimics that of a closed quantum system. Therefore, this nontrivial decoherence-free subspace enables perfect transport over … ” I find the implication in this statement confusing. Closed quantum system doesn’t seem to support perfect transport in the conventional sense.\n3. From the last paragraph on p.8 of the rebuttal, it seems that the authors misunderstood my earlier comment. I meant to say that the notation for y_(…) (\tau) was not used consistently, not that the equations themselves were inconsistent. This inconsistency remains in the revision. For example, eq. (28) and (48) seem to assert the exact same inequality, but with different notation.\n4. A thorough proofreading seems necessary. E.g., on the last line of p.3, left column, “subspace P” should be “subspace Ran P”; after eq. (28) it is written “where … and y_(…)(\tau) …”, but y didn’t appear in eq. (28).\nOverall, once again, I maintain that this paper constitutes an important contribution to the theory of quantum transport in open systems, with both conceptual and technical novelties, and I recommend this paper for publication."}, "author_response": "Response letter\nWe thank both the referees for carefully reading our manuscript and recommending the\npublication of our manuscript. Below we list the summary of changes to the main text and\nthe point-to-point response to the comments by Referee 2 before. We hope that the revised\nmanuscript, together with our reply below, will meet with your approval.\nSummary of Changes\n1. As suggested by Referee 2, we have specified that the claim “The emergence of\ndecoherence-free subspaces inherently supports long-distance and perfect (i.e., 100%)\ntransport” in the Introduction part is shown in Result 2.\n2. To address the concern of Referee 2, we have stressed that the protocols in Ref. [39]\ncan favor perfect transport in closed quantum systems in Result 2.\n3. To address the concern of Referee 2, we have unified the notations in Eqs. (28) and\n(48) to remove the inconsistency.\n4. As suggested by Referee 2, we have replaced “subspace P” with “subspace Ran P”.\n5. Several minor changes have been made to correct typos and improve the readability\nof the manuscript.\nReply to Referee 1\n“I was previously critical of this manuscript; however, the authors’ detailed and thoughtful\nresponse to my comments and those of the second referee has fully addressed my main\nconcerns. The revisions have significantly strengthened the manuscript, and I am\nsatisfied that the work now meets the standards for publication in Nature\nCommunications. I therefore recommend acceptance of the paper in its present form.”\nWe are especially grateful to the referee for supporting the publication of our manuscript.\nThe comments from the referee greatly help us improve the readability and quality of our\nmanuscript.\nReply to Referee 2\nWe are very grateful to Referee 2 for recommending the publication of our work and his/her\ninsightful comments. Below let us reply to each of them.\n“This is a thorough revision, in which most of the comments from my previous report\nhave been sufficiently addressed. The main results and key conceptual novelties have\nbeen further expounded, making the work more accessible to a broader readership.”\nWe thank the referee for recognizing the efforts we made to improve the manuscript.\n“Some minor remaining suggestions:\n1. Regarding the claim “The emergence of decoherence-free subspaces inherently\nsupports long-distance and perfect (i.e., 100%) transport” on p.2: Some reference to\nspecific subsequent text that substantiates this claim would be helpful. For example, it\nseems to me that two different mechanisms are referred to in the sequel, one after the\nstatement of result 2 and another after the statement of result 3. Perhaps these\nstatements can be integrated to a separate section and compared with each other.”\nWe thank the referee for this valuable comment. In the revised manuscript, we have\nspecified that the claim is related to Result 2 in the Introduction part.\n“2. On this note, regarding the first suggested mechanism on p. 3, the authors wrote\n“Consequently, the transport process effectively mimics that of a closed quantum\nsystem. Therefore, this nontrivial decoherence-free subspace enables perfect transport\nover … ” I find the implication in this statement confusing. Closed quantum system\ndoesn’t seem to support perfect transport in the conventional sense.”\nWe thank the referee for this question. Indeed, the perfect transport can be favored in\nclosed quantum systems. The protocols to realize perfect transport in closed bosonic\nsystems have been already raised in the discussion above Theorem 2 in Ref. [39]. We\nhave also further explained this point in Result 2.\n“3. From the last paragraph on p.8 of the rebuttal, it seems that the authors\nmisunderstood my earlier comment. I meant to say that the notation for y_(…) (\tau) was\nnot used consistently, not that the equations themselves were inconsistent. This\ninconsistency remains in the revision. For example, eq. (28) and (48) seem to assert the\nexact same inequality, but with different notation.”\nWe thank the referee for clarifying the earlier comment. We have now unified the notation\nin Eqs. (28) and (48) for consistency.\n“4. A thorough proofreading seems necessary. E.g., on the last line of p.3, left column,\n“subspace P” should be “subspace Ran P”; after eq. (28) it is written “where … and\ny_(…)(\tau) …”, but y didn’t appear in eq. (28).”\nWe thank the referee for the detailed comment. We have thoroughly rechecked our\nmanuscript and made several changes to the inconsistency. We believe that the revised\nmanuscript is ready for publication.\n“Overall, once again, I maintain that this paper constitutes an important contribution to\nthe theory of quantum transport in open systems, with both conceptual and technical\nnovelties, and I recommend this paper for publication.”\nWe thank the referee again for supporting the publication of our work. The comments from\nthe referee greatly help us improve the readability and quality of our manuscript.\nIn conclusion, we have fully taken all the comments raised by the Referee 2 into account\nand revised the manuscript accordingly. Now we believe that our revised manuscript\nclarifies the unclear points and is ready for the publication in Nature Communications.\nSincerely yours,\nHongchao Li1, Cheng Shang2, Tomotaka Kuwahara2,3,4, and Tan Van Vu5\n1Department of Physics, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan\n2Analytical quantum complexity RIKEN Hakubi Research Team, RIKEN Center for\nQuantum Computing (RQC), Wako, Saitama 351-0198, Japan\n3PRESTO, Japan Science and Technology (JST), Kawaguchi, Saitama 332-0012, Japan\n4RIKEN Cluster for Pioneering Research (CPR), Wako, Saitama 351-0198, Japan\n5Center for Gravitational Physics and Quantum Information, Yukawa Institute for\nTheoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-\n8502, Japan\nEmail: lhc@cat.phys.s.u-tokyo.ac.jp"}]

Decision letter