Abstract
In quantum mechanics, a fundamental law prevents quantum communications to simultaneously achieve high rates and long distances. This limitation is well known for pointtopoint protocols, where two parties are directly connected by a quantum channel, but not yet fully understood in protocols with quantum repeaters. Here we solve this problem bounding the ultimate rates for transmitting quantum information, entanglement and secret keys via quantum repeaters. We derive singleletter upper bounds for the endtoend capacities achievable by the most general (adaptive) protocols of quantum and private communication, from a single repeater chain to an arbitrarily complex quantum network, where systems may be routed through single or multiple paths. We analytically establish these capacities under fundamental noise models, including bosonic loss which is the most important for optical communications. In this way, our results provide the ultimate benchmarks for testing the optimal performance of repeaterassisted quantum communications.
Introduction
Today quantum technologies are being developed at a rapid pace^{1,2,3,4}. In this scenario, quantum communications are very advanced, with the development and implementation of a number of pointtopoint protocols of quantum key distribution (QKD)^{5}, based on discrete variable (DV) systems^{6,7,8}, such as qubits, or continuous variable (CV) systems, such as bosonic modes^{9,10}. Recently, we have also witnessed the deployment of highrate opticalbased secure quantum networks^{11,12}. These are advantageous not only for their multipleuser architecture but also because they may overcome the fundamental limitations that are associated with pointtopoint protocols of quantum and private communication.
After a long series of studies that started back in 2009 with the introduction of the reverse coherent information of a bosonic channel^{13,14}, ref. ^{15} finally showed that the maximum rate at which two remote parties can distribute quantum bits (qubits), entanglement bits (ebits), or secret bits over a lossy channel (e.g., an optical fiber) is equal to −log_{2}(1 − η), where η is the channel’s transmissivity. This limit is the Pirandola–Laurenza–Ottaviani–Banchi (PLOB) bound^{15} and cannot be surpassed even by the most powerful strategies that exploit arbitrary local operations (LOs) assisted by twoway classical communication (CC), also known as adaptive LOCCs^{16}.
To beat the PLOB bound, we need to insert a quantum repeater^{17} in the communication line. In information theory^{18,19,20,21}, a repeater or relay is any middle node helping the communication between two endparties. This definition is extended to quantum information theory, where quantum repeaters are middle nodes equipped with both classical and quantum operations, and may be arranged to compose linear chains or more general networks. In general, they do not need to have quantum memories (e.g., see ref. ^{22}) even though these are generally required for guaranteeing an optimal performance.
In all the ideal repeaterassisted scenarios, where we can beat the PLOB bound, it is fundamental to determine the maximum rates that are achievable by two endusers, i.e., to determine their endtoend capacities for transmitting qubits, distributing ebits, and generating secret keys. Finding these capacities not only is important to establish the boundaries of quantum network communications but also to benchmark practical implementations, so as to check how far prototypes of quantum repeaters are from the ultimate theoretical performance.
Here we address this fundamental problem. By combining methods from quantum information theory^{6,7,8,9,10} and classical networks^{18,19,20,21}, we derive tight singleletter upper bounds for the endtoend quantum and private capacities of repeater chains and, more generally, quantum networks connected by arbitrary quantum channels (these channels and the dimension of the quantum systems they transmit may generally vary across the network). More importantly, we establish exact formulas for these capacities under fundamental noise models for both DV and CV systems, including dephasing, erasure, quantumlimited amplification, and bosonic loss which is the most important for quantum optical communications. Depending on the routing in the quantum network (single or multipath), optimal strategies are found by solving the widest path^{23,24,25} or the maximum flow problem^{26,27,28,29} suitably extended to the quantum communication setting.
Our results and analytical formulas allow one to assess the rate performance of quantum repeaters and quantum communication networks with respect to the ultimate limits imposed by the laws of quantum mechanics.
Results
Ultimate limits of repeater chains
Consider Alice a and Bob b at the two ends of a linear chain of N quantum repeaters, labeled by r_{1}, …, r_{N}. Each point has a local register of quantum systems which may be augmented with incoming systems or depleted by outgoing ones. As also depicted in Fig. 1, the chain is connected by N + 1 quantum channels \(\{ {\cal{E}}_i\} = \{ {\cal{E}}_0, \ldots ,{\cal{E}}_i, \ldots ,{\cal{E}}_N\}\) through which systems are sequentially transmitted. This means that Alice transmits a system to repeater r_{1}, which then relays the system to repeater r_{2}, and so on, until Bob is reached.
Note that, in general, we may also have opposite directions for some of the quantum channels, so that they transmit systems towards Alice; e.g., we may have a middle relay receiving systems from both Alice and Bob. For this reason, we generally consider the “exchange” of a quantum system between two points by either forward or backward transmission. Under the assistance of twoway CCs, the optimal transmission of quantum information is related to the optimal distribution of entanglement followed by teleportation, so that it does not depend on the physical direction of the quantum channel but rather on the direction of the teleportation protocol.
In a single endtoend transmission or use of the chain, all the channels are used exactly once. Assume that the endpoints aim to share target bits, which may be ebits or private bits^{30,31}. The most general quantum distribution protocol \({\cal{P}}_{{\mathrm{chain}}}\) involves transmissions which are interleaved by adaptive LOCCs among all parties, i.e., LOs assisted by twoway CCs among endpoints and repeaters. In other words, before and after each transmission between two nodes, there is a session of LOCCs where all the nodes update and optimize their registers.
After n adaptive uses of the chain, the endpoints share an output state \(\rho _{{\mathbf{ab}}}^n\) with nR_{n} target bits. By optimizing the asymptotic rate lim_{n}R_{n} over all protocols \({\cal{P}}_{{\mathrm{chain}}}\), we define the generic twoway capacity of the chain \({\cal{C}}(\{ {\cal{E}}_i\} )\). If the target are ebits, the repeaterassisted capacity \({\cal{C}}\) is an entanglementdistribution capacity D_{2}. The latter coincides with a quantum capacity Q_{2}, because distributing an ebit is equivalent to transmitting a qubit if we assume twoway CCs. If the target are private bits, \({\cal{C}}\) is a secretkey capacity K ≥ D_{2} (with the inequality holding because ebits are specific private bits). Exact definitions and more details are given in Supplementary Note 1.
To state our upper bound for \({\cal{C}}(\{ {\cal{E}}_i\} )\), we introduce the notion of channel simulation, as generally formulated by ref. ^{15} (see also refs. ^{32,33,34,35,36,37} for variants). Recall that any quantum channel \({\cal{E}}\) is simulable by applying a tracepreserving LOCC \({\cal{T}}\) to the input state ρ together with some bipartite resource state σ, so that \({\cal{E}}(\rho ) = {\cal{T}}(\rho \otimes \sigma )\). The pair \(({\cal{T}},\sigma )\) represents a possible “LOCC simulation” of the channel. In particular, for channels that suitably commute with the random unitaries of teleportation^{4}, called “teleportationcovariant” channels^{15}, one finds that \({\cal{T}}\) is teleportation and σ is their Choi matrix \(\sigma _{\cal{E}}: = {\cal{I}} \otimes {\cal{E}}(\Phi )\), where Φ is a maximally entangled state. The latter is also known as “teleportation simulation”.
For bosonic channels, the Choi matrices are energyunbounded, so that simulations need to be formulated asymptotically. In general, an asymptotic state σ is defined as the limit of a sequence of physical states σ^{μ}, i.e., \(\sigma : = \mathop {\mathrm{{lim}}}\nolimits_\mu \sigma ^\mu\). The simulation of a channel \({\cal{E}}\) over an asymptotic state takes the form \(\left\Vert {{\cal{E}}(\rho )  {\cal{T}}(\rho \otimes \sigma ^\mu )} \right\Vert_1\mathop { \to }\limits^\mu 0\) where the LOCC \({\cal{T}}\) may also depend on μ in the general case^{15}. Similarly, any relevant functional on the asymptotic state needs to be computed over the defining sequence σ^{μ} before taking the limit for large μ. These technicalities are fully accounted in the Methods section.
The other notion to introduce is that of entanglement cut between Alice and Bob. In the setting of a linear chain, a cut “i” disconnects channel \({\cal{E}}_i\) between repeaters r_{i} and r_{i+1}. Such channel can be replaced by a simulation with some resource state σ_{i}. After calculations (see Methods), this allows us to write
where E_{R}(·) is the relative entropy of entanglement (REE). Recall that the REE is defined as^{38,39,40}
where SEP represents the ensemble of separable bipartite states and \(S(\sigma \gamma ): = {\mathrm{Tr}}\left[ {\sigma (\mathrm{log}_2\sigma  \mathrm{log}_2\gamma )} \right]\) is the relative entropy. In general, for any asymptotic state defined by the limit \(\sigma : = \mathrm{lim}_\mu \sigma ^\mu\), we may extend the previous definition and consider
where γ^{μ} is a converging sequence of separable states^{15}.
By minimizing Eq. (1) over all cuts, we may write
which establishes the ultimate limit for entanglement and key distribution through a repeater chain. For a chain of teleportationcovariant channels, we may use their teleportation simulation over Choi matrices and write
Note that the family of teleportationcovariant channels is large, including Pauli channels (at any dimension)^{7} and bosonic Gaussian channels^{9}. Within such a family, there are channels \({\cal{E}}\) whose generic twoway capacity \({\cal{C}} = Q_2\), D_{2} or K satisfies
where \(D_1(\sigma _{\cal{E}})\) is the oneway distillable entanglement of the Choi matrix (defined as an asymptotic functional in the bosonic case^{15}). These are called “distillable channels” and include bosonic lossy channels, quantumlimited amplifiers, dephasing and erasure channels^{15}.
For a chain of distillable channels, we therefore exactly establish the repeaterassisted capacity as
In fact the upper bound (≤) follows from Eqs. (5) and (6). The lower bound (≥) relies on the fact that an achievable rate for endtoend entanglement distribution consists in: (i) each pair, r_{i} and \({\mathbf{r}}_{i + 1}\), exchanging \(D_1(\sigma _{{\cal{E}}_i})\) ebits over \({\cal{E}}_i\); and (ii) performing entanglement swapping on the distilled ebits. In this way, at least \({\mathrm{min}}_{i} {D}_{1}(\sigma_{{\cal{E}}_i})\) ebits are shared between Alice and Bob.
Lossy chains
Let us specify Eq. (7) to an important case. For a chain of quantum repeaters connected by lossy channels with transmissivities \(\{ \eta _i\}\), we find the capacity
Thus, the minimum transmissivity within the lossy chain establishes the ultimate rate for repeaterassisted quantum/private communication between the endusers. For instance, consider an optical fiber with transmissivity η and insert N repeaters so that the fiber is split into N + 1 lossy channels. The optimal configuration corresponds to equidistant repeaters, so that \(\eta _{{\mathrm{min}}} = \root {{N + 1}} \of {\eta }\) and the maximum capacity of the lossy chain is
This capacity is plotted in Fig. 2 and compared with the pointtopoint PLOB bound \({\cal{C}}(\eta ) = {\cal{C}}_{{\mathrm{loss}}}(\eta ,0)\). A simple calculation shows that if we want to guarantee a performance of 1 target bit per use of the chain, then we may tolerate at most 3 dB of loss in each individual link. This “3dB rule” imposes a maximum repeaterrepeater distance of 15 km in standard optical fiber (at 0.2dB/km).
Quantum networks under singlepath routing
A quantum communication network can be represented by an undirected finite graph^{18} \({\cal{N}} = (P,E)\), where P is the set of points and E the set of all edges. Each point p has a local register of quantum systems. Two points p_{i} and p_{j} are connected by an edge \(({\mathbf{p}}_i,{\mathbf{p}}_j) \in E\) if there is a quantum channel \({\cal{E}}_{ij}: = {\cal{E}}_{{\mathbf{p}}_i{\mathbf{p}}_j}\) between them. By simulating each channel \({\cal{E}}_{ij}\) with a resource state \(\sigma _{ij}\), we simulate the entire network \({\cal{N}}\) with a set of resource states \(\sigma ({\cal{N}}) = \{ \sigma _{ij}\}\). A route is an undirected path \({\mathbf{a}}  {\mathbf{p}}_i  \cdots  {\mathbf{p}}_j  {\mathbf{b}}\) between the two endpoints, Alice a and Bob b. These are connected by an ensemble of possible routes \(\Omega = \{ 1, \ldots ,\omega , \ldots \}\), with the generic route ω involving the transmission through a sequence of channels \(\{ {\cal{E}}_0^\omega , \ldots ,{\cal{E}}_k^\omega \ldots \}\). Finally, an entanglement cut C is a bipartition (A, B) of P such that \({\mathbf{a}} \in {\mathbf{A}}\) and \({\mathbf{b}} \in {\mathbf{B}}\). Any such cut C identifies a super Alice A and a super Bob B, which are connected by the cutset \(\tilde C = \{ ({\mathbf{x}},{\mathbf{y}}) \in E:{\mathbf{x}} \in {\mathbf{A}},{\mathbf{y}} \in {\mathbf{B}}\}\). See the example in Fig. 3 and more details in Supplementary Notes 2 and 3.
Let us remark that the quantum network is here described by an undirected graph where the physical direction of the quantum channels \({\cal{E}}_{ij}\) can be forward (p_{i} → p_{j}) or backward (p_{j} → p_{i}). As said before for the repeater chains, this degree of freedom relies on the fact that we consider assistance by twoway CC, so that the optimal transmission of qubits can always be reduced to the distillation of ebits followed by teleportation. The logical flow of quantum information is therefore fully determined by the LOs of the points, not by the physical direction of the quantum channel which is used to exchange a quantum system along an edge of the network. This study of an undirected quantum network under twoway CC clearly departs from other investigations^{41,42,43}.
In a sequential protocol \({\cal{P}}_{{\mathrm{seq}}}\), the network is initialized by a preliminary network LOCC, where all the points communicate with each other via unlimited twoway CCs and perform adaptive LOs on their local quantum systems. With some probability, Alice exchanges a quantum system with repeater p_{i}, followed by a second network LOCC; then repeater p_{i} exchanges a system with repeater p_{j}, followed by a third network LOCC and so on, until Bob is reached through some route in a complete sequential use of the network (see Fig. 4). The routing is itself adaptive in the general case, with each node updating its routing table (probability distribution) on the basis of the feedback received by the other nodes. For large n uses of the network, there is a probability distribution associated with the ensemble Ω, with the generic route ω being used \(np_\omega\) times. Alice and Bob’s output state \(\rho _{{\mathbf{ab}}}^n\) will approximate a target state with \(nR_n\) bits. By optimizing over \({\cal{P}}_{{\mathrm{seq}}}\) and taking the limit of large n, we define the sequential or singlepath capacity of the network \({\cal{C}}({\cal{N}})\), whose nature depends on the target bits.
To state our upper bound, let us first introduce the flow of REE through a cut. Given an entanglement cut C of the network, consider its cutset \(\tilde C\). For each edge (x, y) in \(\tilde C\), we have a channel \({\cal{E}}_{{\mathbf{xy}}}\) and a corresponding resource state \(\sigma _{{\mathbf{xy}}}\) associated with a simulation. Then we define the singleedge flow of REE across cut C as
The minimization of this quantity over all entanglement cuts provides our upper bound for the singlepath capacity of the network, i.e.,
which is the network generalization of Eq. (4). For proof see Methods and further details in Supplementary Note 4.
In Eq. (11), the quantity \(E_{\mathrm{R}}(C)\) represents the maximum entanglement (as quantified by the REE) “flowing” through a cut. Its minimization over all the cuts bounds the singlepath capacity for quantum communication, entanglement distribution and key generation. For a network of teleportationcovariant channels, the resource state \(\sigma _{{\mathbf{xy}}}\) in Eq. (10) is the Choi matrix \(\sigma _{{\cal{E}}_{{\mathbf{xy}}}}\) of the channel \({\cal{E}}_{{\mathbf{xy}}}\). In particular, for a network of distillable channels, we may also set
for any edge (x, y). Therefore, we may refine the previous bound of Eq. (11) into \({\cal{C}}({\cal{N}}) \le \mathop {{\mathrm{min}}}\nolimits_C {\cal{C}}(C)\) where
is the maximum (singleedge) capacity of a cut.
Let us now derive a lower bound. First we prove that, for an arbitrary network, \(\mathop {\mathrm{min}}\nolimits_C {\cal{C}}(C) = \mathop {\mathrm{max}}\nolimits_\omega {\cal{C}}(\omega )\), where \({\cal{C}}(\omega ): = \mathop {{\mathrm{min}}}\nolimits_i {\cal{C}}({\cal{E}}_i^\omega )\) is the capacity of route ω (see Methods and Supplementary Note 4 for more details). Then, we observe that \({\cal{C}}(\omega )\) is an achievable rate. In fact, any two consecutive points on route ω may first communicate at the rate \({\cal{C}}({\cal{E}}_i^\omega )\); the distributed resources are then swapped to the endusers, e.g., via entanglement swapping or key composition at the minimum rate \(\mathop {\mathrm{min}}\nolimits_i {\cal{C}}({\cal{E}}_i^\omega )\). For a distillable network, this lower bound coincides with the upper bound, so that we exactly establish the singlepath capacity as
Finding the optimal route \(\omega _ \ast\) corresponds to solving the widest path problem^{24} where the weights of the edges \(({\mathbf{x}},{\mathbf{y}})\) are the twoway capacities \({\cal{C}}({\cal{E}}_{{\mathbf{xy}}})\). Route \(\omega _ \ast\) can be found via modified Dijkstra’s shortest path algorithm^{25}, working in time \(O(\left E \right\mathop {{\log}}\nolimits_2 \left P \right)\), where \(\left E \right\) is the number of edges and \(\left P \right\) is the number of points. Over route \(\omega _ \ast\) a capacityachieving protocol is non adaptive, with pointtopoint sessions of oneway entanglement distillation followed by entanglement swapping^{4}. In a practical implementation, the number of distilled ebits can be computed using the methods from ref. ^{44}. Also note that, because the swapping is on ebits, there is no violation of the Bellman’s optimality principle^{45}.
An important example is an optical lossy network \({\cal{N}}_{{\mathrm{loss}}}\) where any route ω is composed of lossy channels with transmissivities \(\{ \eta _i^\omega \}\). Denote by \(\eta _\omega : = \mathop {\mathrm{min}}\nolimits_i \eta _i^\omega\) the endtoend transmissivity of route ω. The singlepath capacity is given by the route with maximum transmissivity
In particular, this is the ultimate rate at which the two endpoints may generate secret bits per sequential use of the lossy network.
Quantum networks under multipath routing
In a network we may consider a more powerful routing strategy, where systems are transmitted through a sequence of multipoint communications (interleaved by network LOCCs). In each of these communications, a number M of quantum systems are prepared in a generally multipartite state and simultaneously transmitted to M receiving nodes. For instance, as shown in the example of Fig. 4, Alice may simultaneously sends systems to repeaters p_{1} and p_{2}, which is denoted by \({\mathbf{a}} \to \{ {\mathbf{p}}_1,{\mathbf{p}}_2\}\). Then, repeater p_{2} may communicate with repeater p_{1} and Bob b, i.e., \({\mathbf{p}}_2 \to \{ {\mathbf{p}}_1,{\mathbf{b}}\}\). Finally, repeater p_{1} may communicate with Bob, i.e., \({\mathbf{p}}_1 \to {\mathbf{b}}\). Note that each edge of the network is used exactly once during the endtoend transmission, a strategy known as “flooding” in computer networks^{46}. This is achieved by nonoverlapping multipoint communications, where the receiving repeaters choose unused edges for the next transmissions. More generally, each multipoint communication is assumed to be a pointtomultipoint connection with a logical sendertoreceiver(s) orientation but where the quantum systems may be physically transmitted either forward or backward by the quantum channels.
Thus, in a general quantum flooding protocol \({\cal{P}}_{{\mathrm{flood}}}\), the network is initialized by a preliminary network LOCC. Then, Alice a exchanges quantum systems with all her neighbor repeaters \({\mathbf{a}} \to \{ {\mathbf{p}}_k\}\). This is followed by another network LOCC. Then, each receiving repeater exchanges systems with its neighbor repeaters through unused edges, and so on. Each multipoint communication is interleaved by network LOCCs and may distribute multipartite entanglement. Eventually, Bob is reached as an endpoint in the first parallel use of the network, which is completed when all Bob’s incoming edges have been used exactly once. In the limit of many uses n and optimizing over \({\cal{P}}_{{\mathrm{flood}}}\), we define the multipath capacity of the network \({\cal{C}}^{\mathrm{m}}({\cal{N}})\).
As before, given an entanglement cut C, consider its cutset \(\tilde C\). For each edge (x, y) in \(\tilde C\), there is a channel \({\cal{E}}_{{\mathbf{xy}}}\) with a corresponding resource state \(\sigma _{{\mathbf{xy}}}\). We define the multiedge flow of REE through C as
which is the total entanglement (REE) flowing through a cut. The minimization of this quantity over all entanglement cuts provides our upper bound for the multipath capacity of the network, i.e.,
which is the multipath generalization of Eq. (11). For proof see Methods and further details in Supplementary Note 5. In a teleportationcovariant network we may simply use the Choi matrices \(\sigma _{{\mathbf{xy}}} = \sigma _{{\cal{E}}_{{\mathbf{xy}}}}\). Then, for a distillable network, we may use \(E_{\mathrm{R}}(\sigma _{{\cal{E}}_{{\mathbf{xy}}}}) = {\cal{C}}({\cal{E}}_{{\mathbf{xy}}})\) from Eq. (12), and write the refined upper bound \({\cal{C}}^{\mathrm{m}}({\cal{N}}) \le \mathop {\mathrm{min}}\nolimits_C {\cal{C}}^{\mathrm{m}}(C)\), where
is the total (multiedge) capacity of a cut.
To show that the upper bound is achievable for a distillable network, we need to determine the optimal flow of qubits from Alice to Bob. First of all, from the knowledge of the capacities \({\cal{C}}({\cal{E}}_{{\mathbf{xy}}})\), the parties solve a classical problem of maximum flow^{26,27,28,29} compatible with those capacities. By using Orlin’s algorithm^{47}, the solution can be found in \(O(P \times E)\) time. This provides an optimal orientation for the network and the rates \(R_{{\mathbf{xy}}} \le {\cal{C}}({\cal{E}}_{{\mathbf{xy}}})\) to be used. Then, any pair of neighbor points, x and y, distill \(nR_{{\mathbf{xy}}}\) ebits via oneway CCs. Such ebits are used to teleport \(nR_{{\mathbf{xy}}}\) qubits from x to y according to the optimal orientation. In this way, a number nR of qubits are teleported from Alice to Bob, flowing as quantum information through the network. Using the maxflow mincut theorem^{26,27,28,29,47,48,49,50,51,52,53}, we have that the maximum flow is \(n{\cal{C}}^{\mathrm{m}}(C_{{\mathrm{min}}})\) where \(C_{{\mathrm{min}}}\) is the minimum cut, i.e., \({\cal{C}}^{\mathrm{m}}(C_{{\mathrm{min}}}) = \mathop {\mathrm{min}}\nolimits_C {\cal{C}}^{\mathrm{m}}(C)\). Thus, that for a distillable \({\cal{N}}\), we find the multipath capacity
which is the multipath version of Eq. (14). This is achievable by using a non adaptive protocol where the optimal routing is given by Orlin’s algorithm^{47}.
As an example, consider again a lossy optical network \({\cal{N}}_{{\mathrm{loss}}}\) whose generic edge (x, y) has transmissivity \(\eta _{{\mathbf{xy}}}\). Given a cut C, consider its loss \(L_C: = \mathop {\prod}\nolimits_{({\mathbf{x}},{\mathbf{y}}) \in \tilde C} (1  \eta _{{\mathbf{xy}}})\) and define the total loss of the network as the maximization \(L_{\cal{N}}: = \mathop {\mathrm{max}}\nolimits_C L_C\). We find that the multipath capacity is just given by
It is interesting to make a direct comparison between the performance of single and multipath strategies. For this purpose, consider a diamond network \({\cal{N}}_{{\mathrm{loss}}}^\diamondsuit\) whose links are lossy channels with the same transmissivity η. In this case, we easily see that the multipath capacity doubles the singlepath capacity of the network, i.e.,
As expected the parallel use of the quantum network is more powerful than the sequential use.
Formulas for distillable chains and networks
Here we provide explicit analytical formulas for the endtoend capacities of distillable chains and networks, beyond the lossy case already studied above. In fact, examples of distillable channels are not only lossy channels but also quantumlimited amplifiers, dephasing and erasure channels. First let us recall their explicit definitions and their twoway capacities.
A lossy (pureloss) channel with transmissivity \(\eta \in (0,1)\) corresponds to a specific phaseinsensitive Gaussian channel which transforms input quadratures \({\hat{\mathbf{x}}} = (\hat q,\hat p)^T\) as \({\hat{\mathbf{x}}} \to \sqrt \eta {\hat{\mathbf{x}}} + \sqrt {1  \eta } {\hat{\mathbf{x}}}_E\), where E is the environment in the vacuum state^{9}. Its twoway capacities (Q_{2}, D_{2} and K) all coincide and are given by the PLOB bound^{15}
A quantumlimited amplifier with an associated gain g > 1 is another phaseinsensitive Gaussian channel but realizing the transformation \({\hat{\mathbf{x}}} \to \sqrt g {\hat{\mathbf{x}}} + \sqrt {g  1} {\hat{\mathbf{x}}}_E\), where the environment E is in the vacuum state^{9}. Its twoway capacities all coincide and are given by^{15}
A dephasing channel with probability p ≤ 1/2 is a Pauli channel of the form \(\rho \to (1  p)\rho + pZ\rho Z\), where Z is the phaseflip Pauli operator^{7}. Its twoway capacities all coincide and are given by^{15}
where \(H_2(p): =  p\mathop {{\log}}\nolimits_2 p  (1  p)\mathop {{\log}}\nolimits_2 (1  p)\) is the binary Shannon entropy. Finally, an erasure channel with probability \(p \le 1/2\) is a channel of the form \(\rho \to (1  p)\rho + p\left e \right\rangle \left\langle e \right\), where \(\left e \right\rangle \left\langle e \right\) is an orthogonal state living in an extra dimension^{7}. Its twoway capacities all coincide to^{15,54,55}
Consider now a repeater chain \(\{ {\cal{E}}_i\}\), where the channels \({\cal{E}}_i\) are distillable of the same type (e.g., all quantumlimited amplifiers with different gains g_{i}). The repeaterassisted capacity can be computed by combining Eq. (7) with one of the Eqs. (22)–(25). The final formulas are shown in the first column of Table 1. Then consider a quantum network \({\cal{N}} = (P,E)\), where each edge \(({\mathbf{x}},{\mathbf{y}}) \in E\) is described by a distillable channel \({\cal{E}}_{{\mathbf{xy}}}\) of the same type. For network \({\cal{N}}\), we may consider both a generic route \(\omega \in \Omega\), with sequence of channels \({\cal{E}}_i^\omega\), and a entanglement cut C, with corresponding cutset \(\tilde C\). By combining Eqs. (14) and (19) with Eqs. (22)–(25), we derive explicit formulas for the singlepath and multipath capacities. These are given in the second and third columns of Table 1 where we set
Let us note that the formulas for dephasing and erasure channels can be easily extended to arbitrary dimension d. In fact, a qudit erasure channel is formally defined as before and its twoway capacities are^{15,54,55}
Therefore, it is sufficient to multiply by \(\mathop {{\log}}\nolimits_2 d\) the corresponding expressions in Table 1. Then, in arbitrary dimension d, the dephasing channel is defined as
where p_{k} is the probability of k phase flips and \(Z_d^k\left i \right\rangle = {\mathrm{exp}}(2\pi ikd^{  1})\left i \right\rangle\). Its generic twoway capacity is^{15}
where \(H(\{ p_k\} ): =  \mathop {\sum}\nolimits_k p_k \log_2p_k\) is the Shannon entropy. Here the generalization is also simple. For instance, in a chain \(\{ {\cal{E}}_i\}\) of such ddimensional dephasing channels, we would have N + 1 distributions \(\{ p_k^i\}\). We then compute the most entropic distribution, i.e., we take the maximization \(\mathop {\mathrm{max}}\nolimits_i H(\{ p_k^i\} )\). This is the bottleneck that determines the repeater capacity, so that
Generalization to dimension d is also immediate for the two network capacities \({\cal{C}}\) and \({\cal{C}}^{\mathrm{m}}\).
Discussion
This work establishes the ultimate boundaries of quantum and private communications assisted by repeaters, from the case of a single repeater chain to an arbitrary quantum network under single or multipath routing. Assuming arbitrary quantum channels between the nodes, we have shown that the endtoend capacities are bounded by singleletter quantities based on the relative entropy of entanglement. These upper bounds are very general and also apply to chains and networks with untrusted nodes (i.e., run by an eavesdropper). Our theory is formulated in a general informationtheoretic fashion which also applies to other entanglement measures, as discussed in our Methods section. The upper bounds are particularly important because they set the tightest upper limits on the performance of quantum repeaters in various network configurations. For instance, our benchmarks may be used to evaluate performances in relayassisted QKD protocols such as MDIQKD and variants^{56,57,58}. Related literature and other developments^{59,60,61,62,63,64,65,66} are discussed in Supplementary Note 6.
For the lower bounds, we have employed classical composition methods of the capacities, either based on the widest path problem or the maximum flow, depending on the type of routing. In general, these simple and classical lower bounds do not coincide with the quantum upper bounds. However this is remarkably the case for distillable networks, for which the ultimate quantum communication performance can be completely reduced to the resolution of classical problems of network information theory. For these networks, widest path and maximum flow determine the quantum performance in terms of secret key generation, entanglement distribution and transmission of quantum information. In this way, we have been able to exactly establish the various endtoend capacities of distillable chains and networks where the quantum systems are affected by the most fundamental noise models, including bosonic loss, which is the most important for optical and telecom communications, quantumlimited amplification, dephasing and erasure. In particular, our results also showed how the parallel or “broadband” use of a lossy quantum network via multipath routing may greatly improve the endtoend rates.
Methods
We present the main techniques that are needed to prove the results of our main text. These methods are here provided for a more general entanglement measure E_{M}, and specifically apply to the REE. We consider a quantum network \({\cal{N}}\) under single or multipath routing. In particular, a chain of quantum repeaters can be treated as a singleroute quantum network.
For the upper bounds, our methodology can be broken down in the following steps: (i) Derivation of a general weak converse upper bound in terms of a suitable entanglement measure (in particular, the REE); (ii) Simulation of the quantum network, so that quantum channels are replaced by resource states; (iii) Stretching of the network with respect to an entanglement cut, so that Alice and Bob’s shared state has a simple decomposition in terms of resource states; (iv) Data processing, subadditivity over tensorproducts, and minimization over entanglement cuts. These steps provide entanglementbased upper bounds for the endtoend capacities. For the lower bounds, we perform a suitable composition of the pointtopoint capacities of the singlelink channels by means of the widest path and the maximum flow, depending on the routing. For the case of distillable quantum networks (and chains), these lower bounds coincide with the upper bounds expressed in terms of the REE.
General (weak converse) upper bound
This closely follows the derivation of the corresponding pointtopoint upper bound first given in the second 2015 arXiv version of ref. ^{15} and later reported as Theorem 2 in ref. ^{16}. Consider an arbitrary endtoend \((n,R_n^\varepsilon ,\varepsilon )\) network protocol \({\cal{P}}\) (single or multipath). This outputs a shared state \(\rho _{{\mathbf{ab}}}^n\) for Alice and Bob after n uses, which is εclose to a target private state^{30,31} ϕ^{n} having \(nR_n^\varepsilon\) secret bits, i.e., in trace norm we have \(\left\ {\rho _{{\mathbf{ab}}}^n  \phi ^n} \right\_1 \le \varepsilon\). Consider now an entanglement measure E_{M} which is normalized on the target state, i.e.,
Assume that E_{M} is continuous. This means that, for ddimensional states ρ and σ that are close in trace norm as \(\left\Vert {\rho  \sigma } \right\Vert_1\, \le \varepsilon\), we may write
with the functions g and h converging to zero in ε. Assume also that E_{M} is monotonic under tracepreserving LOCCs \(\bar \Lambda\), so that
a property which is also known as data processing inequality. Finally, assume that E_{M} is subadditive over tensor products, i.e.,
All these properties are certainly satisfied by the REE E_{R} and the squashed entanglement (SQ) E_{SQ}, with specific expressions for g and h (e.g., these expressions are explicitly reported in Sec. VIII.A of ref. ^{16}).
Using the first two properties (normalization and continuity), we may write
where d is the dimension of the target private state. We know that this dimension is at most exponential in the number of uses, i.e., \(\log_2d \le \alpha nR_n^\varepsilon\) for constant α (e.g., see ref. ^{15} or Lemma 1 in ref. ^{16}). By replacing this dimensional bound in Eq. (39), taking the limit for large n and small ε (weak converse), we derive
Finally, we take the supremum over all protocols \({\cal{P}}\) so that we can write our general upper bound for the endtoend secret key capacity (SKC) of the network
In particular, this is an upper bound to the singlepath SKC \({\cal{K}}\) if \({\cal{P}}\) are singlepath protocols, and to the multipath SKC \({\cal{K}}^m\) if \({\cal{P}}\) are multipath (flooding) protocols.
In the case of an infinitedimensional state \(\rho _{{\mathbf{ab}}}^n\), the proof can be repeated by introducing a truncation tracepreserving LOCC T, so that \(\delta _{{\mathbf{ab}}}^n = {\boldsymbol{T}}(\rho _{{\mathbf{ab}}}^n)\) is a finitedimensional state. The proof is repeated for \(\delta _{{\mathbf{ab}}}^n\) and finally we use the data processing \(E_{\mathrm{M}}(\delta _{{\mathbf{ab}}}^n) \le E_{\mathrm{M}}(\rho _{{\mathbf{ab}}}^n)\) to write the same upper bound as in Eq. (41). This follows the same steps of the proof given in the second 2015 arXiv version of ref. ^{15} and later reported as Theorem 2 in ref. ^{16}. It is worth mentioning that Eq. (41) can equivalently be proven without using the exponential growth of the private state, i.e., using the steps of the third proof given in the Supplementary Note 3 of ref. ^{15}.
Network simulation
Given a network \({\cal{N}} = (P,E)\) with generic point \({\mathbf{x}} \in P\) and edge \(({\mathbf{x}},{\mathbf{y}}) \in E\), replace the generic channel \({\cal{E}}_{{\mathbf{xy}}}\) with a simulation over a resource state σ_{xy}. This means to write \({\cal{E}}_{{\mathbf{xy}}}(\rho ) = {\cal{T}}_{{\mathbf{xy}}}(\rho \otimes \sigma _{{\mathbf{xy}}})\) for any input state ρ, by resorting to a suitable tracepreserving LOCC \({\cal{T}}_{{\mathbf{xy}}}\) (this is always possible for any quantum channel^{15}). If we perform this operation for all the edges, we then define the simulation of the network \(\sigma ({\cal{N}}) = \{ \sigma _{{\mathbf{xy}}}\} _{({\mathbf{x}},{\mathbf{y}}) \in E}\) where each channel is replaced by a corresponding resource state. If the channels are bosonic, then the simulation is typically asymptotic of the type \({\cal{E}}_{{\mathbf{xy}}}(\rho ) = \mathop {{\lim}}\nolimits_\mu {\cal{E}}_{{\mathbf{xy}}}^\mu (\rho )\) where \({\cal{E}}_{{\mathbf{xy}}}^\mu (\rho ) = {\cal{T}}_{{\mathbf{xy}}}^\mu (\rho \otimes \sigma _{{\mathbf{xy}}}^\mu )\) for some sequence of simulating LOCCs \({\cal{T}}_{{\mathbf{xy}}}^\mu\) and sequence of resource states \(\sigma _{{\mathbf{xy}}}^\mu\).
Here the parameter μ is usually connected with the energy of the resource state. For instance, if \({\cal{E}}_{{\mathbf{xy}}}\) is a teleportationcovariant bosonic channel, then the resource state \(\sigma _{{\mathbf{xy}}}^\mu\) is its quasiChoi matrix \(\sigma _{{\cal{E}}_{{\mathbf{xy}}}}^\mu : = {\cal{I}} \otimes {\cal{E}}_{{\mathbf{xy}}}({\mathrm{\Phi }}^\mu )\), with \(\Phi ^\mu\) being a twomode squeezed vacuum state (TMSV) state^{9} whose parameter \(\mu = \bar n + 1/2\) is related to the mean number \(\bar n\) of thermal photons. Similarly, the simulating LOCC \({\cal{T}}_{{\mathbf{xy}}}^\mu\) is a BraunsteinKimble protocol^{67,68} where the ideal Bell detection is replaced by the finiteenergy projection onto αdisplaced TMSV states \(D(\alpha ){\mathrm{\Phi }}^\mu D(  \alpha )\), with D being the phasespace displacement operator^{9}.
Given an asymptotic simulation of a quantum channel, the associated simulation error is correctly quantified by employing the energyconstrained diamond distance^{15}, which must go to zero in the limit, i.e.,
Recall that, for any two bosonic channels \({\cal{E}}\) and \({\cal{E}}'\), this quantity is defined as
where \(D_{\bar N}\) is the compact set of bipartite bosonic states with \(\bar N\) mean number of photons (see ref. ^{69} for a later and slightly different definition, where the constraint is only on the B part). Thus, in general, if the network has bosonic channels, we may write the asymptotic simulation \(\sigma ({\cal{N}}) = \mathop {{\lim}}\nolimits_\mu \sigma ^\mu ({\cal{N}})\) where \(\sigma ^\mu ({\cal{N}}): = \{ \sigma _{{\mathbf{xy}}}^\mu \} _{({\mathbf{x}},{\mathbf{y}}) \in E}\).
Stretching of the network
Once we simulate a network, the next step is its stretching, which is the complete adaptivetoblock simplification of its output state (for the exact details of this procedure see Supplementary Note 3). As a result of stretching, the nuse output state of the generic network protocol can be decomposed as
where \(\bar \Lambda\) represents a tracepreserving LOCC (which is local with respect to Alice and Bob). The LOCC \(\bar \Lambda\) includes all the adaptive LOCCs from the original protocol besides the simulating LOCCs. In Eq. (44), the parameter n_{xy} is the number of uses of the edge (x, y), that we may always approximate to an integer for large n. We have n_{xy} ≤ n for singlepath routing, and n_{xy} = n for flooding protocols in multipath routing.
In the presence of bosonic channels and asymptotic simulations, we modify Eq. (44) into the approximate stretching
which tends to the actual output \(\rho _{{\mathbf{ab}}}^n\) for large μ. In fact, using a “peeling” technique^{15,16} which exploits the triangle inequality and the monotonicity of the trace distance under completelypositive tracepreserving maps, we may write the following bound
which goes to zero in μ for any finite input energy \(\bar N\), finite number of uses n of the protocol, and finite number of edges E in the network (the explicit steps of the proof can be found in Supplementary Note 3).
Stretching with respect to entanglement cuts
The decomposition of the output state can be greatly simplified by introducing cuts in the network. In particular, we may drastically reduce the number of resource states in its representation. Given a cut C of \({\cal{N}}\) with cutset \(\tilde C\), we may in fact stretch the network with respect to that specific cut (see again Supplementary Note 3 for exact details of the procedure). In this way, we may write
where \(\bar \Lambda _{{\mathbf{ab}}}\) is a tracepreserving LOCC with respect to Alice and Bob (differently from before, this LOCC now depends on the cut C, but we prefer not to complicate the notation). Similarly, in the presence of bosonic channels, we may consider the approximate decomposition
which converges in trace distance to \(\rho _{{\mathbf{ab}}}^n(C)\) for large μ.
Data processing and subadditivity
Let us combine the stretching in Eq. (47) with two basic properties of the entanglement measure E_{M}. The first property is the monotonicity of E_{M} under tracepreserving LOCCs; the second property is the subadditivity of E_{M} over tensorproduct states. Using these properties, we can simplify the general upper bound of Eq. (41) into a simple and computable singleletter quantity. In fact, for any cut C of the network \({\cal{N}}\), we write
where \(\bar \Lambda _{{\mathbf{ab}}}\) has disappeared. Let us introduce the probability of using the generic edge (x, y)
so that we may write the limit
Using the latter in Eq. (41) allows us to write the following bound, for any cut C
In the case of bosonic channels and asymptotic simulations, we may use the triangle inequality
Then, we may repeat the derivations around Eqs. (39)–(41) for \(\rho _{{\mathbf{ab}}}^{n,\mu }\) instead of \(\rho _{{\mathbf{ab}}}^n\), where we also include the use of a suitable truncation of the states via a tracepreserving LOCC T (see also Sec. VIII.D of ref. ^{16} for a similar approach in the pointtopoint case). This leads to the μdependent upperbound
Because this is valid for any μ, we may conservatively take the inferior limit in μ and consider the upper bound
Finally, by introducing the stretching of Eq. (48) with respect to an entanglement cut C, and using the monotonicity and subadditivity of E_{M} with respect to the decomposition of \(\rho _{{\mathbf{ab}}}^{n,\mu }(C)\), we may repeat the previous reasonings and write
which is a direct extension of the bound in Eq. (53).
We may formulate both Eqs. (53) and (57) in a compact way if we define the entanglement measure E_{M} over an asymptotic state \(\sigma : = \mathop {{\lim}}\nolimits_\mu \sigma ^\mu\) as
It is clear that, for a physical (nonasymptotic) state, we have the trivial sequence σ^{μ} = σ for any μ, so that Eq. (58) provides the standard definition. In the specific case of REE, we may write
where γ^{μ} is a sequence of separable states that converges in trace norm; this means that there exists a separable state γ such that \(\left\ {\gamma ^\mu  \gamma } \right\_1\mathop { \to }\limits^\mu 0\). Employing the extended definition of Eq. (58), we may write Eq. (53) for both nonasymptotic σ_{xy} and asymptotic states \(\sigma _{{\mathbf{xy}}}: = \mathop {{\lim}}\nolimits_\mu \sigma _{{\mathbf{xy}}}^\mu\).
Minimum entanglement cut and upper bounds
By minimizing Eq. (53) over all possible cuts of the network, we find the tightest upper bound, i.e.,
Let us now specify this formula for different types of routing. For singlepath routing, we have \(p_{{\mathbf{xy}}} \le 1\), so that we may use
in Eq. (53). Therefore, we derive the following upper bound for the singlepath SKC
where we introduce the singleedge flow of entanglement through the cut
In particular, we may specify this result to a single chain of N points and N + 1 channels \(\{ {\cal{E}}_i\}\) with resource states {σ_{i}}. This is a quantum network with a single route, so that the cuts can be labeled by i and the cutsets are just composed of a single edge. Therefore, Eqs. (62) and (63) become
For multipath routing, we have p_{xy} = 1 (flooding), so that we may simplify
in Eq. (53). Therefore, we can write the following upper bound for the multipath SKC
where we introduce the multiedge flow of entanglement through the cut
In these results, the definition of E_{M}(σ_{xy}) is implicitly meant to be extended to asymptotic states, according to Eq. (58). Then, note that the tightest values of the upper bounds are achieved by extending the minimization to all network simulations \(\sigma ({\cal{N}})\), i.e., by enforcing \(\min_C \to \min_{\sigma ({\cal{N}})}\min_C\) in Eqs. (62) and (66).
Specifying Eqs. (62), (64), and (66) to the REE, we get the singleletter upper bounds
which are Eqs. (4), (11) and (17) of the main text. The proofs of these upper bounds in terms of the REE can equivalently be done following the “converse part” derivations in Supplementary Note 1 (for chains), Supplementary Note 4 (for networks under singlepath routing), and Supplementary Note 5 (for networks under multipath routing). Differently from what presented in this Methods section, such proofs exploit the lower semicontinuity of the quantum relative entropy^{8} in order to deal with asymptotic simulations (e.g., for bosonic channels).
Lower bounds
To derive lower bounds we combine the known results on twoway assisted capacities^{15} with classical results in network information theory. Consider the generic twoway assisted capacity \({\cal{C}}_{{\mathbf{xy}}}\) of the channel \({\cal{E}}_{{\mathbf{xy}}}\) (in particular, this can be either D_{2} = Q_{2} or K). Then, using the cut property of the widest path (Supplementary Note 4), we derive the following achievable rate for the generic singlepath capacity of the network \({\cal{N}}\)
For a chain \(\{ {\cal{E}}_i\}\), this simply specifies to
Using the classical maxflow mincut theorem (Supplementary Note 5), we derive the following achievable rate for the generic multipath capacity of \({\cal{N}}\)
Simplifications for teleportationcovariant and distillable networks
Recall that a quantum channel \({\cal{E}}\) is said to be teleportationcovariant^{15} when, for any teleportation unitary U (WeylPauli operator in finite dimension or phasespace displacement in infinite dimension), we have
for some (generallydifferent) unitary transformation V. In this case the quantum channel can be simulated by applying teleportation over its Choi matrix \(\sigma _{\cal{E}}: = {\cal{I}} \otimes {\cal{E}}(\Phi )\), where Φ is a maximallyentangled state. Similarly, if the teleportationcovariant channel is bosonic, we can write an approximate simulation by teleporting over the quasiChoi matrix \(\sigma _{\cal{E}}^\mu : = {\cal{I}} \otimes {\cal{E}}(\Phi ^\mu )\), where Φ^{μ} is a TMSV state. For a network of teleportationcovariant channels, we therefore use teleportation to simulate the network, so that the resource states in the upper bounds of Eqs. (68)–(70) are Choi matrices (physical or asymptotic). In other words, we write the sandwich relations
with the REE taking the form of Eq. (59) on an asymptotic Choi matrix \(\sigma _{{\cal{E}}_{{\mathbf{xy}}}}: = \mathop {{\lim}}\nolimits_\mu \sigma _{{\cal{E}}_{{\mathbf{xy}}}}^\mu\).
As a specific case, consider a quantum channel which is not only teleportationcovariant but also distillable, so that it satisfies^{15}
where \(D_1(\sigma _{\cal{E}})\) is the oneway distillability of the Choi matrix \(\sigma _{\cal{E}}\) (with a suitable asymptotic expression for bosonic Choi matrices^{15}). If a network (or a chain) is composed of these channels, then the relations in Eqs. (75)–(77) collapse and we fully determine the capacities
These capacities correspond to Eqs. (7), (14), and (19) of the main text. They are explicitly computed for chains and networks composed of lossy channels, quantumlimited amplifiers, dephasing and erasure channels in Table 1 of the main text.
Regularizations and other measures
It is worth noticing that some of the previous formulas can be reformulated by using the regularization of the entanglement measure, i.e.,
In fact, let us go back to the first upper bound in Eq. (49), which implies
For a network under multipath routing we have \(n_{{\mathbf{xy}}} = n\), so that we may write
By repeating previous steps, the latter equation implies the upper bound
which is generally tighter than the result in Eqs. (66) and (67). The same regularization can be written for a chain \(\{ {\cal{E}}_i\}\), which can also be seen as a singleroute network satisfying the flooding condition \(n_{{\mathbf{xy}}} = n\). Therefore, starting from the condition of Eq. (83) with \(n_{{\mathbf{xy}}} = n\), we may write
which is generally tighter than the result in Eq. (64). These regularizations are important for the REE, but not for the squashed entanglement which is known to be additive over tensorproducts, so that \(E_{{\mathrm{SQ}}}^\infty (\sigma ) = E_{{\mathrm{SQ}}}(\sigma )\).
Another extension is related to the use of the relative entropy distance with respect to partialpositivetranspose (PPT) states. This quantity can be denoted by RPPT and is defined by^{31}
with an asymptotic extension similar to Eq. (59) but in terms of converging sequences of PPT states \(\gamma ^\mu\). The RPPT is tighter than the REE but does not provide an upper bound to the distillable key of a state, but rather to its distillable entanglement. This means that it has normalization \(E_{\mathrm{P}}\left( {\varphi ^n} \right) \ge nR_n\) on a target maximallyentangled state \(\varphi ^n\) with \(nR_n\) ebits.
The RPPT is known to be monotonic under the action of PPT operations (and therefore LOCCs); it is continuous and subadditive over tensorproduct states. Therefore, we may repeat the derivation that leads to Eq. (41) but with respect to protocols \({\cal{P}}\) of entanglement distribution. This means that we can write
Using the decomposition of the output state \(\rho _{{\mathbf{ab}}}^n\) as in Eqs. (47) and (48), and repeating previous steps, we may finally write
for a chain \(\{ {\cal{E}}_i\}\) with resource states \(\{ \sigma _i\}\), and
for the single and multipath entanglement distribution capacities of a quantum network \({\cal{N}}\) with resource states \(\sigma ({\cal{N}}) = \{ \sigma _{{\mathbf{xy}}}\} _{({\mathbf{x}},{\mathbf{y}}) \in E}\).
Data availability
All data in this paper can be reproduced by using the methodology described.
Code availability
Code is available upon reasonable request to the author.
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Acknowledgements
This work has been supported by the EPSRC via the ‘UK Quantum Communications HUB’ (EP/M013472/1) and ‘qDATA’ (EP/L011298/1), and by the European Union via Continuous Variable Quantum Communications (CiViQ, Project ID: 820466). The author would like to thank Seth Lloyd, Koji Azuma, Bill Munro, Richard Wilson, Edwin Hancock, Rod Van Meter, Marco Lucamarini, Riccardo Laurenza, Thomas Cope, Carlo Ottaviani, Gaetana Spedalieri, Cosmo Lupo, Samuel Braunstein, Saikat Guha and Dirk Englund for feedback and discussions.
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S.P. developed the theory, carried out the entire work, and wrote the manuscript.
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Correspondence to Stefano Pirandola.
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Pirandola, S. Endtoend capacities of a quantum communication network. Commun Phys 2, 51 (2019). https://doi.org/10.1038/s4200501901473
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