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End-to-end capacities of a quantum communication network

Stefano Pirandola

arXiv:1905.12674v1quant-phcond-mat.othermath-phphysics.optics

TL;DR

The paper addresses the unresolved end-to-end capacity limits of repeater-assisted quantum and private communication. It derives single-letter bounds for adaptive chains and networks, then exactly characterizes distillable cases, including major noise models such as bosonic loss. These results reduce several optimal-capacity problems to classical widest-path or maximum-flow problems and provide benchmarks for repeater networks.

  • Problem

    End-to-end capacities for transmitting quantum information, distributing entanglement, and generating secret keys through quantum repeater chains and networks were not fully understood.

  • Method

    The paper combines quantum-information methods with network-information methods to derive REE-based single-letter bounds and analyze adaptive routing, cuts, and channel simulations.

  • Results

    For distillable chains and networks, widest-path and maximum-flow methods exactly determine secret-key, entanglement-distribution, and quantum-communication capacities across bosonic loss, amplification, dephasing, and erasure models.

  • Takeaways & Limitations

    The resulting formulas establish ultimate performance limits for repeater-assisted quantum communications and enable benchmarking of practical networks.

  • Takeaways & Limitations

    The RPPT measure is tighter than REE but does not upper-bound distillable key, only distillable entanglement.

Abstract

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In quantum mechanics, a fundamental law prevents quantum communications to simultaneously achieve high rates and long distances. This limitation is well known for point-to-point 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 single-letter upper bounds for the end-to-end 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 repeater-assisted quantum communications.

RESULTS

The paper derives end-to-end capacity bounds for adaptive repeater chains and networks, with exact characterizations for important channel families. For lossy channels, optimal rates are governed by the weakest link in a chain and the best route in single-path networks.

  • Repeater chains: Adaptive repeater-chain protocols define capacities for entanglement, quantum communication, and secret-key generation through repeated channel uses and interleaved LOCCs.Entanglement-distribution capacity D2 equals quantum capacity Q2 under two-way classical communication, while secret-key capacity K satisfies K ≥ D2.
  • Repeater chains: Single-letter REE bounds apply to generic repeater chains, while teleportation-covariant channels admit simpler bounds based on their Choi resource states.Bosonic simulations are formulated asymptotically using sequences of physical resource states.
  • Repeater chains: For distillable chains, the exact repeater-assisted capacity is the minimum one-way distillable entanglement among the individual channel resource states.Achievability uses one-way entanglement distillation on each link followed by entanglement swapping.
  • Lossy chains: For a lossy chain, the minimum link transmissivity determines the ultimate rate, and equidistant repeaters maximize that minimum transmissivity.The analysis compares repeater-assisted capacity with the point-to-point PLOB bound as a function of total loss.
  • Single-path networks: For single-path networks, the capacity is determined by the route with maximum end-to-end transmissivity, and distillable networks attain the corresponding upper bound.In lossy networks, this is the ultimate secret-bit rate per sequential network use.

Quantum networks under multi-path routing

Multi-path routing floods a quantum network through sequential multipoint communications, using every edge once, and its capacity is characterized through multi-edge cut bounds and maximum-flow optimization. For distillable networks, these bounds are achievable and yield explicit formulas for several fundamental noise models.

  • Flooding protocols: Flooding routes systems through successive multipoint communications so that each network edge is used exactly once.The protocol alternates network LOCCs with parallel transmissions to progressively reach Bob.
  • Capacity bounds: The multi-path capacity is upper-bounded by minimizing the total resource-state entanglement, or REE, across all network cuts.For teleportation-covariant distillable networks, the resource-state REE can be replaced by the corresponding channel capacities.
  • Capacity bounds: For distillable networks, maximum flow and the max-flow min-cut theorem achieve the multi-path capacity using link rates no larger than the channel capacities.Orlin’s algorithm finds the optimal orientation and rates in O(|P| × |E|) time.
  • Single-path versus multi-path: In a diamond network with equal transmissivity η links, multi-path capacity doubles the single-path capacity.The paper identifies parallel network use as more powerful than sequential use in this example.
  • Noise models: Analytical end-to-end formulas are provided for distillable chains and networks involving loss, quantum-limited amplification, dephasing, and erasure channels.For lossy channels, the two-way point-to-point capacities coincide with the PLOB bound −log2(1 −η).

DISCUSSION

The paper establishes general REE-based single-letter upper bounds for repeater-assisted quantum and private communication, then shows that classical network optimization exactly determines performance for distillable networks. It applies these results to chains and arbitrary networks under several fundamental noise models, while distinguishing general upper bounds from cases where achievability is proved.

  • Contributions: Single-letter relative-entropy-of-entanglement bounds apply to end-to-end capacities in arbitrary quantum networks, including networks with untrusted nodes.The scope covers single- and multi-path routing and arbitrary quantum channels between nodes.
  • Distillable networks: For distillable networks, widest-path and maximum-flow methods make the quantum upper bounds achievable and reduce performance determination to classical network information theory.This applies to secret-key generation, entanglement distribution, and quantum-information transmission.
  • Scope: The general upper bounds need not coincide with the classical lower bounds outside the distillable-network setting.The discussion explicitly separates general bounds from exact characterizations for distillable networks.
  • Applications: The capacities are exactly established for distillable chains and networks affected by bosonic loss, quantum-limited amplification, dephasing, and erasure.Bosonic loss is identified as especially important for optical and telecommunications applications.
  • Method: The upper-bound proof strategy combines entanglement-based weak converses, channel simulation, network stretching, and classical capacity composition for lower bounds.For distillable networks, the lower bounds coincide with REE-based upper bounds.

Network simulation

Network simulation replaces each quantum channel with a resource state and a teleportation-like LOCC, after which stretching converts adaptive protocols into block representations. Entanglement-measure properties and network cuts then produce computable single-letter bounds, with asymptotic bosonic simulations handled by energy-constrained limits.

  • Channel simulation: Each channel is simulated as Exy(ρ) = Txy(ρ ⊗σxy), replacing channel uses with resource states and trace-preserving LOCCs.For bosonic channels, the simulation may be asymptotic and parameterized by resource-state energy.
  • Asymptotic simulations: For asymptotic bosonic simulations, the approximation error vanishes with the simulation parameter for finite input energy, protocol uses, and network size.The error is quantified using the energy-constrained diamond distance and a peeling argument.
  • Network stretching: Network stretching reduces the n-use adaptive output to a trace-preserving LOCC acting on tensor products of edge resource states.The number of resource-state factors depends on edge usage, with flooding protocols using each edge n times.
  • Cut-based bounds: Stretching relative to an entanglement cut reduces the resource representation to the channels crossing that cut.Monotonicity under LOCCs and subadditivity over tensor products then yield single-letter cut quantities.
  • Cut-based bounds: The framework extends the entanglement measure to asymptotic resource states and applies in particular to the relative entropy of entanglement.The resulting formulas support both non-asymptotic and asymptotically simulated channels.

Minimum entanglement cut and upper bounds

The network capacity upper bounds are obtained by minimizing entanglement measures over cuts, with routing-specific formulas for single- and multi-path communication. For teleportation-covariant distillable channels, these bounds become exact capacities.

  • Single-path routing: Single-path routing produces an upper bound based on single-edge entanglement flow through each cut.For a chain, each cut-set contains one edge, reducing the network expression to the corresponding link quantity.
  • Multi-path routing: Multi-path routing produces an upper bound based on multi-edge entanglement flow through network cuts.Flooding uses all available paths, so the cut expression aggregates flow across multiple edges.
  • Cut-based upper bounds: Minimizing over all network simulations and entanglement cuts yields the tightest upper bounds.The optimization extends over resource-state simulations as well as possible cuts.
  • Teleportation-covariant channels: Teleportation-covariant channels can be simulated with Choi or asymptotic Choi resource states, allowing REE-based single-letter bounds.For bosonic channels, the Choi-state construction is formulated asymptotically.
  • Distillable channels: For networks of distillable teleportation-covariant channels, the upper bounds collapse and fully determine the capacities.The relevant capacities are explicitly computed for lossy, quantum-limited amplifier, dephasing, and erasure channels.
  • Regularization and measures: Regularized entanglement measures can tighten the upper bounds, whereas regularization is unnecessary for additive squashed entanglement.The PPT-relative-entropy measure is tighter than REE but bounds distillable entanglement rather than distillable key.

Supplementary Information

The supplementary analysis defines adaptive repeater-chain protocols, extends teleportation stretching across entanglement cuts, and derives single-letter REE bounds for quantum, entanglement, private, and secret-key capacities.

  • Chain model: A repeater chain consists of sequential quantum channels between Alice, repeaters, and Bob, with local registers updated during transmissions.Channels may transmit forward or backward, while two-way classical communication makes the effective task direction-independent.
  • Adaptive protocols: The most general protocol uses adaptive local operations and unlimited two-way classical communication involving every point in the chain.Transmission order may be permuted between uses, and the transmission order together with LOCCs defines the adaptive protocol.
  • Capacity definitions: Under two-way classical communication, entanglement distribution and quantum communication have equal capacities, while secret-key capacity is at least as large.The target states are respectively maximally entangled states and private states.
  • Teleportation stretching: Teleportation stretching replaces channel transmissions with tensor products of resource states followed by a single trace-preserving LOCC.This reduction is applied after simulating the channels and reorganizing the adaptive protocol into block form.
  • Chain stretching: Cutting any chain channel disconnects Alice from Bob and enables a stretched decomposition for arbitrary protocol uses, repeater counts, and energy.The cut partitions the chain into super-Alice and super-Bob and replaces the disconnected channel with its simulation resource.
  • Capacity bounds: Theorem 3 gives a single-letter REE upper bound for all four two-way capacities, simplifying to an exact minimum over individual capacities for distillable chains.For such chains, the capacity is achievable by one-way entanglement distillation followed by entanglement swapping.

Capacities for distillable chains

For distillable repeater chains, the end-to-end capacity is controlled by the weakest link under the relevant channel capacity formula. This yields explicit limits for lossy, amplifying, dephasing, and erasure chains.

  • Lossy chains: The capacity of a lossy chain is determined entirely by the minimum transmissivity ηmin among its links.Adding repeaters does not change this weakest-link dependence.
  • Lossy chains: 1 bit per chain use requires ηmin = 1/2, corresponding to at most 3 dB loss per link and a 15 km maximum link distance at 0.2 dB/km.The same rate can represent one secret bit, ebit, or qubit.
  • Amplifying chains: For quantum-limited amplifying chains, the highest gain gmax determines the repeater-assisted capacity.The capacity is obtained by applying the individual amplifier capacity to the weakest effective link.
  • Dephasing chains: For dephasing spin chains, the maximum phase-flip probability pmax determines capacity through 1 − H2(pmax).The channel probabilities satisfy pi ≤ 1/2.
  • Erasure chains: For erasure spin chains, the maximum erasure probability pmax determines capacity through the corresponding weakest-link formula.The result extends from qubits to qudits of arbitrary dimension using the corresponding two-way capacities.
  • Hybrid chains: The formulas also apply to hybrid distillable chains whose links alternate different channel types.Examples include erasure channels alternated with dephasing or lossy channels.

Quantum repeaters in optical communications

The paper analyzes how quantum repeaters and multiband channels affect optical communication capacities across distance and loss. Equidistant repeaters provide the long-distance advantage, while multiband communication is additive but retains unfavorable rate-loss scaling unless combined with repeaters.

  • Repeater placement: The optimal placement is N equidistant repeaters, creating N + 1 links with identical transmissivities.This follows directly from the capacity formula for the bosonic repeater chain.
  • Scaling regimes: In the repeater-dominant regime, capacity scales logarithmically with the number of repeaters independently of total loss.The contrasting loss-dominant regime instead retains a fundamental rate-loss scaling for highly lossy links.
  • Distance scaling: At long distances, multiband communication cannot compete with repeater-assisted communication because it retains the single-band rate-loss scaling.Multiband channels increase capacity additively, but do not change the unfavorable long-distance scaling by themselves.
  • Combined strategy: The strongest general strategy is to combine multiband communication with quantum repeaters.The paper extends the repeater-chain analysis to multiband channels and derives corresponding capacity expressions.
  • Multiband repeater chains: The multiband repeater-chain capacity is obtained by applying the distillable-chain result to multiband lossy links.Each link has Mi bands and transmissivity ηi, and multiband lossy channels are distillable.
  • Multiband repeater chains: For equidistant multiband repeater chains, capacity depends on both link transmissivity and the minimum bandwidth Mmin along the line.The minimum bandwidth is Mmin := min_i Mi, while high-loss performance reflects an interplay between bandwidth and transmissivity.

SUPPLEMENTARY NOTE 2: QUANTUM NETWORKS

The supplementary note formalizes quantum networks as graphs of memoryless channels and defines adaptive end-to-end communication through sequential or parallel routes. It introduces sequential capacity as the asymptotic optimum over route choices and network LOCC operations.

  • Network model: A quantum network is represented by a finite undirected graph whose vertices are points and whose edges are quantum channels.Each point has a local register used for quantum communication.
  • Routes: An end-to-end route is an undirected path from Alice to Bob through intermediate repeaters and channels.Routes may share repeaters or channels, and a finite network can be reduced to simple paths without cycles.
  • Adaptive protocols: Network transmissions are interleaved with adaptive network LOCCs assisted by unlimited two-way classical communication.The LOCCs act across the local registers of all network points.
  • Routing strategies: Sequential routing uses one route per network use, whereas parallel routing transmits systems through multiple routes simultaneously.Sequential route selection may be stochastic, with route ω chosen according to probability pω.
  • Capacity definition: Sequential capacity is the asymptotic maximum number of quantum, entanglement, or secret bits distributed per sequential network use.The capacity is defined by optimizing over sequential protocols in the large-blocklength, small-error limit.
  • Adaptive routing: With deterministic routing control, the protocol can adaptively select routes and converge to an optimal route ω∗.This is practical when quantum resources are optimized per routed quantum system rather than constrained by resource-use costs.
  • Analytical approach: Teleportation stretching and classical routing methods are used to derive single-letter upper bounds, while point-to-point protocols provide lower bounds for distillable networks.The approach covers both single-path and multi-path routings.

SUPPLEMENTARY NOTE 3: SIMULATION AND STRETCHING OF A QUANTUM NETWORK

The note develops network simulation and teleportation stretching for arbitrary adaptive protocols. Replacing each channel by a resource-state simulation reduces network evolution to LOCC processing of edge-associated resource states, including asymptotic bosonic simulations.

  • Network simulation: Each network channel Exy is simulated by an LOCC Txy acting on a resource state σxy.Applying these simulations to every edge gives a resource representation σ(N) of the network.
  • Teleportation stretching: Teleportation stretching replaces each adaptive channel transmission with an LOCC acting on the prior network state and the corresponding resource state.The original LOCCs and the channel simulation combine into a single trace-preserving LOCC.
  • Stretched output: After iterating over all transmissions, the network output is expressed as LOCC processing of resource states associated with the used edges.The number of copies of σxy equals the number nxy of uses of edge (x,y).
  • Protocol dependence: Sequential protocols use each edge npxy times, while flooding protocols use every edge n times within each end-to-end transmission.Flooding therefore corresponds to parallel use of multiple channels during a network communication.
  • Teleportation covariance: For teleportation-covariant channels, each edge resource state is the channel’s Choi matrix.The network stretching representation can therefore use Choi resources edge by edge.
  • Bosonic channels: Asymptotic stretching remains valid for bosonic channels under finite-use, finite-edge, and finite-energy conditions.The approximation converges in trace norm, with the relevant distance tending to zero as the simulation parameter vanishes.

Network stretching with entanglement cuts

Entanglement cuts simplify the stretched network by assigning resource states within each side of a cut to super-Alice or super-Bob. Only cut-crossing resources remain explicit, enabling capacity bounds from cut flow.

  • Cut-based stretching: Resources on Alice’s or Bob’s side of the cut are absorbed into the corresponding super-party’s local operations.Only resource states associated with cut-set edges remain outside those local operations.
  • Entanglement cuts: An entanglement cut is a bipartition of network points placing Alice and Bob on opposite sides.Its cut-set contains the edges crossing the bipartition and disconnects the network when removed.
  • Reduced output: Tracing out intermediate registers preserves locality between the original Alice and Bob and yields a reduced output state described by a trace-preserving LOCC.This converts the general network stretching decomposition into a cut-based decomposition.
  • Cut decomposition: For any entanglement cut, the network output admits a resource-state decomposition involving only the cut-set resources.For teleportation-covariant networks, these resources are the corresponding channel Choi matrices.
  • Bosonic extension: The cut-stretching construction applies to bosonic networks with asymptotic simulations under finite-use, finite-edge, and finite-energy conditions.These decompositions support the single-letter upper bounds for single- and multi-path capacities.

Converse part (upper bound)

The converse bounds single-path capacities by minimizing an REE-based single-edge flow over entanglement cuts. For distillable networks, this upper-bound framework coincides with the widest-path achievable rate.

  • Converse bound: The single-path capacity is upper-bounded by the minimum REE flow across all entanglement cuts.The bound is single-letter, with asymptotic formulations available for bosonic networks.
  • Proof strategy: The REE cut bound follows from a weak converse, REE monotonicity under LOCC, and subadditivity over tensor products.For sequential protocols, edge-use probabilities are optimized away before the cut minimization.
  • Achievability: The achievable single-path rate is the maximum bottleneck capacity among Alice-Bob routes.Independent point-to-point protocols on consecutive links are combined through network LOCCs.
  • Cut-path equivalence: The widest route equals the single-edge capacity of a minimum entanglement cut.The optimal route is a simple Alice-Bob path within a maximum spanning tree.
  • Graph-theoretic form: For arbitrary weighted undirected networks, the widest-path weight equals the minimum-cut weight.Path weight is defined by its minimum edge weight, while cut weight is defined by its maximum crossing-edge weight.

2. Insert the neighbor-point p with the maximum

For teleportation-covariant and distillable networks, single-path capacities reduce to widest-path optimization and admit exact formulas for several fundamental noise models.

  • Teleportation-covariant networks: Teleportation-covariant networks reduce single-path routing to a widest-path problem followed by a non-adaptive one-way protocol.Consecutive nodes distill entanglement, then entanglement swapping distributes it end to end.
  • Distillable networks: For distillable networks, the single-path capacity equals both the minimum single-edge cut capacity and the maximum route capacity.This result extends the widest-path formulation to quantum communication.
  • Routing algorithm: The optimal route can be found in O(|E| log2 |P|) time, with capacity achieved by one-way distillation followed by entanglement swapping.The route is obtained using a maximum spanning-tree construction.
  • Fundamental noise models: Distillable-network formulas cover bosonic loss, quantum-limited amplification, qubit dephasing, and qubit erasure.These formulas establish single-path limits for quantum communication, entanglement distribution, key generation, and private communication.
  • Lossy networks: For lossy networks, capacity is equivalently determined by the minimum cut transmissivity or the maximum route transmissivity.Each edge has capacity Cxy = −log2(1 −ηxy).

SUPPLEMENTARY NOTE 5: RESULTS FOR MULTI-PATH ROUTING

Multi-path quantum-network capacities are bounded by REE flow across cuts and achieved through conserved network flows, with classical max-flow constructions providing the routing framework.

  • Converse: The multi-path capacity has a single-letter upper bound given by the minimum multi-edge REE flow across entanglement cuts.Bosonic-channel formulations may use asymptotic REE definitions.
  • Lower bound: The minimum multi-edge capacity of the entanglement cuts is achievable by a flooding protocol.The corresponding multi-path routing is found by solving a classical maximum-flow problem in O(|P| × |E|) time.
  • Flow-network transformation: Undirected quantum networks are converted into directed flow networks by orienting Alice’s edges outward, Bob’s edges inward, and splitting intermediate edges into opposite directed edges.The paired directed edges retain the original undirected edge capacity.
  • Directed cuts: The directed cut-set includes only edges directed from the Alice side to the Bob side.Reverse-directed edges crossing the bipartition are excluded.
  • Flow construction: A legal quantum flow conserves the number of qubits received and transmitted at every intermediate node.Opposite directed rates on an undirected edge combine into an effective rate bounded by the edge’s two-way quantum capacity.
  • Achievability: The value of a conserved flow is an achievable end-to-end quantum-communication rate.The same construction extends to private bits and the other quantum tasks.

Formulas for teleportation-covariant and distillable

For teleportation-covariant networks, REE yields multi-path capacity bounds; for distillable networks, these bounds become exact classical max-flow min-cut formulas with explicit noise-model specializations.

  • networks: Teleportation-covariant networks admit a multi-path capacity sandwich expressed through REE-based multi-edge cut quantities.Bosonic networks may require asymptotic finite-energy Choi-state formulations.
  • networks: For distillable networks, the upper and lower bounds coincide, extending max-flow min-cut theory to quantum communications.The capacity equals the minimum multi-edge capacity of the entanglement cuts.
  • networks: Optimal multi-path routing is computable in O(|P| × |E|) time, and a capacity-achieving flooding protocol uses one-way distillation followed by directed teleportation.The classical flow solution determines the network orientation and point-to-point rates.
  • Multi-path capacities of fundamental networks: For lossy networks, the multi-edge capacity of a cut is Cm(C) = −log2 l(C), where l(C) is the product of crossing-edge loss parameters.The network capacity follows by optimizing this cut quantity.
  • Multi-path capacities of fundamental networks: Multiband lossy networks assign each edge a bandwidth Mxy and transmissivity ηxy, representing Mxy independent lossy channels.Equal-transmissivity networks can be expressed using the effective bandwidth Mmin.
  • Multi-path capacities of fundamental networks: Exact multi-path formulas also apply to quantum-limited amplifier, dephasing, and erasure networks.The relevant edge parameters are amplifier gains, dephasing probabilities, and erasure probabilities.

SUPPLEMENTARY NOTE 6: RELATED LITERATURE

The paper’s results were developed alongside distinct approaches to quantum-network capacity bounds and later applied to practical repeater and QKD studies.

  • Related bounds: Azuma et al. derived different single-path private-communication bounds using squashed entanglement rather than channel simulation.Their bounds are reported as less tight for networks connected by teleportation-covariant channels.
  • Related bounds: Azuma and Kato studied multi-path upper bounds but did not consider flooding protocols, where every edge is used exactly once per parallel network use.The paper identifies flooding as essential for deriving its general multi-path capacity upper bound.
  • Subsequent developments: Later work combined this paper’s relative-entropy-of-entanglement channel-simulation approach with squashed-entanglement methods to obtain versatile bounds.Other studies also examined networks composed of Holevo-Werner channels using both approaches.
  • Subsequent developments: Pant et al. explored the superiority of multi-path over single-path protocols using realistic repeater-node models, complementing this paper’s ideal-repeater analysis.The comparison concerns distributing entanglement and secret keys between end-users.
  • Applications: The paper’s performance limits have been used in analyses of relay-assisted twin-field and phase-matching quantum key distribution.These applications connect the theoretical limits to specific QKD protocols.
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