Quantum repeaters: From quantum networks to the quantum internet

Koji Azuma; Sophia E. Economou; David Elkouss; Paul Hilaire; Liang Jiang; Hoi-Kwong Lo; Ilan Tzitrin

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Quantum repeaters: From quantum networks to the quantum internet

Koji Azuma, Sophia E. Economou, David Elkouss, Paul Hilaire, Liang Jiang, Hoi-Kwong Lo, Ilan Tzitrin

arXiv:2212.10820v2Dec 21, 2022quant-ph

TL;DR

Long-distance quantum communication is limited by optical loss, while quantum repeaters remain technologically challenging. This review organizes repeater architectures by loss- and error-suppression mechanisms, surveys experimental progress and near-term protocols, and concludes that practical deployment faces substantial infrastructure and cost barriers.

Problem
Optical loss limits point-to-point quantum communication, and quantum repeater schemes for intercontinental distances remain separated from practical intracity communication by a technological gap.
Method
The review categorizes repeater protocols by their mechanisms for suppressing losses and errors, and surveys architectures, experimental implementations, and near-term protocols targeting repeaterless bounds.
Results
The review covers quantum repeater generations, modified all-photonic schemes, post-pairing measurement-device-independent QKD, entangled-photon sources, and proof-of-concept quantum networks.
Takeaways & Limitations
Quantum repeaters have no fundamental distance limitation in principle, but realizing efficient intercontinental quantum communication remains technologically challenging.
Takeaways & Limitations
Global deployment could require trillions of US dollars, while cryogenic repeater nodes and their maintenance pose major challenges for undersea links.

Abstract

from arXiv · show

A quantum internet is the holy grail of quantum information processing, enabling the deployment of a broad range of quantum technologies and protocols on a global scale. However, numerous challenges exist before the quantum internet can become a reality. Perhaps the most crucial of these is the realization of a quantum repeater, an essential component in the long-distance transmission of quantum information. As the analog of a classical repeater, extender, or booster, the quantum repeater works to overcome loss and noise in the quantum channels comprising a quantum network. Here, we review the conceptual frameworks and architectures for quantum repeaters, as well as the experimental progress towards their realization. We also discuss the various near-term proposals to overcome the limits to the communication rates set by point-to-point quantum communication. Finally, we overview how quantum repeaters fit within the broader challenge of designing and implementing a quantum internet.

I. INTRODUCTION

The quantum internet would connect quantum information processors globally, but quantum repeaters are needed to overcome channel loss and the no-cloning barrier to long-distance communication. This review surveys repeater architectures, experimental progress, photonic and matter-based technologies, and performance-oriented protocols.

A quantum internet is a global network of quantum information processors that can provide remote access to quantum technologies and secure communication.
Quantum key distribution is presented as the only known method for unconditionally secure information transmission, unlike classical security based on computational conjectures.
At 1550 nm, standard optical fiber attenuates signals by 0.2 dB/km, leaving on average 1 of every 100 photons after 100 km.
Quantum repeaters must overcome point-to-point communication limits without trusted relays, because conventional amplifiers cannot copy unknown quantum states.
Repeater implementation draws on atomic ensembles, quantum dots, cavity QED, quantum memories, photonic sources, and light–matter interfaces.
Building a quantum internet is interdisciplinary, combining quantum information with network topology, protocol design, information theory, error correction, mathematics, computer science, and engineering.
The review reorganizes repeater categories by mechanisms suppressing loss and errors, including memoryless, error-corrected, and all-photonic repeaters.

B. Quantum no-cloning theorem

The quantum no-cloning theorem forbids deterministic copying of an unknown quantum state, preventing classical-style signal amplification for quantum communication.

The no-cloning theorem states that no deterministic quantum operation can copy an unknown state |ψ⟩A onto another system as |ψ⟩A ⊗ |ψ⟩B.
The result was originally formulated for pure states and later extended to mixed states through the no-broadcasting theorem.

C. Entanglement

Entanglement is a nonseparable quantum resource underlying teleportation, entanglement swapping, purification, distillation, and quantum error correction. The section develops bipartite and multipartite definitions, measures, Bell states, and graph-state operations.

Entanglement describes composite states that cannot be specified independently through the states of their constituent subsystems.
Entanglement cannot be increased using local operations and classical communication, making it a resource for quantum-network protocols.
Bipartite states: A bipartite pure state is entangled when its Schmidt rank r satisfies r ≥ 2; r = 1 gives a separable state.
Bipartite states: For mixed states, separability includes convex mixtures of product states, while states that cannot be written in that form are entangled.
Bipartite states: Concurrence unambiguously quantifies entanglement for mixed two-qubit states, whereas general higher-dimensional mixed-state entanglement remains difficult to characterize.
Bipartite states: Bell states are maximally entangled and each represents one ebit, sufficient to teleport one qubit of quantum information.
Multipartite states: Multipartite entanglement includes inequivalent GHZ and W classes that cannot be transformed into one another using LOCC.
Graph states: Graph states associate multipartite entanglement with an undirected graph, whose connections transform under local Clifford gates and Pauli measurements.

E. Photonic encodings

Photonic encodings trade communication robustness, gate determinism, measurement capability, and experimental requirements across discrete- and continuous-variable platforms. The review compares time-bin, polarization, path, Fock, coherent/cat, and GKP encodings.

Encoding taxonomy: Photonic encodings are classified as discrete-variable when using finite-dimensional subspaces and continuous-variable when using infinite-dimensional state spaces.
Discrete-variable encodings: Time-bin encoding is suited to fiber communication because it is unaffected by birefringence, but interacting two time-bin qubits is difficult.
Discrete-variable encodings: Polarization encoding supports deterministic single-qubit gates, while linear-optical entangling gates are probabilistic and Bell measurements succeed with probability 1/2 ideally.
Discrete-variable encodings: Path encoding enables deterministic single-qubit gates with beamsplitters and phase shifters, but entangling gates require probabilistic linear-optical operations, measurements, and postselection.
Discrete-variable encodings: Fock encoding permits deterministic generation of selected Bell states with a 50:50 beamsplitter but is sensitive to phase drifts in transmission.
Discrete-variable encodings: Coherent/cat encoding uses coherent states |α⟩ and |−α⟩, with basis flips from a π-phase shifter and discrimination by photon-number-resolving detection.
Continuous-variable encodings: GKP encoding supports deterministic Clifford operations through Gaussian operations, while non-Clifford gates require ancillary states and gate teleportation.
Continuous-variable encodings: Continuous-variable encodings face loss-protection limits because Gaussian operations cannot protect Gaussian states against Gaussian errors, requiring non-Gaussian operations or resources.

III. QUANTUM REPEATERS

Quantum repeaters distribute entanglement across lossy optical links by combining entanglement generation, quantum memories, and entanglement swapping. This review introduces these primitives and shows how repeater concatenation can improve long-distance communication beyond direct transmission.

III. QUANTUM REPEATERS: Quantum teleportation transfers an unknown qubit without moving its physical carrier, using a pre-shared maximally entangled state, a Bell measurement, and classical communication.The receiver applies a measurement-dependent local unitary to recover the original state.
III. QUANTUM REPEATERS: Quantum repeater protocols rely on entanglement generation, entanglement swapping, and quantum memories to distribute Bell pairs across separated stations.Entanglement swapping propagates entanglement between stationary nodes, while memories retain successful links until neighboring links are available.
III. QUANTUM REPEATERS: Quantum communication mainly reduces to distributing Bell pairs efficiently, because teleportation can then transmit arbitrary qubit states by LOCC.Teleportation consumes one ebit and requires two classical bits from sender to receiver.
III. QUANTUM REPEATERS: Direct entanglement generation requires a number of trials that grows exponentially with distance, making direct optical-fiber links inefficient over long distances.A typical 400 km fiber has transmittance about 10^-8, corresponding to a possible key rate on the order of 10 bits per second at a 1 GHz clock rate.
III. QUANTUM REPEATERS: A single midpoint repeater provides a square-root improvement in the number of trials, and concatenated swapping extends this improvement to multiple equally spaced repeater nodes.The idealized DLCZ-like example assumes fiber attenuation is the only error and all other operations are perfect.

L (NQR+1)Latt . (33)

Quantum repeaters must suppress both photon loss and operational errors. The review distinguishes deterministic from probabilistic suppression methods and notes that realistic memory and operation imperfections constrain ideal scaling.

L (NQR+1)Latt . (33): Realistic memory errors and imperfect generation or swapping operations accumulate over longer distances, limiting the idealized loss-only model.Error-suppression mechanisms are therefore needed to address imperfections beyond optical attenuation.
L (NQR+1)Latt . (33): Error-suppression techniques divide into deterministic methods, including quantum error correction and one-way entanglement distillation, and probabilistic methods, including error detection and two-way purification.The classification is based on whether suppression is deterministic or probabilistic.
L (NQR+1)Latt . (33): Quantum error correction encodes logical states redundantly across many physical qubits and corrects errors deterministically without heralding delays.The logical qubit occupies a two-dimensional subspace of a larger Hilbert space.
L (NQR+1)Latt . (33): One-way entanglement distillation converts noisier entangled pairs into nearly maximally entangled pairs using direct one-way LOCC.Only one party communicates results during the distillation process.
L (NQR+1)Latt . (33): Probabilistic methods detect or purify errors through heralding and two-way communication, introducing delays or requiring adaptive operations.Heralded entanglement generation repeats after detected loss, while multiplexing can run trials in parallel.

c. Comparison of deterministic and probabilistic quantum error suppression

Quantum error suppression in repeaters is either probabilistic, which tolerates larger errors but requires signaling and retries, or deterministic, which avoids signaling delay but has finite thresholds. These mechanisms define repeater generations with different performance, technology, and resource trade-offs.

Error-suppression mechanisms: Deterministic suppression requires pure-loss transmittance η > 1/2 and fails for depolarizing strength p > 1/4.Hashing and concatenated coding extend the depolarizing-channel regime to approximately p < 0.18929 and p < 0.19130, respectively.
Error-suppression mechanisms: Probabilistic suppression can operate with nonzero transmission probability and depolarizing strength p < 1/2, but incurs classical signaling delay.Its broader error tolerance comes at the cost of waiting for confirmation that an error-suppression attempt succeeded.
Repeater architecture: Quantum repeaters divide communication into shorter segments so repeater stations can address fiber loss and operational errors over long distances.The relevant operational errors include channel, gate, measurement, and quantum-memory errors.
Error-suppression mechanisms: Probabilistic suppression requires two-way classical signaling for successful progression or retries, whereas deterministic suppression has no corresponding signaling delay.Heralded entanglement generation is a standard loss-detection scheme, while quantum error correction and one-way entanglement distillation provide deterministic suppression.
Generations of quantum repeaters: The three repeater generations combine probabilistic or deterministic suppression of loss and operation errors, with increasing technological difficulty and improved performance.The combination of deterministic loss suppression and probabilistic operation suppression is described as sub-optimal compared with the other three combinations.
First-generation repeaters: First-generation repeaters reduce direct-transfer overhead from exponential to polynomial, but their rate still decreases polynomially with distance because of two-way signaling.Temporal, spatial, and frequency multiplexing can boost their communication rate through quantum-memory degrees of freedom.

2. Second-generation repeaters

Second-generation quantum repeaters use quantum error correction for operation errors while treating photon loss probabilistically, enabling faster long-distance communication than first-generation schemes at higher technological and resource costs.

Architecture: Second-generation repeaters use probabilistic loss suppression and deterministic operation-error suppression, commonly preparing encoded states with CSS codes.Encoded Bell pairs are created through teleportation-based non-local CNOT gates, followed by error correction during encoded entanglement swapping.
Architecture: Encoded Bell pairs extend entanglement through quantum error correction, replacing two-way entanglement purification and distillation procedures.This avoids the time-consuming two-way classical signaling associated with first-generation repeaters.
Rates and resources: Second-generation rates are bounded by heralded entanglement generation and purification between neighboring repeater stations.Reducing station spacing can raise the formal rate bound, but the required number of repeater nodes and quantum memories diverges.
Rates and resources: The total memory requirement depends on code size and station spacing, while CSS code size can increase only poly-logarithmically with total distance.In practice, fault-tolerant initialization of large CSS blocks is challenging; concatenated codes reduce initialization complexity but introduce polynomial code-size scaling.
Comparison: Second- and third-generation repeaters can communicate faster than first-generation repeaters over long distances, but require high-fidelity gates and operation errors below fault-tolerance thresholds.Third-generation repeaters also require smaller spacing because deterministic correction handles only finite loss, up to 50% loss error rates.
Comparison: The preferred repeater generation depends on gate error probability, coupling efficiency, and local gate time across distinct parameter-space regions.The comparison distinguishes regimes favoring first-generation, encoded or unencoded second-generation, and third-generation protocols.

1. Original all-photonic repeaters

The original all-photonic repeater uses repeater graph states with encoded core qubits and outer leaves to distribute entanglement without stationary quantum memories at repeater nodes. Bell measurements connect neighboring graph states, while encoded measurements protect against photon loss.

Repeater graph state: The repeater graph state has a core graph of inner qubits and an outer layer of leaves attached to the core vertices.The core qubits emulate quantum memories, while leaves participate in heralded entanglement generation between neighboring repeater states.
Repeater graph state: Cliques provide the original core-graph structure, while bicliques can suffice when some clique connections are unnecessary.A clique connects every pair of vertices; a biclique connects the two vertex sets without internal connections within either set.
Loss protection: Inner qubits are encoded in redundant tree-graph states so X- and Z-basis measurements can succeed nearly deterministically despite photon loss.The tree-graph code supports loss protection, while local feedforward adapts inner-qubit measurements to outer-qubit outcomes.
Protocol operation: In each clock cycle, receivers perform simultaneous Bell measurements on leaf photons from neighboring repeater graph states to connect their core qubits.Subsequent single-qubit measurements transform the connected state into a linear cluster between Alice and Bob, then into a Bell pair.
Scope: The all-optical scheme requires no quantum memories for applications whose generated entanglement is immediately consumed, such as QKD and nonlocal measurements.Quantum-output applications such as teleportation and distributed computation still require endpoint memories lasting approximately the classical communication time.
Modified proposals: Boosted Bell measurements can raise success probability to 3/4 and improve overheads, but add experimental complexity and cannot achieve unit probability with finite resources.The original proposal sends encoded inner photons to neighboring receivers, whereas a modified design stores them locally in fiber spools.

a. General framework.

Quantum repeater architectures organize loss and error suppression through different resource-generation and entanglement mechanisms. The review covers probabilistic and deterministic graph-state approaches, their resource demands, performance constraints, and the technological gap between long-distance repeater schemes and current point-to-point communication.

General framework: Optical graph-state generation proceeds through unit-resource production, optional growth into meta-units, and iterative stitching into the desired graph state.Dual-rail encodings use type-II fusions for stitching, while GKP states can use continuous-variable CZ gates.
General framework: All-optical and matter-based graph-state generation offer probabilistic and deterministic alternatives, respectively.The review contrasts an all-optical probabilistic approach with a matter-qubit deterministic approach.
General framework: The original all-photonic repeater uses tree graph states whose branching parameters determine the connectivity across successive levels.Its resource-generation procedure includes six single photons and fusion operations producing a 3-partite GHZ state with probability 1/32.
General framework: Deterministic matter-based RGS generation can create the unencoded resource using one emitter and one ancilla, independent of graph size.Deterministic generation of larger encoded RGSs is also described, while generalized constructions target arbitrary graphs with minimal emitters.
Performance and overheads: Realistic performance is constrained by hardware rates, gate durations, source and detector efficiencies, memory errors, and imperfect entanglement operations.For deterministic RGS repeaters, emitter–ancilla CZ-gate duration sets the secret-key-rate bound; strong emitter–waveguide coupling remains experimentally challenging for some proposals.
Milestones: Quantum repeaters have no fundamental distance limitation in the reviewed schemes, but their realization remains technologically challenging relative to intracity point-to-point communication.Intermediate intercity schemes use a single central node and target secret-key-rate scaling proportional to √η rather than η.

A. Adaptive measurement-device-independent QKD

Adaptive MDI QKD uses a central node to store successfully received qubits until matching arrivals enable Bell measurement, improving communication-efficiency scaling from η to √η. Matter-memory implementations face coherence-time constraints, while optical and source-level requirements shape feasible designs.

Adaptive MDI QKD: √η scaling replaces direct-channel η scaling because each qubit traverses only one lossy half-channel before central-node processing.With multiplexing m ∼ (√η)^−1, both sides can supply nonzero qubit pairs with finite probability when intrinsic measurement successes remain constant.
Memory-assisted implementation: The central node stores a qubit received from either side until a qubit arrives from the other side, then performs a Bell measurement.Matter memories can implement the storage and heralding steps, with one memory assigned to each sender.
Memory-assisted implementation: The memory-assisted protocol uses time multiplexing, but dephasing and amplitude damping grow exponentially with storage time and limit the usable multiplexing number.The limitation is specific to matter-memory implementations whose stored states must remain coherent across repeated attempts.
Memory-assisted implementation: For dephasing memories, the key rate follows √η while T2/T ≥ (√η)^−1, then converges to η as η decreases further.The required coherence time scales as e^{L/(2Latt)}T, with Latt = 22 km, although shorter pulse periods can reduce the absolute requirement.
Implementation requirements: Two-mode squeezed states cannot directly supply the teleportation-based QND step because their multiphoton component makes QND success depend on channel transmittance.Proposed memories therefore include single matter qubits in cavities, paired with sources having low multiphoton components.
Twin-field QKD: Twin-field QKD obtains √η scaling through single-photon interference and single-rail encoding, but its original proposal lacked a general security proof.Subsequent variants addressed arbitrary attacks in asymptotic and finite-size settings.

C. The single sequential quantum repeater

Sequential repeater alternatives place memories and detectors in different network locations or combine time multiplexing with single-photon interference. These designs can target √η scaling, but trade simplicity and implementation requirements against measurement-device independence and coherence-time demands.

C. The single sequential quantum repeater: A single sequential repeater places two memory qubits at the central node and detectors at the end nodes, eliminating optical Bell measurement.The central node sequentially sends memory-entangled photons until each end node confirms detection, then performs a Bell measurement.
C. The single sequential quantum repeater: This setup is simple but is not measurement-device independent and requires qualitatively longer coherence times than memory-based adaptive MDI QKD.The memory must cover photon travel and heralding times for both sequential transmission stages.
Experimental status: Experiments with Rubidium atoms demonstrated square-root transmittance scaling, although the achieved rate remained below the direct-transmission fundamental limit.The feasibility of surpassing point-to-point limits depends on hardware parameters.
Time-multiplexed alternatives: Mode-pairing QKD and asynchronous MDI-QKD combine central-node single-photon interference with sequentially time-multiplexed coherent pulses.They are conceptually intermediate between adaptive MDI QKD and twin-field QKD, and both have experimental demonstrations.
Time-multiplexed alternatives: For large N, n = O(N√η) single-photon-interference successes yield O(N√η/2) candidate time-bin pairs for two-photon Bell measurements.The construction assumes phase correlation is preserved between potentially distant time bins.
Experimental requirements: Fiber-based repeater implementations require efficient photon–memory interfaces, while successive repeater generations address memory and loss errors through distillation or quantum error correction.Third-generation designs additionally require high transmission, collection, coupling, and detection efficiencies because QEC tolerates at most 50% erasure probability.

A. Long-lived quantum memories

Long-lived quantum memories are central to repeater performance, requiring suitable coherence, control, register, and photonic-emission properties. Candidate platforms support distinct photon–memory entanglement schemes, whose fidelity, distance scaling, and hardware demands differ.

A. Long-lived quantum memories: Quantum-memory coherence time T2 is the key figure of merit because it determines how long stored quantum information remains usable.Repeater applications also depend on control fidelity, relaxation and dephasing times, registers, collection efficiency, indistinguishability, and spin–photon fidelity.
A. Long-lived quantum memories: Candidate memories include atomic ensembles, trapped ions, diamond color centers, and quantum dots, with reported coherence times spanning milliseconds to seconds.The passage lists ensemble values of 0.2–16 s, 4 ms for ^128Ba+, and 1 s for diamond color centers.
B. Emission of photons entangled with the quantum memory: A Λ-level structure can emit a polarization-entangled photon because two ground states couple to one excited state through orthogonal polarizations.Equal transition coupling is required for a maximally entangled memory–photon state.
B. Emission of photons entangled with the quantum memory: Time-bin entanglement offers an alternative requiring only one strong optical transition, making it available to memories lacking a Λ-level structure.The scheme instead entangles the photon’s emission time bin with the memory qubit.
C. Distant entanglement generation: Cabrillo single-photon heralding and Barrett–Kok two-photon heralding have both generated distant entanglement across several memory platforms.The Barrett–Kok approach reached 1.3 km in a loophole-free Bell-test experiment with NV centers.
C. Distant entanglement generation: Cabrillo’s scheme requires low photon-emission probability for high fidelity, whereas Barrett–Kok tolerates high emission probability but suffers worse long-distance loss scaling.Thus, the preferred scheme depends on emitter efficiency and node separation.
D. Entanglement distillation: Linear-optical entanglement distillation is limited to 25% success, while experiments have demonstrated 65 ± 3% fidelity and a 182 Hz heralded entanglement rate in separated systems.The review notes that 99% distilled Bell-pair fidelity is considered experimentally reachable.
E. Multi-qubit quantum registers and error correction: Second- and third-generation repeaters require multi-qubit registers for logical encoding and error correction, including a trapped-ion Bacon–Shor implementation using nine physical qubits plus four stabilizer qubits.These registers support correction of memory errors and, in third-generation designs, loss errors.

F. Loss mitigation, quantum frequency conversion, and photonic source efficiency

Quantum repeater performance depends on mitigating photon loss, converting photons to telecom wavelengths, improving source and detector efficiencies, and scaling photonic entanglement and repeater architectures. Experiments demonstrate progress across these components, while loss sensitivity and probabilistic generation remain important constraints.

Loss mitigation: Physical qubit losses above 50% cannot be corrected with quantum error correction, making loss reduction critical for QEC-based repeaters.Fiber propagation and coupling losses are major contributors.
Loss mitigation: 1550-nm telecom fiber has a loss coefficient of 0.2 dB per km, while ultra-low-loss fibers reach 0.16 dB per km but are not widely available.Adopting ultra-low-loss fiber would require substantial infrastructure modification.
Quantum frequency conversion: Frequency conversion has been used to shift photons from quantum emitters, including NV centers, quantum dots, atoms, ions, crystals, and ensembles, to telecom wavelengths.The conversion is represented by ωf = ωi−ωl.
Photonic source efficiency: 67% effective collection efficiency has been achieved for multiplexed and actively switched spontaneous parametric downconversion sources, supporting all-photonic approaches.These sources are not suitable for efficient light-matter interfaces based on matter qubits.
Detector efficiency: Commercial SNSPDs reach 95% detection efficiency, while telecom-wavelength superconducting nanowire detectors have demonstrated efficiencies up to 99%.Transition edge sensors additionally provide photon-number resolution for some heralded entanglement schemes.
Photonic graph states: A 12-photon linear cluster state and a 14-photon GHZ state have been demonstrated, but probabilistic fusion gates limit SPDC-generated graph states to a current maximum of 12 photons.The primary challenge is creating large, highly entangled photonic graph states.
Quantum repeater demonstrations: A single repeater node increased MDI-QKD’s secret key rate fourfold over original MDI QKD and exceeded the PLOB bound in key rate versus effective channel transmission.The node used a silicon-vacancy center in a diamond photonic-crystal cavity.
Quantum repeater demonstrations: A memory-enhanced node based on two 87Rb atoms achieved a single-qubit error rate below 11% and can in principle be cascaded.A three-node network with quantum memories was also realized over a maximum inter-node distance of seven meters.

VI. QUANTUM INTERNET

The quantum internet is framed as a staged network architecture supporting communication, cryptography, computation, and sensing. Its development progresses from trusted repeater links toward quantum-memory and fault-tolerant quantum-computing networks, while current stage assignments remain tied to theoretical requirements.

Applications: Quantum networks can support classical and quantum information transmission, cryptographic tasks, communication-complexity advantages, distributed computation, and enhanced sensing.Examples include QKD, quantum fingerprinting, modular quantum computing, clock synchronization, and entanglement-assisted interferometry.
Development stages: The proposed development stages classify quantum internet capabilities according to the functionality available to end nodes.The paper notes that the path to a quantum internet will be long and difficult.
Development stages: Early-stage networks can already provide useful applications, and additional tasks become possible as end-node functionality increases.This makes usefulness possible before the final quantum-internet stage.
Development stages: Trusted repeater networks distribute prepare-and-measure QKD between adjacent nodes but do not transmit quantum information to non-adjacent nodes.These networks can be constructed from individual QKD links.
Development stages: End-to-end prepare-and-measure networks allow nodes to prepare and transmit single qubits to any other node without trust assumptions, with possible post-selection costs.MDI QKD and twin-field QKD belong to this category.
Development stages: Quantum memory networks let end users store quantum information and teleport it to one another, but operations remain on physical qubits without fault tolerance.The minimum storage time is set by transit time between end nodes.
Development stages: Quantum computing networks represent the final stage, supporting large-scale fault-tolerant computation beyond efficient classical simulation.This stage enables protocols including leader election and quantum money.
Development stages: Stage assignments reflect the current theoretical state of the art, and future protocol proposals may reduce the implementation requirements for particular tasks.The paper directs readers to the quantum protocol zoo for a broader protocol-stage overview.

1. An abstract depiction of networks

The paper models quantum networks as graphs whose vertices are processing nodes and whose edges are quantum channels, then evaluates protocols by the entanglement resources they produce and the rates at which they use network resources.

Network model: For analysis, a quantum network is reduced to nodes and quantum communication channels, despite potentially containing routers, switches, and multiplexers.This abstraction isolates the components needed for network-capacity analysis.
Network model: The network is represented as G = (G, g), with G = (V, E) a directed graph and g mapping edges to completely positive trace-preserving quantum channels.Vertices denote nodes and edges denote communication links.
Network model: Node capabilities range from preparing predefined quantum states to operating as universal quantum computers, with noiseless local operations and commonly free classical communication assumed.The assumed classical-communication resource may connect channel-adjacent nodes or, in some settings, all nodes.
Network model: Cuts partition the vertex set and identify directed edges crossing between the two parts through outgoing and incoming cut-sets.The outgoing cut-set is Δ+(V′), and the incoming cut-set is Δ−(V′).
Resource states: Quantum-network applications can be reduced to distributing suitable resource states, such as maximally entangled states, private states, GHZ states, or multipartite private states.These resources support quantum transmission, secret-key generation, secret sharing, and conference key agreement.
Entanglement distribution: An aggregated repeater protocol distributes entanglement by transmitting subsystems through channels and applying local operations and classical communication across repeated rounds.The protocol targets a bipartite state between nodes A and B with bounded error.
Network rates: The figure of merit is average produced entanglement, quantified as ⟨log2 d⟩, corresponding operationally to transmitted qubits or private communication bits.A d-dimensional maximally entangled state supports log2 d qubits, while a d-dimensional private state supports log2 d bits.
Network rates: Network rates divide average entanglement by resources such as channel uses, full network uses, or uses of a path connecting the end nodes.Different resource choices define distinct network capacities, including channel-use, network-use, and single-path capacities.

3. Entanglement based upper bounds

Entanglement measures provide upper bounds on quantum-network capacities by limiting entanglement generation across network cuts. For linear networks, these bounds simplify to channel-wise expressions and constrain noisy quantum-repeater performance.

General framework: The bounds rely on entanglement measures that satisfy monotonicity under LOCC and an inequality limiting entanglement growth when a channel is used.The channel-use inequality adds the channel’s entanglement to the input entanglement as an upper limit.
General framework: Network capacities for distributing a target state can be bounded above using optimization formulae based on suitable bipartite entanglement measures.The framework generalizes earlier bounds for teleportation-simulable channels and arbitrary quantum networks.
Network cuts: For arbitrary networks, the upper bound depends on individual channel entanglements and usage frequencies, minimized over all cuts separating the communicating nodes.The cut argument joins each side of a bipartition into virtual nodes and bounds distribution by the entanglement crossing that cut.
Linear networks: For linear networks, cut-sets are individual channels, which substantially simplifies the upper-bound expressions relevant to repeater protocols.The corresponding per-channel and per-network bounds specialize the general formulas.
Linear networks: For lossy bosonic channels, relative entropy of entanglement gives tight bounds, with ER(Ne) = −log2(1 − ηe).Here ηe is the transmittance of channel Ne, modeling optical-fiber loss.
Noisy repeaters: A memory coherence time below 1.0 × 10−4 s restricts DLCZ-type repeater performance to an exponential in the square root of total distance.The bound applies to a large class of DLCZ-type protocols when memory noise is modeled as a noisy channel.

5. Capacity lower bounds via the aggregated repeater protocol

Aggregated repeater protocols establish achievable lower bounds for distributing entanglement across quantum networks, often matching entanglement-based upper bounds up to prefactors. Their performance can be analyzed through multigraph paths and efficiently computed cut optimizations, while practical deployment remains constrained by cost, hardware, and measurement limitations.

Achievable lower bounds: Aggregated quantum repeater protocols provide general lower bounds that match network upper bounds up to a prefactor.Aggregations of several existing repeater protocols also match the lower bound up to another prefactor for optical networks.
Protocol construction: The protocol first generates near-maximally entangled states independently across channels, representing each resulting state as an edge in a multigraph.For channel e, rates Re below Q(Ne) are achievable, and ne uses yield neRe multigraph edges.
Protocol construction: Entanglement swapping along edge-disjoint paths converts the multigraph resources into maximally entangled states between the target nodes.The number of distributable states equals the maximum number of edge-disjoint connecting paths, characterized using Menger’s theorem.
Achievable rates: For sufficiently large n, any rate below each channel’s quantum capacity is achievable, yielding a lower bound optimized over channel-use fractions.With pe = ne/n, the achievable expression has the same cut-minimization form as the corresponding upper bounds.
Achievable rates: When E(Ne) = Q(Ne) for every channel, the lower and upper bounds coincide; this holds for networks composed only of lossy bosonic channels.The lower bounds replace channel entanglement by quantum capacity, while the upper bounds use entanglement measures.
Computation: The relevant cut-minimization problems can be solved by linear programming in polynomial time in the number of graph nodes.The optimization uses only the individual channel-entanglement values and supports network resource allocation.
Practical constraints: Quantum repeaters are essential for an efficient quantum internet, whose realization also raises issues of cost, energy, cryogenic operation, and photonic measurement efficiency.The review identifies global-scale investment, limited memory lifetimes, cryogenic repeater deployment, and Bell-measurement success as practical boundaries.