Equidistant Repeater Chains in Quantum Networks

• Equidistant repeater chains are quantum network designs that partition a communication channel into identical segments with uniform loss and operational characteristics.
• They employ techniques like teleportation stretching and entanglement cuts to derive and optimize capacity formulas across various channel models.
• This structure mitigates exponential loss, enabling scalable long-distance quantum communication for protocols such as QKD and entanglement distribution.

An equidistant repeater chain is a quantum network topology in which the total communication distance between two end users is divided into identical segments separated by evenly spaced quantum repeaters. Each segment, or elementary link, experiences the same channel characteristics (such as transmission loss or decoherence), thereby balancing performance and synchronizing protocol operations throughout the network. This architecture is foundational for achieving scalable, long-distance quantum communication, supporting tasks such as quantum key distribution (QKD), entanglement distribution, and quantum information transmission. The equidistant structure enables optimal use of quantum network resources, simplifies performance analysis, and underpins rigorous capacity results and protocol design.

1. Fundamental Principles and Capacity Formulation

The key idea behind equidistant repeater chains is to split a long, high-loss channel into multiple shorter segments, each serviced by a quantum repeater node. This division increases the end-to-end rate and robustness of entanglement or quantum information transmission because the per-segment performance is strictly better than that of a direct, unsegmented channel. If the total channel transmissivity is $\eta$ and there are $N$ repeaters forming $N+1$ links, each link has transmissivity $\eta_s = \eta^{1/(N+1)}$ (Pirandola, 2016).

For bosonic lossy channels (optical fiber or free-space), the fundamental two-way assisted capacity per use is

$\mathcal{C}(\eta) = -\log_2(1 - \eta)$

and the end-to-end capacity of an equidistant repeater chain is

$$\mathcal{C}_{\text{loss}}(\eta, N) = -\log_2[1 - \eta^{1/(N+1)}].$$

This result generalizes to other distillable quantum channel models:

• Quantum-limited amplifiers: The capacity is defined by the maximal inverse gain among segments,

$$\mathcal{C}_{\text{amp}} = -\log_2(1 - g_{\max}^{-1}).$$

• Dephasing channels: Capacity per segment is $1 - H_2(p)$ (with $H_2$ the binary entropy).
• Erasure channels: Capacity per segment $1 - p$, with $p$ the erasure probability.

For a chain of identical, distillable channels, the end-to-end capacity is given by the minimum per-segment capacity: $$\mathcal{C}(\{\mathcal{E}_i\}) = \min_i \mathcal{C}(\mathcal{E}_i).$$ This reflects that the worst-performing link determines the overall performance.

2. Teleportation Stretching, Channel Simulation, and Entanglement Cuts

The formal derivation of these capacities relies on two main quantum information techniques (Pirandola, 2016):

1. Channel Simulation and Teleportation Stretching: Any quantum channel $\mathcal{E}$ can be simulated by local operations and classical communication (LOCC) acting on a fixed resource state (often the Choi matrix of $\mathcal{E}$). An adaptive protocol over the entire chain is "stretched" so the output state depends only on the product of all resource states and a final LOCC.
2. Entanglement Cut and Relative Entropy of Entanglement (REE): The capacity is upper bounded by the minimal REE of the resource state associated with any link. For a chain of distillable channels, this upper bound is tight, and the optimal protocol is to distill at each link's capacity and perform entanglement swapping along the chain.

The cut-set approach directly links the bottleneck segment to the overall capacity, justifying the optimization and optimality of equidistant chains where all segments have equal, maximized performance.

3. Scaling Laws, Regimes, and Practical Implications

Equidistant repeater chains impose favorable scaling laws, crucial for overcoming exponential loss in long-distance quantum communication. Specifically, instead of the point-to-point exponential PLOB bound,

$$\mathcal{C}_{\text{direct}}(\eta) = -\log_2(1 - \eta),$$

the capacity with $N$ repeaters decays only logarithmically per segment. With strong enough repeaters (e.g., $\eta_s \sim 0.5$, the 3dB regime), nonzero end-to-end rates are maintained even at high total loss (Pirandola, 2016). Amplifier chains and dephasing/erasure channels admit analogous results, with equidistant placement ensuring balanced bottlenecks across the network.

Routing in general quantum networks follows the same principles: single-path (widest path/bottleneck capacity) and multi-path (maximum flow/min-cut) protocols still associate the optimal rate to the worst-performing cut, emphasizing the utility of evenly balancing resources—a natural property of equidistant chains.

4. Implementation in Protocols and Physical Platforms

The equidistant structure naturally supports protocol modularity—standardized operations on each segment (entanglement generation, purification, swapping) are applied sequentially. For instance, coherent-state–based protocols in cavity QED repeat the same controlled phase shift, displacement, coherent-state discrimination, and purification at every segment (Gonţa et al., 2016).

In optical platforms, this regularity simplifies hardware requirements—identical fiber lengths, memory times, and signal handling per node. In hybrid architectures (e.g., buffered automated chains with NV$^-$ routers, rare-earth ion–doped memories, and multiplexed photon-pair sources), every elementary link is programmed for uniform delay, greatly simplifying synchronization and matching between quantum modules (Askarani et al., 2021).

Unitary one-way repeaters, essential for future error-corrected architectures, can also be arranged in equidistant chains. In this setting, precisely constructed Hamiltonians act identically at each node to coherently transfer or correct errors segment by segment (Miatto et al., 2017).

5. Optimization, Memory Allocation, and Performance Trade-offs

The regularity of equidistant chains underpins analytical treatments and optimization of protocol performance. The Markov chain approach enables exact or approximate calculation of waiting time distributions and entanglement fidelity for arbitrarily long chains (Brand et al., 2019, Dai et al., 2021). Variant protocols (e.g., those including intermediate entanglement distillation or memory cut-offs) can be efficiently analyzed recursively. Memory and resource allocation can be optimized globally, ensuring that with only moderate (often constant) memory per node, polynomial—as opposed to exponential—decay of rate with total chain length is achievable.

Furthermore, memory queuing and communication delays—dominant factors in long chains—can be bounded and managed effectively, with constant or slowly growing memory requirements even as the number of segments increases (Dai et al., 2021).

Cut-off optimization further maximizes secret key rates, with uniform cut-off policies across all levels shown to be nearly optimal for equidistant chains (Li et al., 2020). This facilitates practical implementations: identical timing or fidelity thresholds can be programmed network-wide.

6. Robustness to Imperfections and Asymmetry

While the idealized equidistant placement delivers theoretical optimality, analysis demonstrates that network performance is resilient to small deviations from perfect symmetry (Avis et al., 2023). Key performance metrics—success probability, end-to-end fidelity, distribution time, QBER—are only affected to second order in small asymmetries. Even moderate disparities in link lengths (characterized by an asymmetry parameter) result in only minor reductions in secret key rates or entanglement throughput. This resilience allows tolerance for deployment imperfections in fiber infrastructure, lessening the need for precise node positioning.

However, persistent asymmetry or poor balancing can create true bottlenecks that degrade overall capacity, justifying the preference for equidistant design wherever practical.

7. Extensions to Networked and Hybrid Quantum Architectures

The conceptual clarity and algorithmic tractability of equidistant chains have facilitated extensions to networked configurations with multiple paths, hybrid routers, multiplexing strategies, and advanced memory technologies. In buffered, frequency-multiplexed automated repeater chains, equidistant segmentation underlies the timing of spectral and temporal multiplexing as well as compatibility with deterministic quantum routers and quantum buffers (Askarani et al., 2021).

Analytical equivalence has been shown between multipartite GHZ routers (star networks) and standard two-user repeater chains, with network rate scaling and memory saturation phenomena transferable between models (Kunzelmann et al., 23 May 2025). Hybrid integrations, including cavity–magnon systems, further benefit from the uniform, modular structure of equidistant chains, supporting dynamic parameter tuning and modular network scaling (Khan et al., 6 Jul 2025).

Table: Key Capacity Formulas for Equidistant Repeater Chains

Channel Model	Per-Segment Capacity	End-to-End Capacity (Equidistant, $N$ repeaters)
Bosonic Loss ($\eta$)	$-\log_2(1-\eta_s)$	$-\log_2[1-\eta^{1/(N+1)}]$
Amplifier ($g$)	$-\log_2(1-g^{-1})$	$-\log_2(1-g_{\max}^{-1})$
Dephasing ($p$)	$1 - H_2(p)$	$\min_i [1-H_2(p_i)]$
Erasure ($p$)	$1 - p$	$\min_i (1-p_i)$

This table highlights the dependence of the overall rate on either the balanced, per-segment capacities (equidistant case) or the worst-performing segment (arbitrary chain).

Summary

Equidistant repeater chains constitute an optimal, theoretically justified structure for repeater-assisted quantum communications, enabling efficient scaling, straightforward protocol design, and maximal achievable end-to-end capacities under a broad range of channel models. Application of quantum-information theoretic tools (teleportation stretching, channel simulation, entanglement cuts) and classical network optimization methods (widest path, max-flow) substantiates the value of equidistant segmentation in both protocol performance and implementation scalability. These principles underpin current and future quantum networking architectures, from point-to-point links to large-scale, heterogeneous quantum internets.