Can quantum networks overcome distance and loss?

The question in perspective

A classical repeater can read and regenerate a signal. An unknown quantum state cannot be copied in that way. Quantum networks instead distribute entanglement, use heralded successful links, and connect them through swapping or error-corrected protocols. The central question is whether the delivered rate and fidelity improve once all waiting and hardware costs are counted.

Loss sets a benchmark

For a simple optical attenuation model, transmissivity over distance \(L\) is \(\eta=e^{-L/L_{\rm att}}\), where \(L_{\rm att}\) is an attenuation length. In an ideal bosonic pure-loss channel, the asymptotic repeaterless secret-key capacity with unlimited two-way classical assistance and no input-energy constraint is

\[K_0(\eta)=-\log_2(1-\eta)\quad\text{bits per optical mode}.\]

For small \(\eta\), \(K_0\approx\eta/\ln2\). The rate therefore falls exponentially with distance. This is a benchmark for that channel and resource model; it is not an unassisted qubit capacity or a rate per second.

Consider two independently attempted segments. In a toy model, each segment succeeds with probability \(p\) per round, and a memory stores the first success while the second segment continues. Let the waiting times \(T_1,T_2\) be independent geometric random variables, counted from one. Then

\[\mathbb E[\max(T_1,T_2)]=\frac2p-\frac1{2p-p^2}.\]

This follows from \(\max(T_1,T_2)+\min(T_1,T_2)=T_1+T_2\). Each waiting time has mean \(1/p\). The minimum has success probability \(1-(1-p)^2\) per round. For \(p\ll1\), the expected completion time is approximately \(3/(2p)\) rounds. Requiring both links to succeed simultaneously instead gives mean \(1/p^2\). At \(p=0.1\), these are about 14.74 and 100 rounds.

Storage creates its own cost. If coherence decays as \(e^{-t/T_2}\), a stored link loses coherence while waiting. The completion-time improvement alone does not establish an improved usable-entanglement or secret-key rate.

What is established

The repeaterless capacity bound gives a precise rate–loss benchmark. The quantum repeater review develops architectures that use additional nodes and resources to change the trade-off. Exceeding the benchmark requires comparing the same channel uses and assistance model, with repeater resources reported separately.

Converting modes into a rate per second also requires clock rate, multiplexing, coupling and detection efficiencies, classical signalling time, and protocol success probabilities. A heralded event is not automatically a high-fidelity delivered pair.

What remains unresolved

Which combination of memory lifetime, optical interfaces, swapping fidelity, and multiplexing yields a scalable advantage for a target network task? Memory cutoffs, asynchronous scheduling, and error correction interact. Optimizing each component in isolation can choose a poor end-to-end protocol.

A concrete next investigation

Reproduce: sample the two geometric waiting times and recover the exact completion-time mean. Keep attempted modes and elapsed rounds as separate counters.

Check: include decoherence during \(|T_1-T_2|\) waiting rounds and report the delivered fidelity distribution, not only the mean delivery rate.

Extend: optimize a memory cutoff or multiplexing schedule under a fixed hardware budget. A useful result improves a specified end-to-end metric and states when it exceeds the appropriate direct-link benchmark.

Selected references

1. Fundamental Limits of Repeaterless Quantum Communications

   Stefano Pirandola, Riccardo Laurenza, Carlo Ottaviani, Leonardo Banchi. 2015. Research paper

   Two-way assisted capacity benchmarks for pure-loss and other channels; specify modes and assistance.

2. Quantum repeaters: From quantum networks to the quantum internet

   Koji Azuma et al.. 2022. Review

   Connects repeater architectures with the challenge of distributing quantum information over long distances.

Years refer to first arXiv submission. Publication details are available in the linked records.