Sending one photon straight down D km of fiber succeeds with probability P = e^(−D/L), where L is the attenuation length — losses compound exponentially, so past a few hundred km direct transmission is essentially hopeless.
A repeater chain splits D into N+1 shorter segments and generates entanglement independently on each one. Because each hop is short, its own round-trip heralding time is short too, so in the same wall-clock time a segment gets roughly N+1 attempts instead of the single shot a direct link gets — dramatically boosting each segment's success chance within one round. Once every segment succeeds, each repeater node performs an entanglement swap (a joint Bell-state measurement on its two local photons), stitching the two shorter entangled pairs into one longer one without ever moving a photon the full distance. Chain them all and you get one pair spanning the entire link.
P_direct = e^(−D/L)
d_seg = D / (N+1)
p_seg = e^(−d_seg/L)
P_seg(round) = 1 − (1 − p_seg)^(N+1)
P_repeater = P_seg(round)^(N+1) × η_swap^N
- Total distance — the source-to-destination separation to bridge.
- Repeater nodes — more nodes → shorter, more reliable segments, but one more swap that can fail each time.
- Swap efficiency — the Bell-measurement success probability at each repeater; imperfect optics and detectors make this <100%.
Real-world relevance: this segment-and-swap trick is the whole reason a practical quantum internet is considered feasible at continental scale — without repeaters, entanglement distribution dies off exponentially within tens to a few hundred kilometres of fiber.