Photon efficiency–coherence time product rule for NV-centre and trapped-ion quantum repeater chains

The fidelity of a first-generation quantum repeater is controlled by a single hardware quantity, whose identification explains a striking equivalence between the two leading qubit platforms. By simulating nitrogen-vacancy (NV) center and trapped-ion (171Yb+) repeater chains over an identical 200km fiber channel—anchored to optical time-domain reflectometry (OTDR) field measurements of a deployed single-mode link and statistically extended to 66.08 dB total loss - we find that the two platforms deliver statistically indistinguishable raw fidelity at three segments (NV: Ftel = 0.605; trapped ion: Ftel = 0.589), despite five-fold opposite photon-efficiency and coherence-time specifications. A two-dimensional sensitivity analysis reveals why: at fixed channel loss, repeater fidelity is controlled by the product νphoton·T2 of photon efficiency and coherence time, not by either quantity alone. Both platforms share νphoton·T2 = 0.20s and therefore perform identically; any two platforms on the same iso-product line will likewise coincide. This design rule yields a concrete hardware target: reaching the BB84 key-distribution threshold at three segments requires νphoton·T2 ≥ 2.44s, a twelve-fold improvement, and at the throughput-optimal segment count gate and measurement errors cap fidelity near 0.806 regardless of memory quality. The comparison is built on the NetSquid discrete-event simulator with experimentally validated parameters, full three-basis state tomography, and density-matrix DEJMPS purification; the last reveals that realistic dephasing-biased chain states purify substantially better than the Werner-state recurrence predicts. Trapped ions purify more effectively (Bell fidelity 0.608 versus 0.509 after one round) owing to superior gate and read out fidelity, while NV centers generate entangled pairs 4–5×faster. The photon–memory interface, captured by the νphoton·T2 product, is the universal bottleneck limiting both platforms.