QKD Secure Key Rate
Decoy-state BB84 secure key rate (GLLP), QBER, single-photon gain, maximum distance and optimal mean photon number versus channel loss, with the PLOB repeaterless bound.
Secure key rate
223 kb/s
2.23e-4 bit per pulse
QBER E_μ
3.34 %
single-photon error e₁ = 3.32 %
Channel loss / overall η
10.5 dB
η = η_ch·η_Bob = 4.01e-3
Gain Q_μ / single-photon gain Q₁
1.92e-3 / 1.19e-3
detections per pulse
Maximum distance (key > 0)
142.0 km
at 29.82 dB channel loss, μ fixed
Optimal μ here
0.479
key 2.23e-4 bit per pulse
PLOB bound (channel)
135 Mb/s
0.135 bit per pulse, any protocol
Mean optical power
-72.1 dBm
6.15e-11 W = μ·f_rep·hc/λ
Secure key per pulse vs distance
QBER vs distance
Key rate (GLLP with infinitely many decoy states, Ma et al. 2005):R = ½·{−f·Q_μ·H₂(E_μ) + Q₁[1 − H₂(e₁)]}per pulse, where Q_μ = 1 − (1 − Y₀)e^(−ημ), E_μQ_μ = ½Y₀e^(−ημ) + e_d(1 − e^(−ημ)), Q₁ = Y₁μe^(−μ) and e₁ = (½Y₀(1 − η) + e_dη)/Y₁ with Y₁ = Y₀ + η − Y₀η. A background click alone gives a random bit and a photon click an error e_d; Ma et al.'s eqs. (10)–(11) are the same for Y₀ ≪ 1. The first term pays for error correction; without it the key never ends.
Defaults are the GYS experiment of Ma et al.'s Table 1 (0.21 dB/km, η_Bob = 4.5 %, Y₀ = 1.7×10⁻⁶, e_d = 3.3 %, f = 1.22, μ = 0.48), whose key ends at 142 km. For a free-space link, set the attenuation to zero and put the geometric and atmospheric loss in the fixed loss. The PLOB bound caps any repeaterless protocol over a channel of transmittance η_ch. Finite-key effects are not included.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Asymptotic decoy-state BB84 (infinite decoys and key), GLLP rate with error-correction cost f·Q_μ·H₂(E_μ), Poisson source, threshold detectors. Reproduces the 142 km GYS limit of Ma et al. 2005.
- Ma X., Qi B., Zhao Y., Lo H.-K. (2005). Practical decoy state for quantum key distribution. Phys. Rev. A 72, 012326. Eqs. (1), (5)–(11); Table 1 (GYS); Fig. 2, 142.05 km. doi:10.1103/PhysRevA.72.012326
- Pirandola S., Laurenza R., Ottaviani C., Banchi L. (2017). Fundamental limits of repeaterless quantum communications. Nat. Commun. 8, 15043. Bound −log₂(1 − η). doi:10.1038/ncomms15043