Ultra-long-haul digital coherent PSK Y-00 quantum stream cipher transmission system: supplement

This document provides supplementary information to “Ultra-long-haul digital coherent PSK Y-00 quantum stream cipher transmission system.”


Quantum-noise masking number in PSK Y-00 cipher
The quantum-noise masking number Γ Q in a PSK Y-00 cipher system was derived using semi-classical theory. An ideal heterodyne detection is assumed here. The variance of shot noise σ shot 2 at ideal heterodyne detection with a local oscillator (LO) is calculated as shot 2 = 2e bias , #(S1) where e, i bias , and B are the electric charge, bias current of a PD, and electrical signal bandwidth, respectively. The bias direct current is obtained as follows: bias = ( S + L ), #(S2) where S, P S , and P L , are the PD responsivity, optical power of the signal, and LO. The optical powers are defined for a single polarization. The signal current of the heterodyne detection i sig is expressed as where ω S , ω L , and ϕ(t) are the angular frequencies of the signal, LO, and modulated phase, respectively. The initial phase difference between the signal and LO is omitted for simplicity. The angle of uncertainty imposed by Δφ shot , as shown in Fig. S1, is calculated from Eqs. (S1)-(S3).
As the optical power of the LO is much larger than the signal power, P S + P L ≈ P L is assumed. Meanwhile, the angle of adjacent signals of the cipher Δθ basis is obtained from the order of data modulation M and bit resolution of phase randomization m.

Shot noise 2σ shot
Δφ shot The PD responsivity S is calculated as where h, ν 0 , and η q are the Planck constant, signal frequency, and quantum efficiency of the PD, respectively. Subsequently, using Eqs. (S5)-(S7), the quantum-noise masking number is defined as the ratio of Δφ shot and Δθ basis .

Symbol error ratio (SER) of detecting PSK Y-00 cipher without a seed key
One can estimate the SER when an eavesdropper performs an ideal heterodyne measurement of the cipher. Only the signal masking by shot noise is considered. Figure S2 shows the model of multilevel phase detection. A i , Δ basis , and σ shot represent the phase level of the ith signal, distance between adjacent signals, and standard deviation of shot noise, respectively. The SER of the detection P SER_eve is calculated as follows: In the model, the quantum-noise masking number Γ Q = 2σ shot / Δ basis . Subsequently, Eq. (S9) is expressed as a function of the masking number Γ Q . _ = 1 √2Γ #(S10)

Relation between bit error ratio (BER) P BER and signal-to-noise ratio per bit E b /N 0
When data modulation is quadrature phase shift keying (M = 4), a BER P BER is obtained from a signal-to-noise ratio per bit E b /N 0 as follows:

Distance between adjacent signals
Standard deviation of shot noise

Phase
Using this equation, E b /N 0 is calculated as 2.2 (3.42 dB) for a BER of 1.8 × 10 -2 which is a typical threshold of soft decision forward error correction (SD-FEC) with an overhead of 20 %.