Evaluation of the phase randomness of a light source in quantum-key-distribution systems with an attenuated laser

Phase-randomized light is one of the key assumptions in the security proof of the Bennett-Brassard 1984 (BB84) quantum-key-distribution (QKD) protocol using an attenuated laser. Though the assumption has been believed to be satisfied for conventional systems, it should be reexamined for current high-speed QKD systems. The phase correlation may be induced by the overlap of the optical pulses, the interval of which decreases as the clock frequency increases. The phase randomness was investigated experimentally by measuring the visibility of interference. An asymmetric Mach-Zehnder interferometer was used to observe the interference between adjacent pulses from a gain-switched distributed feedback laser diode driven at 10 GHz. Low visibility was observed when the minimum drive current was set far below the threshold, while interference emerged when the minimum drive current was close to the threshold. The theoretical evaluation on the impact of the imperfect phase randomization provides target values for the visibility to guarantee the phase randomness. The experimental and theoretical results show that secure implementation of decoy BB84 protocol is achievable even for the 10-GHz clock frequency by using the laser diode under proper operating conditions.

Figure 1

A schematic illustration of a measurement apparatus for the phase correlation between the adjacent pulses. The delay time of the asymmetric interferometer is adjusted to the pulse period. The phase difference $\phi$ between the paths can be modulated.

Figure 2

The imbalance of the quantum coin, defined by $\Delta =[1-F({\rho }_Z,{\rho }_X)]/2$, as a function of standard deviation of phase distribution. Solid lines represent $\Delta$ for the difference of central phase value ${\theta }_0=0$, broken lines for ${\theta }_0=\pi$.

Figure 3

Distinguishability $[1-F({\rho }_1,{\rho }_2)]/2$ between the signal and decoy states as a function of the standard deviation of phase. Calculation was done for the partially phase randomized states of average photon number 0.5 (signal) and 0.1 (decoy). Solid line represents the calculated fidelity for ${\theta }_0=\pi$. Dashed-dot line stands for the distinguishability between the completely phase randomized states, and dashed line between coherent states with ${\theta }_0=\pi$.

Figure 4

Waveforms of the LD pulses. The values of the normalized minimum excitation are as follows: (a) $\Lambda =-1.6$, (b) $\Lambda =0.074$, (c) $\Lambda =0.49$, and (d) $\Lambda =2.6$.

Figure 5

Interference fringes for several values of the normalized minimum excitation: (a) $\Lambda =-1.6$, (b) $\Lambda =0.074$, (c) $\Lambda =0.49$, and (d) $\Lambda =2.6$.

Figure 6

Magnified view of the interference fringe. (a) $\Lambda =-1.6$ and (b) $\Lambda =-1.2$. Solid lines denote the fitted curve. The fitted values of the visibility were 0.004 and 0.02.

Figure 7

Visibility of the interference as a function of the normalized minimum excitation $\Lambda$. For $\Lambda >0$, the LD was always turned on. The LD was reversely biased at the bottom of the pulse, when $\Lambda <-1$.

Figure 8

Probability of correct decision in the optimal discrimination of the signal and decoy states as a function of the standard deviation of phase. The state discrimination was done for the partially phase-randomized states of average photon number 0.5 and 0.1. Inconclusive results are allowed in the state discrimination. Squares represent $P_{\text{inc}}=0.983,{\theta }_0=0$, circles $P_{\text{inc}}=0.983,{\theta }_0=\pi$, and triangles $P_{\text{inc}}=0.712,{\theta }_0=\pi$. Dotted line and dashed line refer to the probability of correct decision on the phase-randomized states under the conditions of $P_{\text{inc}}=0.983$ and $P_{\text{inc}}=0.712$, respectively.