Secure quantum communication in the presence of phase- and polarization-dependent loss

Silicon photonics holds the promise of the miniaturization of quantum communication devices. Recently, silicon chip optical transmitters for quantum key distribution (QKD) have been built and demonstrated experimentally. Nonetheless, these silicon chips suffer substantial phase- and polarization-dependent loss (PDL) which, if unchecked, could compromise the security of QKD systems due to overestimation of the secret key rate. Here, first, we restore the security by regarding single photons without phase and polarization dependence as untagged and secure qubits. Next, by using a postselection technique, we implement a secure QKD protocol that provides a high key generation rate even in the presence of severe phase- and polarization-dependent loss. Our solution is simple to realize in a practical experiment, as it does not require any hardware modification.

Figure 1

(a) Schematic of the QKD source used for the PDL measurement. A CW laser is attenuated using a variable optical attenuator (VOA). The setup further consists of a phase modulator (PM) for phase randomization and an intensity modulator (IM) to generate pulses. The polarization of the weak coherent pulses is modulated by a polarization modulator (PolM) which is based on a structure proposed in [16] using a Faraday mirror (FM). Two electronic polarization controllers (EPCs) are included in the source for polarization alignment. (b) Variation of power at the output of the source with respect to the voltage applied to the phase modulator in the polarization modulator. The maximum variation of the output power for different polarization states is around 0.2 dB.

Figure 2

By numerical simulation, we obtain the optimal value of ${\mu }_s$ that satisfies Eq. (20). When PDL increases, the optimal ${\mu }_s$ increases as well in order to generate more single photons. The experimental parameters are listed in Table 1.

Figure 3

(a) Optimal values of ${\mu }_s$ that maximize the secret key rate according to our numerical simulations. (b) Optimal values of $P$. The value of PDL increases from 0 to 10 dB (from top to bottom). (c) Deviation between the numerically optimal value of $P$ and the theoretical optimal value, $P_{\mathrm{optimal}}$, obtained from Eq. (18). As the deviation is rather small, this verifies the validity of our heuristic argument.

Figure 4

Comparison between (a) a QKD system with a postselection scheme ($P=P_{\mathrm{optimal}}$, ${\mu }_s={\mu }_{s,\mathrm{optimal}}$) and (b) a QKD system without a postselection scheme ($P=1$, ${\mu }_s={\mu }_{s,\mathrm{optimal}}^{\prime }$). The secret key rate can be largely increased if the system suffers large PDL. The value of PDL increases from 0 to 10 dB (from top to bottom) in (a) and (b).

Figure 5

The deviation of $Y_1$ is proportional to the intensity of the decoy states. Large PDL values lead to large deviations. The value of PDL increases from 0 to 10 dB (from bottom to top).

Figure 6

Here, we consider the two-decoy-state method (decoy + vacuum) with ${\mu }_w=0$ and numerically optimize the secret key rate over the free parameters, ${\mu }_s$ and ${\mu }_v$. Also, we use the optimal value of $P$ given by Eq. (18). (a) We estimate the secret key rate with different PDL values. The value of PDL increases from 0 to 10 dB (from top to bottom). (b) We compare the two-decoy-state method (bottom) with the asymptotic case (top) described in Fig. 4. The value of PDL is 10 dB. The two-decoy-state method works well, since the resulting secret key rate is close to the asymptotic case.

Figure 7

Simulation result of the finite data size effect. We select the total number of pulses to be between $N={10}^{10}$ and $N={10}^{14}$ (from left to right). The value of PDL is 10 dB. As expected, the secret key rate approaches the asymptotic case if we increase the data size.