Passive sources for the Bennett-Brassard 1984 quantum-key-distribution protocol with practical signals

Most experimental realizations of quantum key distribution are based on the Bennett-Brassard 1984 (the so-called BB84) protocol. In a typical optical implementation of this scheme, the sender uses an active source to produce the required BB84 signal states. While active state preparation of BB84 signals is a simple and elegant solution in principle, in practice passive state preparation might be desirable in some scenarios, for instance, in those experimental setups operating at high transmission rates. Passive schemes might also be more robust against side-channel attacks than active sources. Typical passive devices involve parametric down-conversion. In this paper, we show that both coherent light and practical single-photon sources are also suitable for passive generation of BB84 signal states. Our method does not require any externally driven element, but only linear optical components and photodetectors. In the case of coherent light, the resulting key rate is similar to the one delivered by an active source. When the sender uses practical single-photon sources, however, the distance covered by a passive transmitter might be longer than that of an active configuration.

Figure 1

(Case A) Example of an active setup with four laser diodes controlled by a random number generator (RNG). Each source emits one different BB84 signal state [e.g., $H$ (horizontal), $V$ (vertical), $L$ (circular left), or $R$ (circular right) polarization states]. BS denotes a beam splitter. (Case B) Example of an active setup with one laser diode together with a polarization modulator controlled by a RNG. This modulator rotates the state of polarization of the incoming pulses to generate BB84 signal states. (Case C) Basic setup of a passive QKD transmitter with linear optics elements. One or more laser diodes produce different signal states that are sent through a linear optics network. Depending on the detection pattern observed in the detectors $D_i$, different signal states are actually generated. No RNG or active modulator is necessary in this last scenario.

Figure 2

Basic setup of a passive BB84 QKD source with strong coherent light. The mean photon number of the signal states ${\rho }_{+{45}^{°}}$ and ${\rho }_{-{45}^{°}}$ can be chosen very high; for instance, $\approx$${10}^8$ photons. The parameter $t$ represents the transmittance of the BS; it satisfies $t\ll 1$.

Figure 3

Graphical representation of the valid regions for the angle $\theta$. These regions are marked in gray. They depend on an acceptance parameter $\Omega \in [0,\pi /4]$.

Figure 4

Lower bound on the secret key rate $R$ given by Eq. (11) in logarithmic scale for the passive source illustrated in Fig. 2 (dashed line). The signal states ${\rho }_{\mathrm{out},\theta }$ are given by Eq. (6). For simulation purposes, we consider the following experimental parameters: the dark count rate of Bob’s detectors is ${\epsilon }_B=1\times {10}^{-6}$, the overall transmittance of Bob’s detection apparatus is ${\eta }_B=0.1$, the loss coefficient of the channel is $\alpha =0.2$ dB/km, $q=1/2$, and the efficiency of the error correction protocol is $f(E)=1.22$. The solid line represents a lower bound on $R$ when Alice employs an active source. The inset figure shows the value for the optimized parameters $\mu$ (dashed line) and $\Omega$ (solid line) in the passive setup.

Figure 5

Alternative implementation scheme with only one pulsed laser source. The delay introduced by one arm of the interferometer is equal to the time difference between two consecutive pulses. The polarization rotator $R_{-{45}^{°}}$ changes the $+{45}^{°}$ linear polarization of the incoming pulses to $-{45}^{°}$ linear polarization. The intensity modulator (IM) blocks either all the even or all the odd optical pulses in mode $a$.

Figure 6

Basic setup of a passive BB84 QKD phase-encoding transmitter with strong coherent light. The mean photon number of the signal states, ${\rho }_{\mathrm{source}}$, can be chosen very high; for instance, $\approx$${10}^8$ photons.

Figure 7

Basic setup of a passive BB84 QKD transmitter with practical SPS. Each laser diode emits pulses prepared in a different BB84 polarization state: $H$ (horizontal) and $V$ (vertical) linear polarizations, and $L$ (left) and $R$ (right) circular polarizations. The parameter $t$ represents the transmittance of the BSs. All detectors shown in the figure denote threshold single-photon detectors.

Figure 8

Lower bound on the secret key rate $R$ given by Eq. (11) in logarithmic scale for the passive setup illustrated in Fig. 7 with perfect on-demand SPS, i.e., $p_1=1$ in Eq. (14). In the simulation we consider the following experimental parameters: the dark count rate of Alice’s and Bob’s detectors is ${\epsilon }_A={\epsilon }_B=1\times {10}^{-6}$, the overall transmittance of Bob’s detection apparatus is ${\eta }_B=0.1$, the loss coefficient of the channel is $\alpha =0.2$ dB/km, $q=1/2$, and the efficiency of the error correction protocol is $f(E)=1.22$. We further assume the channel model described in Appendix , where we neglect any misalignment effect. Otherwise, the actual secure distance will be smaller. The figure includes three cases, depending on the actual value of the efficiency ${\eta }_A$ of Alice’s detectors, and we optimize the transmittance $t$ of the BSs on Alice’s side for each case. In particular, we assume ${\eta }_A=1$ (dashed line), ${\eta }_A=0.5$ (thin solid line), and ${\eta }_A=0.1$ (dash-dotted line). The solid line represents a lower bound on $R$ when Alice employs an active source that emits single-photon pulses with probability 1.

Figure 9

Lower bound on the secret key rate $R$ given by Eq. (11) in logarithmic scale for the passive setup illustrated in Fig. 7 with imperfect SPS. In particular, we consider that the photon number distribution $p_n$ of the sources is given by $p_0=0.0099$, $p_1=0.9882$, and $p_2=1-p_0-p_1=0.0019$. We further assume the same experimental data used in Fig. 8. We study three cases, depending on the actual value of the efficiency ${\eta }_A$ of Alice’s detectors, and we optimize the transmittance $t$ of the BSs on Alice’s side for each case. Specifically, we assume ${\eta }_A=1$ (dashed line), ${\eta }_A=0.5$ (thin solid line), and ${\eta }_A=0.1$ (dash-dotted line). The solid line represents a lower bound on $R$ when Alice employs an active source with photon number distribution $p_n$ in combination with a BS. In this last case, we optimize the value of the transmittance of the BS as a function of the distance.

Figure 10

Lower bound on the secret key rate $R$ given by Eq. (11) in logarithmic scale for the passive setup illustrated in Fig. 7 with imperfect SPS. The photon number distribution $p_n$ of the sources is given by $p_0=0.2$, $p_1=0.785$, and $p_2=1-p_0-p_1=0.015$. We further assume the same experimental data used in Fig. 8. We study three cases, depending on the actual value of the efficiency ${\eta }_A$ of Alice’s detectors, and we optimize the transmittance $t$ of the BSs on Alice’s side for each case. In particular, we assume ${\eta }_A=1$ (dashed line), ${\eta }_A=0.5$ (thin solid line), and ${\eta }_A=0.1$ (dash-dotted line). The solid line represents a lower bound on $R$ when Alice employs an active source with photon number distribution $p_n$ in combination with a BS. In this last case, we optimize the value of the transmittance of the BS as a function of the distance.

Figure 11

Alternative implementation scheme with only one photon source. The polarization rotators $R_V$, $R_L$, and $R_R$ change the horizontal polarization of the incoming pulses to vertical polarization, left circular polarization, and right circular polarization, respectively.