Bennett-Brassard 1984 quantum key distribution using conjugate homodyne detection

Optical homodyne detection has been widely used in continuous-variable (CV) quantum information processing for measuring field quadrature. In this paper we explore the possibility of operating a conjugate homodyne detection system in “photon counting” mode to implement discrete-variable (DV) quantum key distribution (QKD). A conjugate homodyne detection system, which consists of a beam splitter followed by two optical homodyne detectors, can simultaneously measure a pair of conjugate quadratures $X$ and $P$ of the incoming quantum state. In classical electrodynamics, $X^2+P^2$ is proportional to the energy (the photon number) of the input light. In quantum optics, $X$ and $P$ do not commute and thus the above photon-number measurement is intrinsically noisy. This implies that a blind application of standard security proofs of QKD could result in pessimistic performance. We overcome this obstacle by taking advantage of two special features of the proposed detection scheme. First, the fundamental detection noise associated with vacuum fluctuations cannot be manipulated by an external adversary. Second, the ability to reconstruct the photon number distribution at the receiver's end can place additional constraints on possible attacks from the adversary. As an example, we study the security of the BB84 QKD using conjugate homodyne detection and evaluate its performance through numerical simulations. This study may open the door to a family of QKD protocols, complementary to the well-established DV-QKD based on single-photon detection and CV-QKD based on coherent detection.

Figure 1

Conjugate optical homodyne detection. ${\text{BS}}_{1-4}$, symmetric beam splitter; ${\text{BD}}_{1-2}$, balanced photodetector; LO, local oscillator.

Figure 2

(Simulation results) The probability distributions $P_Z\left(z\right|n)$ corresponding to a different input photon number $n$.

Figure 3

(Simulation results [25]) Detection efficiency ${\eta }_D$, dark count probability ${\upsilon }_D$, and the ratio $R={\eta }_D/{\upsilon }_D$.

Figure 4

The mutual information $I_{\mathrm{AB}}$ in three different cases: Perfect SPDs, independent detection mode, and differential detection mode. We assume the quantum channel is standard telecom fiber with an attenuation coefficient of $\gamma =0.2dB/\mathrm{km}$.

Figure 5

Secure-key rates using the standard security proof [29]. Green dashed line, independent detection mode; blue solid line, perfect SPDs.

Figure 6

A schematic diagram of the virtual detection model. Mod, polarization modulator for basis selection; PBS, polarizing beam splitter; ${\text{S}}_0$ and ${\text{S}}_1$, virtual ideal nondemolition SPDs; ${\text{D}}_0$ and ${\text{D}}_1$, real noisy detectors used by Bob. One key idea is to use the real detectors' outputs $\left\{B_i\right\}$ to estimate the quantum bit error rate of the virtual detectors' outputs $\left\{B_i^{\left(V\right)}\right\}$.

Figure 7

Secure-key rates using the improved security analysis (independent detection mode). $E_d$, error probability due to polarization misalignment. As a comparison, the secure-key rate of the BB84 QKD implemented with perfect SPDs at $E_d=0$ is also presented.

Figure 8

Optimal detection threshold $\tau$. Note there is a jump of $\tau$ around 8.2 km when $E_d=0.05$ (see discussion in the main text).

Figure 9

Secure-key rates using the improved security analysis (differential detection mode). No secure key can be generated when $E_d=0.05$. As a comparison, the secure-key rates of the BB84 QKD implemented with perfect SPDs at $E_d=0$ are also presented.

Figure 10

Secure-key rates using the tighter bound in Appendix (independent detection mode).

Figure 11

Secure-key rates using the tighter bound in Appendix (differential detection mode).