Unconditional security of the Bennett 1992 quantum key-distribution protocol over a lossy and noisy channel

We show that the security proof of the Bennett 1992 protocol over loss-free channel given by Tamaki, Koashi, and Imoto [Phys. Rev. Lett. 90, 167904 (2003)] can be adapted to accommodate loss. We assumed that Bob’s detectors discriminate between single-photon states on one hand and vacuum state or multiphoton states on the other hand.

Figure 1

The estimated phase error rate ${\overline{n}}_{\text{ph}}∕n_{\text{fil}}$ (dot-dashed line), the actual phase error rate (solid line), and bit error rate (dotted line) in case of $p=0.01$. (a) is for $L=0$ and (b) is for $L=0.5$. We assume that quantum channel is a depolarizing channel with loss.

Figure 2

(a) The optimum key generation rate $G$ for the case $\gamma =\beta$ and (b) the optimum value of ${\mid ⟨{\phi }_0\mid {\phi }_1⟩\mid }^2$. Here we assume that the quantum channel is the depolarizing channel with loss. The dotted line, the dot-dashed line, and solid line represent the case where $L=0$, $L=0.2$, and $L=0.5$, respectively.