Robust unconditionally secure quantum key distribution with two nonorthogonal and uninformative states

We present an unconventional form of decoy-state technique to make the single-photon Bennett 1992 protocol robust against losses and noise of a communication channel. Two uninformative states are prepared by the transmitter in order to prevent the unambiguous state discrimination attack and improve the phase-error rate estimation. The presented method does not require strong reference pulses, additional electronics, or extra detectors for its implementation.

Figure 1

(Color online) Plots of the secure gain $G$ as a function of the channel loss rate $L$ for the standard B92 protocol (black circles) and for the B92-1 (red circles). The B92-1 scales linearly with $L$, while the B92 protocol scales nearly quadratically and ceases to provide a positive gain at about $L=0.6$. The curves are drawn assuming a depolarizing channel with $p=0.01$ (see the text). On the top right, the states of traditional B92 protocol (black arrows) and the extra uninformative states (red arrows) used in the two modifications of the B92 protocol (see the text).

Figure 2

(Color online) Relative phase-error rate upper bound ${\overline{\Lambda }}_{ph}$ (top diagram) and secure gain $G$ (bottom diagram) of the B92-1 versus the squared scalar product of the two signal states ${\left|⟨{\phi }_0|{\phi }_1⟩\right|}^2$ for channel transmission $\eta =\left\{0,10,20,30\right\}\text{dB}$ and depolarizing rate $p=0.01$. The vertical dotted lines at $0.322\left(\theta =55.4°\right)$ and $0.538\left(\theta =42.8°\right)$ mark, respectively, the maximum of $G$ and the minimum of ${\overline{\Lambda }}_{ph}$.