Optimal ratio between phase basis and bit basis in quantum key distributions

In the original Bennett-Brassard 1984 protocol, the bit basis and the phase basis are used with equal probability. Lo et al. [J. Cryptology 18, 133 (2005)] proposed modifying the ratio between the two bases by increasing the final key generation rate. However, the optimum ratio has not yet been derived. In this Rapid Communication, in order to examine this problem, the ratio between the two bases is optimized for exponential constraints, given Eve’s information distinguishability and the final error probability.

Figure 1

(Color online) Asymptotic key generation rate: The upper line shows $\max R_A$, while the lower line shows $\max R_S$.

Figure 2

(Color online) Optimal choice rate for the check bits: The upper line is $\mathrm{argmax}{R}_S$, while the lower line is $\mathrm{argmax}{R}_A$.

Figure 3

(Color online) Optimal choice rate of × basis.