Efficient quantum key distribution using Fibonacci-number coding with a biased basis choice

The conjugation relation between the Lucas sequence and the Fibonacci sequence leads us to devise some simple modifications which essentially increase the efficiency and the capacity of the quantum key distribution scheme proposed by Simon et al. (2013) [8]. The first major ingredient of our scheme is the sources for preparing entangled OAM states based on a Vogel spiral with the same Fibonacci numbers, which can prepare two types of entangled states. The second major ingredient is the distribution of significantly different probabilities to the entangled state photons and measurement bases during both transmission and reception, thus reducing the fraction of discarded data. The third major ingredient is that we use the close relationships among the Lucas sequence, Chebyshev maps and k-Chebyshev maps to greatly enhance the capacity of each entangled particle for key generation without the limit of spiral and OAM bandwidths. Lastly, by combining the ideas proposed by Lo et al. (2005) [9] and Xue et al. (2002) [10], we divide the accepted data into different subsets in terms of the types of measurements chosen and estimate an error rate for each subset separately to guarantee the security of our proposed scheme.