Robustness of a quantum key distribution with two and three mutually unbiased bases

We study the robustness of various protocols for quantum key distributions. We first consider the case of qutrits and study quantum protocols that employ two and three mutually unbiased bases. We then derive the optimal eavesdropping strategy for two mutually unbiased bases in dimension 4 and generalize the result to a quantum key distribution protocol that uses two mutually unbiased bases in arbitrary finite dimensions.

Figure 1

Mutual information for Alice and Bob $\left[I\left(\mathrm{AB}\right)\right]$ and Alice and Eve as a function of the disturbance $D$ for three-dimensional quantum states in a scheme with two $\left[I2\left(\mathrm{AE}\right)\right]$, three $\left[I3\left(\mathrm{AE}\right)\right]$, and four $\left[I4\left(\mathrm{AE}\right)\right]$ mutually unbiased bases. The latter curve was derived in Ref. 6.

Figure 2

Mutual information for Alice and Bob $\left[I\left(\mathrm{AB}\right)\right]$ and Alice and Eve as a function of the disturbance for four-dimensional quantum states in a scheme with two $\left[I2\left(\mathrm{AE}\right)\right]$ and five $\left[I5\left(\mathrm{AE}\right)\right]$ mutually unbiased bases. The latter curve was conjectured in Ref. 6.