Error tolerance of two-basis quantum-key-distribution protocols using qudits and two-way classical communication

We investigate the error tolerance of quantum cryptographic protocols using $d$-level systems. In particular, we focus on prepare-and-measure schemes that use two mutually unbiased bases and a key-distillation procedure with two-way classical communication. For arbitrary quantum channels, we obtain a sufficient condition for secret-key distillation which, in the case of isotropic quantum channels, yields an analytic expression for the maximally tolerable error rate of the cryptographic protocols under consideration. The difference between the tolerable error rate and its theoretical upper bound tends slowly to zero for sufficiently large dimensions of the information carriers.

Figure 1

$2d$-state QKD protocols: The tolerable error rate $D_{2\mathrm{CC}}$ (dashed line) and its theoretical upper bound $D_{\mathrm{th}}$ (solid line) as functions of the dimension $d$. Secret-key distillation is impossible in the regime (I), while it may be possible for error rates below $D_{\mathrm{th}}$. In the regime (II), a secret key can be distilled by means of the key-distillation procedure considered here. Inset: The gap between the two regimes $\delta \left(d\right)=D_{2\mathrm{CC}}-D_{\mathrm{th}}$ is plotted as a function of the dimension. The symbols (triangles, circles, and squares) correspond to prime dimensions.