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. 
Georgios M. Nikolopoulos, Kedar S. Ranade, and Gernot Alber