  The security of a cryptographic key that is generated by communication
through a noisy quantum channel relies on the ability to distill a shorter
secure key sequence from a longer insecure one. For an important class of
protocols, which exploit tomographically complete measurements on entangled
pairs of any dimension, we show that the noise threshold for classical
advantage distillation is identical with the threshold for quantum entanglement
distillation. As a consequence, the two distillation procedures are equivalent:
neither offers a security advantage over the other.
