Quantum Cryptography on Noisy Channels: Quantum versus Classical Key-Agreement Protocols

When the 4-state or the 6-state protocol of quantum cryptography is carried out on a noisy quantum channel, then the raw key has to be processed to reduce the information of a spy down to an arbitrarily low value, providing Alice and Bob with a secret key. In principle, quantum algorithms as well as classical algorithms can be used for this processing. A natural question is: Up to which error rate on the raw key is a secret-key agreement at all possible? Under the assumption of incoherent eavesdropping, we find that the quantum and classical limits are precisely the same: As long as Alice and Bob share some entanglement, both quantum and classical protocols provide secret keys.