[urgoroger]

Yes, this is pretty much the basis to quantum key distribution protocols, and probably one of the first studied uses of quantum information in general. Consider BB84 (http://www.cse.wustl.edu/~jain/cse571-07/ftp/quantum/, https://en.wikipedia.org/wiki/BB84, apparently the first quantum cryptography protocol), which would allow the transmission of a one time pad with an any eavesdropper being unnoticed with provably exponentially decreasing probability in terms of key length. The one time pad can then be used to securely transmit data.

Thus it is possible to transmit arbitrary (classical) data with exponentially decreasing probability that we do not detect an eavesdropper. Alice and Bob can communicate with each other knowing that there is provably pretty much no chance that anyone else knows what they said to each other.