Resource requirements of private quantum channels

Shannon [Sha48, Sha49] in celebrated works had shown that n bits of shared key is necessary and sufficient to transmit n-bit classical information in an information-theoretically secure way. Ambainis, Mosca, Tapp and de Wolf in [AMTdW00] considered a more general setting, referred to as Private quantum channels, in which instead of classical information, quantum states are required to be transmitted and only one-way communication is allowed. They show that in this case 2n bits of shared key is necessary and sufficient to transmit an n-qubit state. We consider the most general setting in which we allow for all possible combinations i.e. we let the input to be transmitted, the message sent and the shared resources to be classical/quantum. We develop a general framework by which we are able to show tight bounds on communication/shared resources in all but one of these cases and this includes the results of Shannon and Ambainis et al.