Resource requirements of private quantum channels and consequence for oblivious remote state preparation

Shannon 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 quant-ph/0003101 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 of these cases and this includes the results of Shannon and Ambainis et al. As a consequence of our arguments we also show that in a one-way oblivious Remote state preparation protocol for transferring an n-qubit pure state, the entropy of the communication must be 2n and the entanglement measure of the shared resource must be n. This generalizes on the result of Leung and Shor which shows same bound on the length of communication in the special case when the shared resource is maximally entangled e.g. EPR pairs and hence settles an open question asked in their paper regarding protocols without maximally entangled shared resource.