Resource Requirements of Private Quantum Channels

Authors: Rahul Jain

Publish Date: 2010/10/02

Volume: 25, Issue: 1, Pages: 1-13

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Abstract

Shannon Bell Syst Tech J 27623–656 1948 Bell Syst Tech J 28656–715 1949 in celebrated work had shown that n bits of shared key are necessary and sufficient to transmit nbit classical information in an informationtheoretically secure way using oneway communication Ambainis Mosca Tapp and de Wolf in Proceedings of the 41st Annual IEEE Symposium on Foundation of Computer Science pp 547–553 2000 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 oneway communication is allowed They show that in this case 2n bits of shared key is necessary and sufficient to transmit an nqubit state We consider the most general setting in which we allow for all possible combinations in oneway communication ie 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 simultaneously tight bounds on communication/shared resources in all of these cases and this includes the results of Shannon and Ambainis et alAs a consequence of our arguments we also show that in a oneway oblivious remote state preparation protocol for transferring an nqubit pure state the entropy of the communication must be 2n and the entanglement measure of the shared resource must be n This generalizes the result of Leung and Shor Phys Rev Lett 90 2003 which shows the same bound on the length of communication in the special case when the shared resource is maximally entangled eg EPR pairs and hence settles an open question asked in their paper regarding protocols without maximally entangled shared resource

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