A remark on quantum key distribution with two way communication: the classical complexity in decoding the CSS code can be removed

摘要

So far all the proven unconditionally secure prepare and measure protocolsfor the quantum key distribution(QKD) must solve the very complex problem ofdecoding the classical CSS code. In the decoding stage, Bob has to compare hisstring with an exponentially large number of all the strings in certain codespace to find out the closest one. Here we have spotted that, in anentanglement purification protocol(EPP), the random basis in the statepreparation stage is only necessary to those check qubits, but uncessary to thecode qubits. In our modified two way communication EPP(2-EPP) protocol, Aliceand Bob may first take all the parity checks on $Z$ basis to reduce the bitflip error to strictly zero with a high probability, e.g., $1-2^{-30}$, andthen use the CSS code to obtain the final key. We show that, this type of 2-EPPprotocol can be reduced to an equivalent prepare and measure protocol. In ourprotocol, the huge complexity of decoding the classical CSS code is totallyremoved.