Equivalence of three quantum algorithms: Privacy amplification, error correction, and data compression
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Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study quantum algorithms of these three types (PA, EC, and DC) in the one-shot scenario, and show that they all become equivalent if modified properly. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.
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