Reliability Function of Classical-Quantum Channels

Reliability function, defined as the optimal error exponent describing the exponential decay of decoding error probability when the communicating rate is below the capacity of the channel, is one of the fundamental problems in information theory. In this work, we determine the reliability function for a general cq channel. The main contribution is a lower bound for the error exponent which is characterised by the Renyi divergence in Petz's form. It turns out that the lower bound matches the upper bound given by Dalai (IEEE Transactions on Information Theory 46, 2256 (2000)) when the rate is not very low. Thus the reliability function is obtained by combining these two bounds in a proper range of communicating rate. The approach to derive the lower bound makes use of tricks on types and an observation by Renes (arXiv: 2207.08899) that channel code can be constructed from data compression scheme for uniform distribution relative to side information, whose solution to the error exponent problem is in turn determined by its dual problem -- privacy amplification, for which the exact error exponent is known.