Abstract

This talk will combine two recent results of mine. 1. arXiv:1704.05703: We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai’s expression is stronger in general classical-quantum channels. Second, we establish a sphere-packing bound for classical-quantum channels, which significantly improves Dalai’s prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(logn/n), indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions. 2. arXiv:1701.03195: In this work, we study the optimal decay of error probability when the transmission rate approaches channel capacity slowly, a research area known as moderate deviation analysis. Our result shows that the reliable communication through a classical-quantum channel with positive channel dispersion is possible when the transmission rate approaches the channel capacity a rate slower than 1/sqrt(n). The proof employs a refined sphere-packing bound in strong large deviation theory, and the asymptotic expansions of the error-exponent functions. Moderate deviation analysis for quantum hypothesis testing is also established. The converse directly follows from our channel coding result. The achievability can be proved by using a recent noncommutative concentration inequality.