In this talk, we study the interplay between the transmission rate, blocklength, and error probability in channel coding problems. First, we discuss crucial properties of the error exponent function, which characterizes how fast the optimal error probability in classical coding over classical-quantum channels decay.
Second, we establish lower bounds on the optimal error probability, commonly termed quantum sphere-packing bounds. Our result significantly improves the existing 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(log n / n), indicating our sphere-packing bound is almost exact in the high rate regime.
Finally, we study the situation when the transmission rates approach channel capacity slowly, a research topic known as moderate deviation analysis. We prove that the optimal error probability vanishes under this rate convergence, and show how strong large deviation techniques are employed in this error regime.