# Quantum Sphere-Packing Bounds with Polynomial Prefactors

## TL;DR

This paper advances quantum sphere-packing bounds by establishing polynomial prefactors, clarifying the relationship between bounds, and providing nearly exact error probability characterizations for certain classical-quantum channels.

## Contribution

It introduces a new sphere-packing bound with polynomial prefactors and characterizes error probabilities for symmetric channels, improving upon previous bounds.

## Key findings

- Established a variational representation of exponents using Golden-Thompson inequality.
- Improved Dalai's prefactor from subexponential to polynomial order.
- Showed the error exponent gap vanishes at high rates for constant composition codes.

## Abstract

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 $o(\log n / n)$, 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.