This paper shows that the capacity of a classical-quantum channel with arbitrary (possibly mixed) signal states equals the maximum of the entropy bound over all prior distributions. This result completes the work of Hausladen, Jozsa, Schumacher, Westmoreland, and Wooters, who proved the equality for pure states. The capacity of a quantum channel is defined as the limit of the average information per symbol as the number of channel uses tends to infinity. The main result is that the capacity is given by the maximum of the difference between the entropy of the average state and the sum of the entropies of individual states, over all prior distributions. This confirms the old physical wisdom that the entropy bound is used to evaluate the quantum capacity. The proof relies on the idea of projecting onto typical subspaces and estimating the error probability, which is more complex than in the case of pure states. The key idea is that the entropy bound provides a lower bound for the capacity, and the result shows that this bound is tight for general states. The paper also discusses the concept of typical subspaces and the choice of suboptimal decision rules, and provides an estimate for the error probability. The result is important because it shows that the quantum capacity is not simply the sum of the capacities of individual states, but depends on the prior distribution. This is in contrast to classical channels, where the capacity is additive. The paper concludes that the capacity of a quantum channel with arbitrary signal states is given by the maximum of the entropy bound over all prior distributions.This paper shows that the capacity of a classical-quantum channel with arbitrary (possibly mixed) signal states equals the maximum of the entropy bound over all prior distributions. This result completes the work of Hausladen, Jozsa, Schumacher, Westmoreland, and Wooters, who proved the equality for pure states. The capacity of a quantum channel is defined as the limit of the average information per symbol as the number of channel uses tends to infinity. The main result is that the capacity is given by the maximum of the difference between the entropy of the average state and the sum of the entropies of individual states, over all prior distributions. This confirms the old physical wisdom that the entropy bound is used to evaluate the quantum capacity. The proof relies on the idea of projecting onto typical subspaces and estimating the error probability, which is more complex than in the case of pure states. The key idea is that the entropy bound provides a lower bound for the capacity, and the result shows that this bound is tight for general states. The paper also discusses the concept of typical subspaces and the choice of suboptimal decision rules, and provides an estimate for the error probability. The result is important because it shows that the quantum capacity is not simply the sum of the capacities of individual states, but depends on the prior distribution. This is in contrast to classical channels, where the capacity is additive. The paper concludes that the capacity of a quantum channel with arbitrary signal states is given by the maximum of the entropy bound over all prior distributions.