 maximum: C = max H(X;) Y). (23.41) The coding theorem states that: Given a noisy transmission channel, there exist binary codes of sufficiently long codeword size, with 2nR codewords of sufficient length n, and code rate R < C, for which an originator message can be transmitted to a recipient with arbitrary low error probability. 23.2 The Holevo-Schumacher-Westmoreland capacity theorem It is clear that Shannon', coding theorem and the HSW theorem are conceptually very similar. As in QIT, the typical set of codewords (or of typical sequences) is defined under the condition (Eq. (13.25)): (23.42) where p(x) is the codeword probability distribution associated with the symbol source X, and the entropy H(X). This condition parallels that in Eq. (23.29), with p(x) H(X) (S). In the classical case, the dimension of the typical set is upper- bounded by 2n(H+s). while in the quantum case, the bound is 2n((S)+8), see Eq. (23.30). Because of this analogy, it is clear that the HSW theorem owes a great deal to Shannon's classical analysis, even if its background assumptions are hardly reducible to any classical conception. Then we may ask: "Is quantum channel capacity in any way reducible to Shannon's original concept?" The following discussion, although with no pretence at providing any academic proof, may guide towards some form of a satisfactory answer; we expected no! We may now conclude this section with a discussion about the connection between the HSW theorem and the Holevo bound described in Chapter 21, and clarifying the difference between X and the classical channel capacity C. Indeed, letting X' = S ((o)) - (S(o)), we recognize from Chapter 21 that X' is exactly the definition of the Holevo bound for the quantum source o, and the HSW theorem seems simply to state that the channel capacity is given by X = max X'. Then, apart from the maximization issue, we may wonder what is conceptually new and useful in this HSW theorem that has not already been captured into the Holevo bound concept. The answer to this apparent p