of the HSW theorem but only provide a background illustrating the similarity with its classical counterpart. The resemblance with the channel capacity X and the Holevo bound, as described in Chapter 21, and with the classical mutual information H(X; Y), as described in Chapter 5, are both discussed. For advanced reference, a hint is provided as to the meaning of the still not fully explored concept of quantum coherent information. Several examples of quantum channel capacity, derived from direct appli- cations of the HSW theorem, along with the solution of the maximization problem, are provided. Noisy quantum channels The notion of "noisiness" in a classical communication channel was first introduced in Chapter 12, when describing channel entropy. Such a channel can be viewed schemat- ically as a probabilistic relation between two random sources, X for the originator, and Y for the recipient. These sources are defined by symbol alphabets, X = {x1, x2} and Y = {y1, for instance, and their associated probabilities p(xi), p(yj). An originator message is, thus, a string of n symbols Xi E X randomly selected from X according to the distribution p(xi), which is transformed at the recipient's end into a string of n symbols yi E Y randomly selected from Y according to the distribution p(yi). An ideal or noiseless channel is such that the knowledge of any symbol Xi input from the origi- nator absolutely conditions the knowledge of the symbol yj obtained at the recipient's end. Thus, for any originator symbol Xi, there must be a recipient symbol Yj, such that p(yj|xi) = 1 and p(yj|xii#i) = 0. The set of conditional probabilities p(yj|xi) can be put in the form of a transition matrix P(Y|X) and in the case of a noiseless channel with Quantum channel noise and channel capacity a two-symbol alphabet (binary channel) we must have (23.1) In the nonideal case where the channel is corrupted by noise, we have p(yj)xi)=1-8, where 0<E< 1 defines the symbol error probability (the probability of mistaking one symbol for another). The 