al information H(X;Y)=H(Y)-H(YX) (23.47) does in Shannon's classical theory. In particular, it can be shown that given the entropy difference between source and originator, AS=S(o) - S(p), (23.48) 23.3 Capacity of some quantum channels the following property is always satisfied: AS+SE(P,E)> (23.49) This last result shows that the entropy change caused by the quantum operation E on both the quantum channel and the "outside world" is always nonnegative, which recalls the second law of thermodynamics, according to which the entropy of a closed system may only increase on any physical transformation or evolution from its initial state. Capacity of some quantum channels In this section, I apply the HSW theorem to evaluate the capacity of different types of quantum channel. We assume that the originator source has, for symbols, the pure states which are associated within a quantum message (or codeword) with some probability distribution Pi. For convenience, we introduce (23.50) to which, after the HSW theorem, the channel capacity X = max(x') corresponds. Before touching upon any examples, it is interesting to consider the case of ideal, constant or "noiseless" channels. This is the case where E (pi) =pi and, hence, from Eq. (23.50): =(Ixa)-, (23.51) where we used the property that for any pure state Pi the VN entropy S(pi) is zero. The result in Eq. (23.49) shows that in this ideal case, X' actually represents the VN entropy of the originator's message source S(p)=S((pi)). The "ideal channel" capacity is given by Xideal = max(Xideal), or (23.52) This maximization problem, thus, addresses the question of how much classical infor- mation can be conveyed through any given message p before transmission through any quantum channel. Intuitively, we expect that for nonideal channels we have the property Xideal, (23.53) meaning that, obviously, noisy transmission does not improve the information of the message source. This is just like in the classical case, with the mutual information satis- fying H(X; Y) < H(X), and with th