e capacity C = max[H(X; Y)] </ max[H(X)] = 1 bit. The following examples will illustrate the fact that the quantum channel capacity X cannot, indeed, exceed the highest-possible VN entropy of the originator source, i.e., Xideal max[S(p)]. Quantum channel noise and channel capacity Example 23.1: Depolarizing channel In Eq. (23.8) we have seen that this channel is defined by the operation (23.54) Assuming the orthogonal symbols P1 = |0) <01 and P2 = |1) (1), we obtain (2mm) (23.55) and the corresponding VN entropy (23.56) As we have seen in the previous section, the VN entropy of the recipient symbols is S[E(p1)] = S[8(p2)] = f(p/2), h hence, s)s() (23.57) From Eq. (23.50), we finally obtain x'(p, (23.58) In the above equation, it is clear that the first term in braces, corresponding to the definition in Eq. (23.56), is maximized if we choose the two symbols to have equal probabilities p1=1-p2=1/2, - which gives (23.59) 23.3 Capacity of some quantum channels Probability p Figure 23.1 Quantum capacity x(p) of the depolarizing channel as a function of the depolarizing parameter p. Thus, we obtain the channel capacity X: (23.60) As expected from the analysis in the previous section, the capacity vanishes for p = 1, which corresponds to a "useless channel," projecting any random symbol Pi onto o = E(pi) = I/2 as shown from Eq. (23.54). The highest capacity x(0) = 1 bit is obtained for p = 0, in which case the quantum channel E is constant or "ideal," or "noiseless," i.e., o = E(pi) = Pi, but this case is of no interest here. Figure 23.1 shows a plot of the depolarizing channel capacity as a function of the depolarizing proba- bility parameter p. It is seen from the figure that, as expected, the channel capacity monotonously decays as the depolarizing parameter increases from p = 0 to p = 1. To provide a practical engineering interpretation of the above result, assume, for instance, that p = 0.1, which yields f(p/2) 2 0.25 (see Fig. 4.7), and, thus, defines a channel capacity of X = 0.75 bit. The HSW theorem states that