ed bits are uncorrelated with the received bit yi, hence H(yily1,y2, yi-1, Xn) = H(yilX"). Furthermore, the knowledge of yi is not con- ditioned to that of the input message bits X1, x2, Xn except for the bit Xi of the same rank i in the codeword, hence H(yily1, y2,..., yi-1, X") = H(yilxi). Substituting the result in Eq. (L8) into Eq. (L7) yields: (L10) Next, we look for an upper bound in the right-hand side in Eq. (L8). We observe that H(Y") < -1 H(yi), the equality standing if the distribution p(yi) was uniform, in which case we would have H(yi) = H(Y) and H(Y") = n(((H(Y). Applying the inequality to Eq. (L10) we finally obtain (L11) This result shows that the mutual information between the two transmitted or received codeword sources is less than or equal to the sum of mutual information between the transmitted or received bits in the binary channel. Since, by definition, H(x;;ii) < Appendix L max H(x{;yi) = C, where C is the binary-channel capacity, we also have H(X";Y") <nC. (L12) This result establishes that the mutual information between extended sources Xn, Yn, where bits are passed through the channel n times, is no greater than n times the binary- channel capacity. The two properties A and B can now be used (finally!) to establish the converse proof of the CCT. As before, we assume that the 2nR input message codewords are chosen at random with a uniform distribution, hence H(X") = nR. By definition of the mutual information, H(X"; Yn), and introducing the majoring properties A and B, we obtain nR = = (L13) R S PeR+ Since the starting assumption is that the error probability of the code, Pe, vanishes for 8, the above result asymptotically becomes: R C C, (L14) which represents a necessary condition and, hence, proves the converse of the CCT. Appendix M (Chapter 16) Bloch sphere representation of the qubit In this appendix, I show that qubits can be represented by a unique point on the surface of a sphere, referred to as a Bloch sphere. As seen from the main text, any qubit can be represented as the vect