titutes a nonintuitive situation that cannot be experienced from any "classical viewpoint." The formal proof of the Holevo bound is presented in Appendix X. The proof is tractable but, unfortunately, only to a certain extent! As described in this appendix, indeed, the full proof requires the introduction of an additional entropy property referred to as strong subadditivity. Such a property, which has not be described in the earlier subsection, rests on two additional theorems, whose demonstration is rather mathe- matically involved. For this reason, and for the purpose of these chapters, we shall take "strong subadditivity" as a granted postulate. Going through the derivations in Appendix X, however, should not be viewed as a vain exercise. Rather, in addition to coming very close to an intuitive (albeit incomplete) proof of the Holevo bound, it also represents an opportunity to familiarize oneself with the concept of quantum operations, of which quantum measurements with POVM operators represent a very representative illustration. I shall now illustrate the consequences of the Holevo bound through a few basic examples. Example 21.4: A "useless" quantum communication channel Assume that the originator is able to prepare a quantum system in two possible orthog- onal states |91) = (|0) +|1))/2 and 99 = (|0) - |1))/2, but with some relative uncertainty, according to a probability law defined by p1 = p and p2 1 - p. Not being 100% certain, these two states are mixed states, having the corresponding and The property of "strong subadditivity" states that for any three quantum systems A, B, C, the following inequalities hold: S(A)+S(B)<S(A,C)+S(B,C) S(A, +S(B) S(A, S(B, C). Quantum information theory equal density operators: (21.76) The mean density operator is p = ppi = p1 P2. Clearly, the VN entropy is the same for the three states p, P1, P2, i.e., S(p) = S(p1) = S(p2) = -2 X (1/2) g(1/2) = 1, hence, the Holevo bound X is zero, meaning H(X; Y) = 0. Such a quantum communication channel has no mutual information availab