 Px, which is made according to the message's probability distribution Px. Here, the word "preparation" means setting up the conditions for a given quantum system to be exactly in the state Px. The operation of decoding is, for the recipient, the action of performing "measurements" on each of the received quantum states. According to Chapter 17, such measurements can be made through a collection of n Hermitian operators {Em}m=1...n referred to as a POVM set. The outcome of any measurement is a real positive number y belonging to some alphabet Y of size n. We note that from this description, X, Y are classical sources with X, y as associated random events, while the communication channel is quantum. I shall now focus on the mutual information, H(X;Y) associated with the above- described quantum communication channel. A key property is that the mutual informa- tion is bounded by a maximum, X, referred to as a Holevo bound, and defined by H(X;Y)<s(p)-PxS(px) x (21.74) 21.3 Quantum communication channel and Holevo bound where p=x = = (p) is the mean density operator, as averaged over the coding possibilities. Based on this notation, and combining Eq. (21.74) with the general property of concavity of entropy in Eqs. (21.50) and (21.51), the Holevo bound condition also gives H(X;Y) (), (21.75) where H(X) is the Shannon entropy of the originator message source. In a clas- sical communication channel, the mutual information cannot exceed the entropy of the originator source, namely H(X; Y) = H(X) - H(X|Y) with H(X|Y) > 0 (see Chapter 5), hence, H(X; Y) < H(X). Thus, Eq. (21.74) is granted. However, the inter- mediate Holevo bound X in this equation is nontrivial, except in the case X = H(X), corresponding to the specific situation where Px have support on orthogonal states. In the contrary case, we have X < H(X) and, hence, the condition H(X; Y) < H(X). In such a condition, there exists no possibility for the recipient to completely recover the originator's information H(X) through any set of measurements on Y. This cons