 Holevo bound is less than unity (the originator's source entropy) and there is no possibility for the original information to be fully retrieved by the recipient. Clearly, such a limitation is attributable to the fact that the set of density operators used to code the symbols does not have orthogonal support, meaning that they have a finite overlap, defined formally as 1(q1/92)12 = cos2 O # 0. The case O = /2, where the two states |91), 92) are orthogonal, reduces the quantum channel to a classical one, as described in Example 2. In summary, the key lesson to retain from the above description and examples is that it is possible to transmit (or "convey") classical information through a quantum communication channel. A prerequisite is that the alphabet of quantum symbols used code information must be made of pure states, in which case the Holevo bound X equals the VN entropy or average quantum information per symbol (x = S(p)). A second, optional, requirement is that the alphabet states be orthogonal, in which case the Holevo bound is maximized and is equal to the originator's source entropy (x = H(X)). In any case, the mutual information H(X; Y), i.e., the amount of information that the originator and the recipient may be able to share, satisfies H(X; Y) < X. The concept of the quantum communication channel will be revisited in Chapter 23, which considers the effect of channel corruption by quantum noise. Exercises 21.1 (B): Show that the quantum system with density matrix (basis representation {|0), |1)}) p=1(H) 21.4 Exercises corresponds to the pure state 21.2 (B): Show that the density operator p = 0)(01 of a pure state |0) can be trans- formed into the basis {l+), |-)} with the definition (using here the sign + for clarity in the summation): 21.3 (B): Show that for any density operator p, the property tr(p2) applies with equality only for the case of pure states. 21.4 (M): Given a quantum system with the density matrix p=1(i1). determine the corresponding state 1/4/> in the eigenstate basis and the von Neu- m