le whatsoever in reserve and, in the sense of information theory, is, therefore, useless. Formally, this use- lessness characteristic is because the qubit states used for the message quantum states, albeit orthogonal as "symbols, have strictly identical supports (or eigenstate spaces). Basically, the originator does not have any control on his or her source to prepare any "communicable" information. Based on the fact that each of the proposed qubits con- tain exactly one classical bit (cbit), and that the two are orthogonal, this conclusion of uselessness sounds like a disillusionment concerning the communication potential. But it is now clear that their mixed-state nature render the communication attempt useless, no matter what the distribution defined by the parameter p. Example 21.5: A quantum communication channel reduced to classical The originator (based on the previous experience) is now able to prepare his or her quantum system with the same orthogonal qubits, i.e., |91) = (|0) + |1) 2 and 192) = (|0) - |1)) / 25, but this time as pure states, contrary to the previous example. This means that in each possible preparation, there is absolute certainty that the outcome is either |91) or 92), as required. Then the two preparations that can be chosen have corresponding density-operators: (21.77) In the above, the arrows mean in each case that the density matrix is transformed from the basis {|0), |1)} into the eigenstate basis {|+), |-)} = {|91), |q2)}. The origi- nator then selects the qubits according to the message's probability law, here defined by p1 = p and p2 = 1 - p, where p is a parameter. The mean density operator is, therefore, (21.78) Clearly, we have for the VN entropyS(p)=-[plogp-(1-p)log(1-p)]=f(p) = = H(X) and S(p1) = S(p2) = 0 (pure states). Thus, the Holevo bound is x=f(p) = H(X) meaning H(X; Y) < H(X) as in the case of a classical communication channel. We note that the strict equality for the Holevo bound, X = H(X), stems from the fact that 21.3 Quantum communication channel and Holevo boun