d the message states P1, P2 have orthogonal support. This result shows that the quantum communication channel is able to convey information that can be up to H(X), which is the classical entropy of the originator's message source. Interestingly, the quantum "symbols" carry no information by themselves, since S(p1) = S(p2) = 0 are pure states. As we know, H(X) is maximized for the choice p = 0.5, corresponding to a coin-tossing probability law. Example 21.6: General case This example represents an interesting generalization of the previous situation. The originator is able to prepare single qubits as pure states, and the message to be encoded is a long string of 1 or 0 cbits with a uniform probability distribution. If the cbit is 0, the qubit symbol is chosen, for instance, to be |0), and if the cbit is 1, the qubit symbol is chosen to be cos 0|0) + sin 0|1), where O E [0, is an arbitrary parameter. The density operators of the message's qubit symbols are = <01 + sin 0 <1|) O sin 0|0) (1| + cos O sin 0|1) <0| + sin² 0|0) <01 (21.79) Each qubit symbol having the same probability p1 = P2 = 1/2, we obtain for the mean density operator: (21.80) It is easily found through the characteristic equation that the eigenvalues of p are 21,2 = (1 + cos 0)/2, or A1 = s2(0/2) and 22 = sin2(9/2)=1-21. = The VN entropy of the originator's quantum source (p) is, therefore: S(p) = - A2 log 22 = -21 log -(1-21) log(1-21) (21.81) = f(21) = f[cos2(0/2)] Since S(p1)=S(p2)= 0 (pure states) the Holevo bound is (21.82) A plot of the above-defined Holevo bound is provided in Fig. 21.1. It is seen that a maximum is reached for X = 0.5 or O = /2, which corresponds to a choice for the Quantum information theory Figure 21.1 Holevo bound X as a function of the argument A, showing a maximum at O = /2. second qubit of 192) =||) or P2 = |1) (1). In this case, H(X; Y) < X = H(X) = 1, meaning that information can be transmitted through the quantum channel up to 1 bit, corresponding to the originator's source entropy. In any other case, or 0 # /2, the