 PQR in the state (X11) We may now trace out the system over Q to obtain P'PR and, hence, to be able to derive later on the mutual information S(pp; PR) existing after the measurement. This tracing gives (X12) In the above, we may substitute tr(Eypx) = p(y|x) as a conditional probability, because {Ey} form a POVM operator set satisfying the completeness relation. Using Bayes's theorem (see Chapter 1), we havep(x,) y) = p(y|x)px for the joint probability, which, Holevo bound from Eq. (X12), yields: (X13) Because of the above tensor-operator form, we might as well express the post- measurement joint state in the tensor-state form: (X14) It is clear, then, that the joint VN entropy S'(pp, PR) and mutual information S'(pp; PR) of the post-measurement system are simply given by S'(pp, pr) = H(X,Y) (X15) S'(pp;pr) = H(X;Y), just as in a classical communication channel with an originator source X and a recipient source Y, correlated together with joint probability p(x, y). The next and final step is to relate the post-measurement mutual information S'(pp; PR) to that characterizing the system before the measurement, which we call S(pp;pq). To this effect, we shall first assume the following inequality (the proof being provided at the end of this appendix): S(pp;PR) S S(pp;pg) (X16) This property states that in quantum channels, mutual information may not necessarily be conserved, neither be increased by measurement. We obtain S(pp;pg) as: - pg), (X17) with (from Eqs. (X2) and (X3)): S(pp) (X18)¹ S(pp, = S To obtain this result, define Rx = |x)(x/ Q Px. We have, then, Appendix X From Eqs. (X17) and (X18), we obtain (X19) Combining the results in Eqs. (X15), (X16), and (X19), we finally obtain (X20) which establishes the proof of the Holevo bound stated in Eq. (X1). To reach such a conclusion, the inequality in Eq. (X16), namely, S'(pp; PR) < S(pp;pq), was made as a key assumption. The following discussion, destined for the demanding, provides the key elements proving such an assumption. This proof is, how- ever, not comp