al illustration of the PKC algorithm. Appendix U (Chapter 21) Klein's inequality This appendix provides a basic demonstration of Klein's inequality, according to which the relative entropy S(pl)o) between two quantum systems characterized by the density operators p and o is nonnegative. Assume the following eigenstate-basis decompositions for the density operators: Substituting the first of the above definitions into the relative entropy, we obtain S(plo) = tr(plogp) - tr(p) The last term in the right-hand side can then be developed using the eigenstate property (ilp : (i/pi, which gives (i/p log oli)=pi(i log o/i). Furthermore, since o is diagonal we have, from the second definition in Eq. (U1), logo = Log = qklk) <k and, therefore, Substituting these results into Eq. (U2), we obtain Klein's inequality Then we use the property according to which (log y) < log(y) for any function y or, in the discrete case, log yk < log Xkyk, where {xk} is a probability distribution. Such a property stems from the concavity of the logarithm function (see Exercise 5.10). Letting xk(i) = |i|k)|2, we note that xk(i) > 0 and (i) = 1, which makes xk(i) a probability distribution. Thus, we have = logri. Combining, then, Eqs. (U4) and (U5) we finally obtain In the term log(pi/ri), we recognize the classical definition of relative entropy or the Kullback-Leibler distance D(p)rr) between two discrete probability distributions {pi}, {ri}. As we have seen in Chapter 5, D(p)rr) is nonnegative, which justifies the last inequality introduced in the right-hand side in Eq. (U6). The above, thus, proves Klein's inequality or the nonnegativity of S(plo). Appendix V (Chapter 21) Schmidt decomposition of joint pure states This appendix describes the so-called Schmidt decomposition. Such a decomposition applies to the case of a composite system made of two subsystems A, B, which is in a pure joint state 1/s ), namely, 1/>/ = = (VA)/YB) = 1VAVB). According to Schmidt decomposition, the joint state can be expressed in the form where Xi (i = 1...n) are 