mination of subsystem entropy. Subadditivity inequality and quantum joint entropy Assume a composite system AB made of two subsystems A, B. Let PAB, PA, PB be the density operators of the system and its subsystems. The corresponding entropies S(PAB), S(PA), S(pB) are then linked by the subadditivity inequality: S(PAB)<S(PA)+S(pB), (21.28) where the equality stands when PAB = PA x PB, which corresponds to two subsystems with uncorrelated information. The entropy of the composite system, S(PAB), which is the quantum joint entropy, is, therefore, less than or equal to the sum of the subsys- tem entropies. There is no surprise here, since this property appears to be the quan- tum equivalent of that concerning the Shannon entropy H(X, Y) of joint sources (see Chapter 5), i.e. H(X,Y)H(X)+H(Y). (21.29) The apparent equivalence between the classical and quantum definition of joint entropy, as intuitive and innocuous as it may appear, is in fact deceiving. We shall immediately realize this fact by considering the case where AB is in a pure state. From the previous property, we must have S(PA) = S(PB), but also S(PAB) = 0, since AB is in a pure state. Thus, regardless of the information contained in subsystems A or B, which may be finite, the joint system AB contains no information! Such a possibility is definitely nonclassical and counterintuitive. Indeed, referring to the Venn diagrams in Fig. 5.2, Quantum information theory where H(X), H(Y) are represented by sets with H(X, Y) = H(X) UH(Y) there is no possibility for H(X, Y) = (empty set), except in the limiting case H(X) = H(Y) = . I shall now prove the subadditivity inequality introduced in Eq. (21.28). Basically, it is a consequence of Klein's inequality, which was discussed previously. We have shown that for any pair of quantum systems of equal dimension and having density operators p, o, Klein's inequality can be translated into S(p)t(plogp)tr(plogo) (21.30) S(p) < - tr(plog o). Substituting p = PAB and =PADPB into the last inequality yields: S(PAB) (21.31) It is l