ystem, as expected in the uncorrelated case. Again, while we observe a nice parallel between the classical and the quantum definitions for conditional entropy, we should know for a fact (from preceding analysis) that it is deceptive! Indeed, in the case where AB is in a pure state, we have S(PAB) = 0 and, from Eq. (21.39), S(PA/PB) = S(PB) < 0 (as per the nonnegativity of S(PB)) and, likewise, S(PB/PA) = -S(PA) < 0. Hence, the quantum conditional entropy can be negative, which is definitely a nonclassical feature. Since, in this case, S(PA) = S(PB), we also (21.42) which contrast with the correlated case expressed in Eq. (21.41). Triangle inequality The subadditivity inequality in Eq. (21.28) provided an upper bound to the joint entropy S(PAB), namely S(PAB) < S(PA) + S(PB). The triangle inequality, also referred to as the Araki-Lieb inequality, provides a lower bound to S(PAB), according to |S(PA)-S(PB)<S(PAB). (21.43) To prove this property, assume the existence of a third system R, such as the composite system ABR in a pure state. The system R is referred to as a purifying system for AB (see Appendix W). From the subadditivity inequality we find S(PAR)<S(PA)+S(pR). (21.44) Because ABR is in a pure state, we also find that S(PAB) = S(PR) and S(PAR) = S(pB) (see earlier subsection on composite system in pure state). Substituting these last two equalities into Eq. (21.44) we obtain S(PAB) S(pB) - S(PA). Since A and B play a symmetric role, we also have S(PAB) > S(PA) - S(PB), and, hence, S(PAB) > +1S(PA) - S(pB)|, which proves the property in Eq. (21.43). It can be shown (but the proof is not to be considered here) that the equality in Eq. (21.43) is given by the condition PAR = PA x PR, which means that the A, R information is uncorrelated and, therefore, that all possible correlation of A information with the rest of the world is exclusively with B. 21.2 Relative, joint, and conditional entropy, and mutual information Concavity of entropy and entropy of system in random states Assume a quantum system that can b