e learnt from Chapter 5, the mutual information H(X; Y) measures the amount of information correlation between the two event sources X, Y, as defined by H(X;Y)=H(X)+H(Y)-H(X,Y). (21.37) If the sources are uncorrelated, the equality in Eq. (21.29) stands and H(X;Y) Eq. (21.37) is zero. We may define quantum mutual information S(PA;PB) through S(PA;PB)=S(PA)+S(PB)-S(PAB). (21.38) As in the classical case, S(PA;PB) represents the measure of information correlation between the two quantum subsystems A, B. In the absence of any correlation (PAB = PA X PB), the equality in Eq. (21.28) stands, and S(PA;PB) is zero. From Eq. (21.28), we note that S(PA;PB) > 0 in all cases, meaning that the quantum mutual information is always nonnegative. In the case where the joint system AB is in a pure state, we have S(PAB) = 0 and S(PA) = S(PB), which implies that S(PA;PB) = 2S(pA) = 2S(pB). Such a result is definitely nonclassical. Indeed, referring to the Venn diagrams in Fig. 5.2, where H(X), H(Y) are represented by sets with H(X; Y) = H(X) H(Y), there is no possibility for H(X; Y) = 2H(X) or H(X; Y) = 2H(Y), except in the limit- ing case H(X) = H(Y) = . If two quantum systems have correlated information, we shall say from now on that their quantum states are entangled. The end of this section will provide examples of entangled states. Conditional entropy Given two subsystems A, B and their composite system AB, the conditional entropy S(PA/PB) is defined as S(PA/PB)=S(PAB)-S(PB), (21.39) and, likewise, S(PB/PA) = S(PAB) - S(PA). This definition is the quantum equivalent of the classical case concerning joint events from two sources X, Y, namely (see Chapter 5): H(XY)=H(X,Y)-H(X), (21.40) Quantum information theory and, likewise, H(Y|X) = H(X, Y) - H(Y). Where the information in the two subsys- tems is uncorrelated (S(PAB) = S(PA) + S(pB)), we have, from Eq. (21.39): S(PA/PB) = S(pA) (21.41) S(PBIPA) = S(PB), which translates that the information of one subsystem is not affected by the knowledge of the information in the other subs