bsystem B. Likewise, we obtain (21.60) For the sake of curiosity, let us now calculate the tensor product PA x from Eqs. (21.59) and (21.60). We obtain (21.61) <01| + 25|10) (10| + 5|11) (11|) AB. 21.2 Relative, joint, and conditional entropy, and mutual information It is clear by comparison with Eq. (21.56) that PAB # PA x , meaning that, in the present example, the tensor product PA x of the reduced density operators PA, PB does not correspond to the density operator of the composite system. This is not a surprise, since we established earlier that the joint state 14 AB) cannot be reduced to a tensor product. Pushing curiosity further, we may calculate the quantum information, S(PA), S(PB), S(PA x PB) associated with the reduced density operators PA, , and PA x , to find: S(PA) = S(pB) -(1/6)log(1/6)-(5/6)log(5/6) = 0.650, PB) -2x (5/36) log(5/36) - (1/36) log(1/36) - (25/36) log(25/36) 1.300 S(PA) + S(pB). (21.62) Based on the information knowledge of the composite system, S(PAB) = 1.251, as established in Eq. (21.56), we find that: S(PAB) = 1.251 <S(PA)+S(PB) (21.63) =1.300. This result is consistent with the subadditivity inequality established earlier and for- malized in Eq. (21.28). It is also consistent with the triangle inequality defined in Eq. (21.43). We may, thus, conclude, as a postulate but not as a formal demonstration, that S(PA), S(PB) indeed represent the quantum information contents of the two subsys- tems A, B, despite the fact that the density operator of the composite system, PAB, does not reduce to the tensor product PA x . Without realizing it, this example and postulate helped make great progress in the analysis of information correlation! Indeed, we can now apply the definition of quantum mutual information in Eq. (21.38) to obtain in this example: S(PA;PB)=S(PA)+S(PB)-S(PAB) =1.300-1.251 (21.64) =0.049. We can, thus, conclude that quantum information in subsystems A, B is correlated, as expressed by the mutual information S(PA;PB) = 0.049, corresponding to about 0.05 classical bit. Fro