agonal matrix elements (p)kk = pink and (log(p))xx=log(p)X). = Using these different properties, we obtain S((p)) = =-tr((p)log(p)) (21.49) which proves Eqs. (21.46) and (21.47). As we have seen, the property thus estab- lished rests on the assumption that the set of operators Pi has support on orthog- onal subspaces. If such a condition is not met, the equality in Eq. (21.47) or Eq. (21.48) does not hold and the originator-source entropy H(X) is only an upper bound, namely: S((p))-S(p))<H(X), (21.50) or, formally: H(X). (21.51) Here, for the sake of conciseness and to keep the momentum, we shall not go through the detailed demonstration of the inequalities in Eqs. (21.50) and (21.51). The key point is to remember that the equality in these relations is not guaranteed, except in the case where the subspaces defined by Pi are orthogonal. In all cases, the entropy H(X) remains the upper bound. This general property for the concavity of entropy shall be used in the next section, concerned with the so-called Holevo bound. This tedious inventory of entropy properties, which has been limited here to the most elementary ones, now calls for a few illustrative examples. I shall consider three basic possibilities for a composite quantum system AB made of two subsystems A, B, namely: 3 Indeed, we have pi2*12k). 21.2 Relative, joint, and conditional entropy, and mutual information (1) when the subsystem information is uncorrelated, (2) when it is correlated, and (3) when the composite system is in a pure state. The last two examples will also highlight the interest and implications of using partial tracing, and illustrate the concept of state entanglement. Example 21.1: Subsystems with uncorrelated information Consider, for instance, two subsystems A, B in respective qubit states:4 (21.52) The joint state of the composite system is assumed to be (10)+1))A(10)+11)+12))B (21.53) It is clear from the above definitions that the three systems A, B, AB are expressed in their respective eigenstate bases and, thus, have density matr