e in any random mixed state /Vi), according to some known probability distribution Pi. Each of these random states is associated with a density operator Pi. The concavity of a function f(x) corresponds to the property (f(x)) < f((x)). Here, I shall establish that entropy is a concave function of the density- operator variable Pi, namely, (S(p)) < S((p)), (21.45) or, formally: (21.46) Furthermore, if one assumes that the set of operators Pi has support on orthogonal subspaces (meaning that the set of all possible eigenstates form an orthonormal basis), an exact relation also exists between S((p)) and (S(p)), which nicely relates to the Shannon entropy of a classical source H(X) = - Pi log Pi, according to: S((p))=(S(p))+H(X), (21.47) or, formally: (21.48) The property expressed in Eq. (21.47) or Eq. (21.48) can easily be interpreted as follows: the quantum information of a system whose states are random equals the mean infor- mation, as averaged over the individual states, plus the information on the probability distribution, which is the Shannon entropy. Note that since H(X) is nonnegative, this property also establishes the concavity of the entropy, as expressed in Eq. (21.45) or Eq. (21.46). In the case where the probability distribution is uniform, the Shannon entropy H(X) is maximal, meaning that there is maximal uncertainty as to which quan- tum state the system is in. Such a situation also maximizes the entropy S((p)) of the system. The deterministic case where the system has only one possible quantum state corresponds to H(X) = 0 and a minimum entropy. The demonstration of the property expressed in Eq. (21.48) is relatively simple. Indeed, let 12k) and 2' be the eigenstates and eigenvalues for each quantum state associated with the density operator Pi. Hence Pi has diagonal matrix elements (pi)kk = 2k Next, given any i, k we observe that 12ki and pink are eigenvectors and eigenvalues of Quantum information theory = making {12*)} an orthonormal basis for (p), regardless of the choice of i. Hence, (p) has di