ng the system A in mixed state (VA) into a composite system AR, whose joint state /VAR) is a pure state. The state (VAR) is referred to as the purification of (VA), or the purification of PA. The principle of state purification is relatively simple to explain. First, assume for the mixed state the following decomposition over the orthonormal basis {|i A }}: Second, assume a reference system R of same dimension of A and same basis {|ir}}, meaning (iR/JA) = Sij. Then define the joint state 1VAR): IVAR) The above-defined joint state 1VAR) is a pure state, because it cannot be defined by any other possible state (OAR) of AR. We can show this by using the most general definition for OAR): Then we have (VAR/OAR) = dij(KA|AA) (kr/jr) k i j#i kijii which establishes that (VAR), |OARY are, indeed, orthogonal. State purification Since (VAR) is a pure state, the density operator of the composite system AR is As the next step, we execute the partial tracing over R according to the generic definition in Eq. (21.25), which here reads (B = R): trRUti(jkRR) hence, tr ( RAR) = = PA. This result shows that the partial trace over R of the composite system AR yields the density operator PA, which corresponds to the definition of state purification. Appendix X (Chapter 21) Holevo bound In this appendix, I formally establish that the mutual information H(X; Y) in a quantum communication channel is bounded by a maximum, X, referred to as Holevo bound, and defined by where p = Expxpx = (p).In the above definition, Px represents the density matrix of any quantum state used for the transmission, as selected from an "alphabet" X according to some probability distribution {px}. The retrieval of information is made by POVM measurements, which define a source of random outcomes Y of size N. The formal proof of the Holevo bound requires one to assume three quantum systems, P, Q, R. The first system, P, corresponds to the quantum source used by the originator to "prepare" the states Px, namely with density operator: with {|x)} being the orthono