y quite different from that assumed in the Holevo bound derivation. Furthermore, however tempting, we ought not to view X' as representing some elaborated definition of "quantum mutual information" between the sources p and o of two quantum systems Q (originator) and Q' (recipi- ent). In Chapter 21, we have described the quantum mutual information S(PA; PB) for a composite system AB as (23.44) where PA = tr B(PAB) and PB = tr A(PAB) are the partial-traced operators. Life in QIT would be SO much simpler if max[S(pg;pg)) could represent the quantum channel capacity X, but this is definitely not the case! To cut through any other explanation, it is not possible to define the composite system QQ with density operator PQQ', simply because after the operation E the original system Q does not exist anymore as it has been transformed into Q'! The physical parameter that can make sense is the difference AS = S(pg) - S(pg)=S(o) - S(p), - see further on. The quantum counterpart of mutual information is, indeed, far more complex, and its description is beyond the scope of this chapter. Here, to quench our thirst, I may provide just a brief hint of a new concept referred to as quantum coherent information. Appendix W shows that given a quantum system Q in a mixed state 140) with density operator PQ = p, it is possible to define a reference quantum system R, such that the composite system QR is in a pure state /VOR), namely, for which PQR = 1/QQR) (VOR). Such an operation is referred to as state purification. Upon the action of the quantum operation E, the systems Q, R are then transformed into the systems Q', R' with, in particular, PQ = o = E(p). One defines the entropy exchange of the action of E on the composite system QR that transforms it into the composite system Q'R' as SE(P,E)=S(pgr). (23.45) Then one defines the quantum coherent information as the difference I(p,E)=S(o)-SE(p,E). (23.46) In today's state of the art of QIT, the quantum coherent information I(p, is only "conjectured" as playing the same role as the mutu