pecific codeword OM corresponds to a set of N' = N eigen- values = eigenvectors = such that OM has the diagonal form (23.29) For instance, |M> = |aja2a2a3> is represented by the density operator PM = P1@P2@P2QP3. 23.2 The Holevo-Schumacher-Westmoreland capacity theorem with AMK = 1. Next, define (S) = (S(o)) as the mean value of the per-symbol VN entropy of the received codeword, according to: (23.30) Given any parameter E > 0, and based on the concept described in Chapter 22, we can define the E-typical subspace So for OM, based on the parameters AMK, (S) and spanned by the eigenvectors |AMK) according to the condition: 1108A (23.31) Let PM be the projector onto this E-typical subspace S. Based on Eq. (22.30), we know that the dimension of this subspace, tr(PM), is upper-bounded according to tr(PM)<2"(5)-8), (23.32) We also know that for any OM with sufficiently long length n and for any 8 such that 0<8<1, the probability p = p(OM E SL) of projecting onto the typical subspace is lower-bounded according to: p=tr(oMPM))1-8=Pmin, (23.33) where the brackets have the meaning of an expectation value, i.e., the projection prob- ability as averaged over all possible message codewords OM. This result shows that the probability that any received codeword OM belongs to the typical subspace has a lower bound Pmin, which is nonzero. I shall now introduce the notion of error probability, which is the probability that a POVM measurement of O'M by the recipient fails to identify OM as one of the codewords from the typical set S. Consider, indeed, that given the typical set S=={OM} of size 2nR there is a corre- sponding set {EM} of POVM measurement operators (see Chapter 17) of the same size. The POVM set is completed with the measurement operator E0 for all other possible codewords, defined as (23.34) The probability PM of the recipient identifying a specific message M in So while using a POVM measurement EM is given by PM=tr(oMEM) (23.35) and in the opposite case, where such identification is unsuccessful, the error probabilit