e typical set of classical bit sequences, of size 2nH(X) and the typical subspace So of quantum state messages, of dimension close to 2ns(p). Consider, now, the quantum message block PM = pon of length n defined in Eq. (22.2), where p is the density operator associated with any of the individual mes- sage symbols, as defined in Eq. (22.1). The 2n eigenvalues and eigenstates of PM are M1, M2 1111), 1122) /un), respectively. To recall, the eigenvalues Mi represent the probability that the message is in the state lui). Formally, the typical subspace So is defined by the set of eigenstates {|Mi}} for which the eigenvalues Mi satisfy the double inequality: (22.24) where S = S(p) and E is a given positive real, which can be arbitrary small. We note that this double inequality is conceptually identical to that in Eq. (13.26), corresponding to the formal definition of typical sequences. Let me immediately illustrate this definition of the typical subspace by means of the n = 3 example used in the previous section. For reading clarity, we recall here the 22.2 Schumacher's quantum coding theorem Table 22.2 Lower bounds for the parameter E, defining different possibilities for the typical subspace So and its complement S21 Typical subspace (S2) and complement (S21) 0.6219 0.372 0.1067 0.1067 0.475 0.1067 0.0183 1.323 0.0183 0.0183 0.0031 2.171 density matrix of PM: (22.25) which shows that the 2n eigenvalues Mi of PM are of the form Mi=u,v dady with utu=n. Taking the base-2 logarithm of Eq. (22.24), we have (22.26) We note that the inequality is conceptually identical to that in Eq. (13.25), correspond- ing to the formal definition of typical sequences. Substituting n = 3, S = 0.6008, da = cos2 /8 = 0.853, db = sin2 /8 = 0.146, 11=23, M2 M5 = 6 = M7 = hadz, and M8 = 23 in Eq. (22.26), we obtain the lower bounds for E listed in Table 22.2. We observe that E = 0.372 (eigenvalue M1 = 23) defines a first, 1D typical subspace S21, spanned by Idahada). A second possible boundary, E = 0.475 (eigen- values M1 and M2=M3 = 14 = 222b)