, = defines a 4D typical subspace S22 spanned by {12ahaha), 12ahahb), 12ahbha), 12bhahal}. A third possible boundary, E = 0.475 (eigen- values M1, M2 and M5 = 6 = M7 = had?) defines a 7D typical subspace S23 spanned by { 12ahahb), 12ahbha), 12bhada), 12ahbdb), , 12bhadb)}. The remain- ing subspace, spanned by 12b2bdb), corresponds to the smallest atypical subspace, Quantum data compression Table 22.3 Parameter 8 and fidelity F corresponding to the typical subspace choices for So and related dimension dim(s2). dim(S) 0.6219 0.378 0.622 0.9419 0.058 0.923 0.9969 0.003 0.996 In the previous section, we heuristically used S22 as "the" typical subspace, but it is now clear that the other subspaces S21 and S23 are also eligible under the intrinsic boundary definition in Eq. (22.24) or Eq. (22.26) for the corresponding eigenvalues, as defined by the parameter E. Given the highest value of E in the typical subspaces S = S1, S2, S33, we shall call any "codeword" lui) = 12,2k21 E So as "E-typical." To recall, the probability p(S2) that any message belongs to the typical subspace S given by the sum of the associated eigenvalues Mi. Formally, if E is the projector onto S2, we have (22.27) where 8 is a positive real and smaller than unity. Table 22.3 shows the values of the parameter 8 corresponding to the typical subspaces shown in Table 22.2, and the corre- sponding fidelity as defined in the previous section by = (1 - 8)2 +823 Eq. (22.20). It is seen from the table that the fidelity increases as the parameter 8 decreases and as the dimension of the typical subspace dim(s2) = tr(E), or the number of E-typical codewords increases. We note that the case So = S21, which has only one codeword, is a poor choice, since the fidelity is F = 62.2%, corresponding to a "useless channel" almost half of the time. And what valuable information can be transmitted with only one codeword, independently of this consideration? The best choice is So = S33, which has seven codewords (out of 23 = 8 codeword possibilities), since the fidelity rea