 8.193 x 10-1 2.03683 2.63771 20219 1.406 x 10-16 2.17068 2.77156 is shown in Fig. 22.3. It is seen that convergence towards 100% fidelity is relatively rapid (e.g., F 22 99% or better, for n > 70). Interestingly, such a convergence goes through sawtooth-like oscillations. These are explained by the irregular variations of the E(n, p) cluster size and amplitude previously observed in Fig. 22.2. The last open issue to discuss is the dimension of the typical subspace, dim(S2), corre- sponding to a given message length N. From Eq. (22.28), we know that dim(S) is bounded according to (1 - 8)2n(S-8) < dim(S2) < 2n(Sts). which asymptotically becomes 2nds as 0 and with sufficiently long messages. In the most general case (E > 0, any message length n), we have (22.40) 22.3 A graphical and numerical illustration E 0.29681 Message length Figure 22.3 Fidelity as a function of message length (n = 19 to n = 100), assuming the E-typical subspace So with E < 0.29681. =0.29681 Message length Figure 22.4 Evolution of the parameter see (n, S2) with increasing message length n. Using the property log2(1 - 8)/n - 8/n(1n 2) 2 0, we can introduce the parameter s for which the double inequality in Eq. (22.40) is very nearly equivalent to: (22.41) Figure 22.4 shows the evolution of the parameter s(n,S2) with the message length n, as calculated from our previous example, up to n = 500. For each n, the value dim(S) is estimated by summing the values of N (number of E-typical codewords for which E < 0.29681), displayed in the third column in Table 22.4. For clarity, the plot shows successive data with an increment of one up to n = 100, then with an increment of 10 up to n = 200, then with an increment of 50. The figure shows that the parameter see (n, S2) asymptotically converges towards the limit E = 0.29681, which defines the typical subspace So for each N. We may, thus, write "=S+E=R (22.42) and since E can be chosen arbitrarily small, the limiting value for dim(S2) is ens Quantum data compression To summarize, the numerical example dev