antum coding, both graphical and numerical. This will help visualize how the dimension of the typical subspace So asymptotically converges towards 2ns as the message length n increases and if the parameter E may be chosen to be arbitrarily small. The numerical application will also illustrate that there is still a large degree of freedom over the choice of E and the typical subspace possibilities to compress quantum messages. Here, we shall assume the same parameters as in the example used in the two previous sections, i.e., S(p) = 0.6008 for the per-symbol VN entropy, and the two eigenval- ues da = 0.8535, Nb = 0.1465. For a message length n = 3, we have 23 = 8 possible messages defining an 8D eigenspace A. Any of these messages are based on the qubit permutations of four codeword types of the form (dadaha), 12ahahb), 12a2bdb), 12b2bdb) with the respective measurement probabilities 23, in descending order. Such probabilities also correspond to the = M5 = 6 = U7 = hadz, and M8=23 = According to Eq. (22.37), which I reproduce here for convenience, we can associate a parameter E with each of these four groups of eigenvalues: (22.37) Quantum data compression E (n, 0) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Message length n Figure 22.1 Lower bound for the parameter E(n,p), calculated from Eq. (22.39), in the range = 1 to n = 20. Substituting the above values of Mi with n = 3 and S(p) = 0.6008, we obtain four lower bounds for E, namely, to call them by the same letter: E = 0.372, 0.475, 1.322, 2.170. As we have seen in the previous section, this calculation does not tell us much as to how to define a typical subspace. Furthermore, only the first value E = 0.372 gives a parameter R < 1 (namely R = S E = 0.6008 + 0.3724 = 0.9732 < 1), and there is nothing that we can conclude about any compression potential from such a result! But if we increase the message length n, and calculate all possible E values from Eq. (22.37), we obtain a much broader range of possibilities for the parameter R. Given the messag