ches the maximum; F = 99.6%. But we have now reached a confusing situation where there apparently exist several possible choices for the typical subspace, which are all consistent with previous eligibility criteria. We, thus, need to develop the analysis further, bearing in mind that we want to show that the dimension of the typical subspace is close to2ns(p) In the above n = 3 example, we have 2nS(p) = 23x0.6008 = 3.488 2 4 = dim(S22), which indicates that S22 is, in fact, the correct choice! But to reach such a conclusion, we must consider messages of asymptotically increasing length, and we will have to forget the previous n = 3 example, despite its usefulness in demonstrating the possibility of message compression. As we have seen earlier, the two parameters E, 8, defined in Eqs. (22.26) and (22.27), determine a lower bound for the probability p(S2) = tr(pME) of a given message codeword to be E-typical. More generally, we may state that given the two parame- ters E, 8, there exists a message length n sufficiently long to provide the condition that p(S2) is at least 1 - 8. To keep the focus, we shall not worry here about formally 22.2 Schumacher's quantum coding theorem demonstrating such a statement. Suffice it to take for granted as a postulate that, given E, it is possible to make p(S2) arbitrarily close to unity (with 8 chosen sufficiently small), if the message length n is sufficiently large. Consistently, the dimension of the typical subspace, dim(S2) = tr(E), also increases. But can we show to what extent? To answer the above question, I shall now introduce a key theorem concerning the dimension of the typical subspace. Given a message length n, the VN entropy S(p) of the quantum-symbol source, the two parameters E, 8, then the dimension dim(S2) of the corresponding typical subspace So is bounded according to: (22.28) The proof for this is actually straightforward. First, we sum each term in the double inequality in Eq. (22.24) from i = 1 to = dim(S) to obtain dim(S2) (22.29) Second, we substitute the t