tum data compression Consider, next, the case n = 5, based on the same assumptions for S(p), Na, Ab. We have [S(p) X 51 = [3.004] = 4 bits, which means that the minimum length of the compressed message is four qubits. The associated compression factor is Nn=5 = 4/5 = 0.80, = to compare with Nn=4 = 3/4 = 0.75 and Nn=3 = 2/3 = 0.66 in the previous cases. Thus, as we expect the fidelity to be higher with messages of length n = 5, the compression performance is poorer than in the previous cases. Consider, next, the case n = 6, based on the same assumptions for S(p), Na, 2b. We have [S(p) X 6] = [3.608] = 4 bits, which means that the minimum length of the compressed message is four qubits. Thus, a compression factor of Nn=6 = 4/6 = 0.66 is achievable, which represents compressing the 6-qubit original message into a 4-qubit one. As the next section describes, the compression code requires the recip- ient to manipulate the transmitted states 1/comp1) x |00) or 1/comp2) x |00), with the first case corresponding to the typical or most likely subspace. I will also clarify how the dimension of the typical subspace can be defined formally. Schumacher's quantum coding theorem In this section, I shall formalize Schumacher's quantum coding theorem. The driving concept is that it is possible to encode a message with high fidelity when using quantum states from the typical subspace S. The key property of So is that it asymptotically reaches a dimension close to 2ns(p). It is useful to look back at Section 13.2 concerning typical sets. To recall, the typical set represents roughly 2nH(X) bit strings of length n, referred to as typical sequences. Such typical sequences roughly contain nq 1 bits and n(1 - q) 0 bits, with H(X) = f(q) being the source entropy of the sequence bits, assumed to be generated independently, (H(X") = nH(X)). The fundamental property is that when n becomes large, any typical sequence asymptotically has the probability 2-nH(X) of being observed. Thus, there is a one-to-one conceptual correspondence between th