wo properties 1 8 <p(s) and p(s) < 1 into Eq. (22.29) and then obtain dim(s2) < 2n(Sts) (22.30) which is in Eq. (22.28) and, thus, proves the typical subspace theorem. Our exploitation of the above theorem will be made purposefully simple, the goal being to convey only an intuitive proof of Schumacher's quantum coding theorem (usually called Schumacher's noiseless channel coding theorem) and, thus, avoid lengthier and more academically involved developments. Define the parameter R such that R = S + E, with the condition R < 1 (or 0<E< 1 - S). From the typical subspace theorem, the dimension of the typical subspace S satisfies dim(S2) < 2nR Thus, we may use up to 2nR E-typical codewords, such that for any 8 > 0 the condition p(s) > 1 - 8 is satisfied for message lengths n sufficiently large. In this case, we may say that the compression scheme is "reliable," i.e., corresponds to arbitrary high fidelity. Under these conditions, Schumacher's quantum coding theorem states that there exists a reliable compression code, with arbitrarily high transmission fidelity, and compression factor n = R. We note that R is strictly higher than the VN entropy S, which sets an ultimate limit to the achievable compression factor (R > S). The second part of the theorem is that any compression code with R < S is not reliable. Here, again to avoid lengthy mathematical developments, I shall only concentrate on the reliable compression code, and provide a simple description. Quantum data compression The quantum symbols forming the message are now assumed to be the eigenstates (2i) of the density operator p, and the possible messages are the eigenstates lui) of the operator PM = pon. The originator then proceeds to encode these into states of the (22.31) where is a compressed state of nR < n qubits, and |Orest)=00..0)0)9 represents the remainder of the block, of length q = n(1 - R) qubits. The originator then makes "fuzzy" measurements on each of the qubits from the remainder state. If outcome is 00 0, then IM') has been projected onto the