 the typical space So is EV (V|E+ = EVIVE, which provides a first component of p, corresponding to the typical message states 1/type. The second component of p, corresponding to the other atypical message states, can be heuristically written junk where = I is the identity matrix. Clearly, I-E is the projector on the atypical space St Thus,(1||(I-E)\V) equals one if 1/5/ is an atypical state, with corresponding projector Idahaha) (hahadal, and (V|(I-E)VY is equal to zero otherwise. The transmitted and Quantum data compression reconstructed message, thus, has the complete density operator (hadadal (22.19) =EVIVE + Pjunk From the definition in Eq. (22.8), we obtain the fidelity F=(YOY) =(VEV\VEV)- (22.20) = P(S2)2 Substituting 8 = 0.058 and (20202) = 0.6219 from Eq. (22.10) into Eq. (22.20), we obtain F=(1 -0.058)2 + 0.058 x 0.629 = 0.923. = (22.21) The result F = 0.923 compares well with the fidelity value F = 0.853 obtained earlier. To recall, this is the fidelity obtained by "guessing" any missing qubit is likely to be 12a). Thus, the alternate approach of transmitting only the first two qubits as uncoded, and leaving the recipient the task of "guessing" the third missing one (likely to be (2a)) is F = 0.853. Such a guess-based coding for compression does not have a fidelity as high as that of the above coding, for which F = 0.923. The example has shown that it is possible to compress a quantum message of n-qubit size (n = 3) into a coded version of m-qubit size (m = 2), with relatively high fidelity. Here, the compression factor that was achieved is n = m/n = 2/3 = 0.666 = 66.66%. The next section will establish that for messages of asymptotically increasing length, the achievable compression factor is bounded according to (22.22) where S(p) is the VN entropy carried by each of the message symbols. This result is known as Schumacher's (quantum coding) theorem. In this example, the per-qubit entropy is S(p) = 0.6008, corresponding to a best compression factor of n = 60.08%. Therefore, for 3-qubit messages based o