 typical subspace S, and is sent through the quantum channel. If the outcome is anything different, the originator sends an nR qubit message 1/comp2), as a "junk" substitute, which is determined by the same rotation U such that U-1 (1 / comp2) x |0rest)) = /Mjunk) E S2. As in the previous (n = 3) example, the recipient just needs to append |0rest) to the received message, followed by the inverse transformation U-1 on this n-qubit state. If E is the projector onto the typical subspace S2, and lui) the original message that has been sent, the corresponding density operator pi of the received/uncompressed message takes a form similar to that in Eq. (22.19): (22.32) + Pjunk The fidelity of the transmission must be averaged over all the message possibilities, each with probability Mi, as follows: (22.33) which, by substituting the result in Eq. (22.32), yields (22.34) The last inequality is caused because the junk contribution is nonnegative; i.e., tr(pjunk) > 0. Then we use the property that for any real x we have x2 > 2x - 1 because (x - 1) 2 0 . Thus, each term x2 = in the above right-hand side 22.3 A graphical and numerical illustration summation is greater than or equal to 2x - 1 = - 1, which yields (22.35) = 2tr(pME) - 1 2p(s) - 1. From Schumacher's (quantum coding) theorem, for any 8 > 0 and sufficiently long messages we have p(S2) > therefore F2p(s2)-12(1-8)-1=1-28 (22.36) This final result indicates that for sufficiently long messages, the fidelity F can be made arbitrarily close to unity (consistent with the limit 8 0), which establishes that the compression code is "reliable." It is possible to show (but not described here for the sake of brevity) that any other coding choice, namely with R < S, or R = S - E (with E > 0, 8 > 0), makes the fidelity of sufficiently long messages to be asymptotically bounded according to F < 8, which is essentially zero as 8 A graphical and numerical illustration of Schumacher's quantum coding theorem In this section, I shall provide an original illustration of Schumacher's qu