t-phase-flip channel As a fourth example of noisy quantum operation, we define next the bit-phase-flip channel in the operator-sum representation: E(p)=pYpY+(1-p)p (23.22) =UppU0+U1pU1, 23.2 The Holevo-Schumacher-Westmoreland capacity theorem with the corresponding elements: (23.23) It is again left as an exercise to show that the originator message 19) = a|0) + B|1) is transformed into |q') = (B|0)-a|1) - with probability p and is left invariant otherwise. The action of the noisy channel is to flip both the phase and the complex amplitudes. It is easily found that for any originator messages p = |0) <01, p = |1) (1), p = 1+) <+1 the net entropy difference is AS = S(o) - S(p) - = f(p). It is clear that for p = |0) <0|, p=|1)(1) this difference is only attributable to the bit-flipping effect, while for l+) <+1 it is attributable to the phase-flipping effect.1 Other noise processes, which are beyond the scope of these chapters but very important in the analysis of time evolution of qubits in quantum channels, concern amplitude damping and phase damping. The Holevo-Schumacher-Westmoreland capacity theorem In this section, I shall introduce the concept of quantum channel capacity, based on the Holevo-Schumacher-Westmoreland (HSW) theorem. The HSW theorem defines the channel capacity X under which, at a rate R < X, it is possible to transmit quan- tum messages with arbitrary small probability of error or information loss. It can be viewed conceptually as the elegant quantum counterpart of Shannon's channel cod- ing theorem, which was described in Chapter 13. Here, the term "capacity" refers, in fact, to the same concept as in classical information, while assuming that such information is encoded in quantum messages that are passed through noisy quantum channels. The formal demonstration of the HSW theorem being particularity involved and tedious, it will not be considered here. For the general interest and purpose of this chapter, it will suffice to state the HSW theorem "as is:" given a quantum-symbol source {pi} wit