f(1-p)=f(p), with the same result being obtained with p = |1) (1). The maximum uncertainty (AS = or one cbit), corresponds to the case p = 0.5, meaning that, according to Eq. (23.13), there is a 50% chance that the qubit amplitudes are flipped when passing through the channel. Interestingly, there is no entropy change (AS = 0) if the qubit amplitudes are deterministically flipped (p = 1). This is because there is no difference in information Quantum channel noise and channel capacity contents if the cbit polarity (the definition of 0 and 1, or |0) and |1)) is systematically flipped. Phase-flip channel As a third example of noisy quantum operation, I define next the phase-flip channel in the operator-sum representation: E(p)=pZpZ+(1-p)p=UopUo+U1pU1, (23.18) with the corresponding elements: (23.19) It is again left as an exercise to show that the originator's message /q) = a|0) + B|1) is transformed into |q') = = a|0) - B|1) with probability p and is left invariant otherwise. Thus, the action of the noisy channel is to flip (with probability p) the relative phase eis randomly between the complex amplitudes a, B from ei0 to eir In the case of qubit states /q) = |0), |1) it is found that the entropy change is zero (AS = 0), clearly because no phase change can affect a single qubit. In the case lq) = |+), or p=|+> (+1, one finds the recipient's message: =p-)-1+(1-p)l+)(+1, (23.20) which in the basis {l+>,|-)} is diagonal, i.e., (23.21) hence, the net entropy differenceAS=S(o)-s(p)=f(p)-0=f(p),with = the same result being obtained with p=|-)(-1. The maximum uncertainty (AS = 1 or one cbit), corresponds to the case p = 0.5, meaning that, according to Eq. (23.18), there is a 50% chance that the relative qubit phase is flipped by eir when passing through the channel. Interestingly, there is no entropy difference (AS=0) if the phase is deterministically changed (p = 1). This is because there is no difference in information contents if the cbit polarity (the definition of 0 and 1 or |+) and |-) is systematically flipped. Bi