s- sion, which also holds in the general case where p is nondiagonal, shows that the effect of noise is to transform with equal probability q/3 the originator's mes- sage p into either XpX, or YpY or ZpZ, and to leave it invariant with probability 23.1 Noisy quantum channels Bit-flip channel As a second type of noisy quantum operation, I introduce next the bit-flip channel, as defined in the operator-sum representation: E(p)=pXpX+(1-p)p (23.13) =UppUo+U1pU1, where, as usual, p is some probability distribution, and where we have the following channel-operator elements: (23.14) It is straightforward to establish that the action of the first operation term XpX in Eq. (23.13) is to transform the originator's message p into the recipient message o = E(p) with the flipped coefficients 011 = P22 and 022=011. Such a transformation has probability p of occurring, while the originator's message is left unchanged (o = p) with probability 1 - p. Effecting the calculation with the qubit message /q) = a|0) + B|1) (a = = is also straightforward to obtain the channel transformation: Thus, if the originator's message is the pure-state qubit |0) (i.e., a = 1, B = 0), the recipient receives the message (23.16) Clearly, the symbol-error (or cbit-error) probability is E = p, meaning that the channel has the probability p of outputting the errored message |1) when |0) was sent. The same conclusion applies in the opposite case, which illustrates the notion of bit flipping as a noise effect in quantum channels. It is left as an interesting exercise to analyze the effect of bit-flip transformation in the case where p is nondiagonal. As expected, the outcome of the exercise is that an originator's qubit message 1q) = a|0) + B|1) is "flipped" through the noisy channel into a recipient's message lq) = B|0) + a|1) under an outcome probability p. Finally, the entropy difference between a pure-state originator's message p = |0) <01 and the recipient's message o = (1 - p)|0)(0|+p|1)(1) is as=S(o)-s(p)=[-(1-p)log(1-p)-plogp]-0 = = - - (23.17) = 