Eq. (23.7) is referred to as the operator-sum representation of the channel, with Uk being the channel-operator elements. Consider next the case of nonideal, or noisy quantum channels. Clearly, such channels can be modeled by a quantum operation E(PM), whose outcome OM must be random. Here, I shall describe four basic types of noisy quantum channels, assuming that the message is made of a single qubit. Depolarizing channel The first type of noisy quantum channel, called a depolarizing channel, is defined as follows: e(p)=p+(1-p)p. (23.8) where p is some probability distribution and I is the identity matrix. According to this definition, there is a probability 1- - p that the originator message p is left invariant by the channel, and a probability p that it is transformed into the message o = I/2. Thus, the messages p' = |0)<01 and p" = |1) (1), corresponding to the pure basis states |0) and |1), are transformed into (23.9) The above-defined quantum operation E is referred to as the "depolarizing" channel, because it transforms a pure basis state p', p" into a uniformly mixed state o = I/2 with a certain probability p. From the above results, it would appear that the operation E(p) is in fact equivalent to the following pure-state transformation: /1-210 (23.10) However, in view of the recipient's qubit definitions in Eq. (23.10), such an interpre- tation is ambiguous, since the relative phases () between the |0) and |1) qubits are undefined. Then which are the actual recipient qubits? The answer can be obtained by analyzing the quantum operation defined in Eq. (23.8). We observe that there is a probability E=1- - p that the communication channel accurately transmits the original message P. Thus, E represents the probability of the recipient obtaining nonerrored states Quantum channel noise and channel capacity /9 in = corresponding to original states. The probability that the recip- ient states are errored is p. In this event, the communication channel "ignores" p and "chooses" instead to transmit o = I/2, which i