n of the Pauli matrix X is to flip the states |0), |1) or X|0) = |1) and X|1) = |0), yielding (q/X/q) = 0 and F=V1-p = - in the first case. The fidelity, thus, ranges from F = 1 (noiseless channel, p=0) to F = 0 ("certain" bit-flip channel, p=1). In the second case, the action of X leaves the states 1+), |-) invariant, i.e., X|+) = (X|0) - - |0))/V2ei)=|-),which yields Kq/X|q}l = 1 and, hence, F=1. The fidelity is, thus, maximal, even if the channel is "noisy" and no error-correction coding is implemented! Consider, next, the case with the error-correction code implementation according to Eq. (24.2), i.e., lg) = alô) + B|1), with p== We now have a noisy quantum channel that has the capability of corrupting one, two, or more qubits, which calls for a definition of E(p) that is, in fact, different from Eq. (24.6). For each corruption pattern X, we need to introduce the corresponding quantum operation Ex(p). Here are the different corruption patterns with their operations: (24.8) Exactly one error: Ea(p) = p(1 (24.9) +XQIOXIpXQIQX Exactly two errors: (24.10) Exactly three errors: (24.11) Exactly four errors: (24.12) Quantum error correction which makes up the overall channel definition p' E(p) = E0(p) + Ea(p)+Eb(p)+Ec(p)+Ed(p). (24.13) Here, there is no point in developing the full expression of the recipient's qubit by substituting p = 19) = (a0)+i))((0+B(i),which generates no less than 4 X 16 = 64 operator terms! Rather, it is sensible to conclude directly that after error- correction decoding, which would ideally detect and correct all possible error patterns, the transmitted qubit state takes the form: (p')* = E*(p) = p)4 + 4p (1 p)³ 6p2 (1 = 19)(q1. (24.14) If the QECC only has the capability of correcting up to a single error (using the projectors Pi=1,2,3,4), we may write (24.15) with the missing terms corresponding to patterns of more than one error (see further in discussion about correcting higher-order error patterns). In this case, the fidelity is given by (24.16) = Fmin. In the above, Fmin represents a 