lower bound for the fidelity corresponding to an error- correction capability of up to one qubit. We may compare the minimum fidelity Fmin to that corresponding to the transmission of |0), |1) without error-correction coding, and for which F = 1 - p, as established earlier. It is found numerically that Fmin > F under the condition p < Pmax = 0.23, as illustrated in Fig. 24.2. This result means that for channels with p > 0.23, there is no point or merit whatsoever in implementing this quantum repetition code! In comparison, the 3-qubit repetition code yields a minimum fidelity of Fmin = (1-p)3 3p(1 - p)2 (which is straightforward to establish), to which the condition p < 0.5 = Pmax corresponds. The fact that the channel bit-flip prob- ability p may not exceed some threshold value Pmax, and that this threshold decreases as we may increase the number of repeated qubits, nicely illustrates the intrinsic limitations of the quantum repetition code. As we have seen from the above description, implementing the repetition code with longer codewords increases the chance of obtaining more than one (bit-flip) error. How about the possibility of correcting these higher-order error patterns? We may easily convince ourselves that, unfortunately, the repetition code does not offer such a possibility. Indeed, consider, for instance, the projection operator P = |0011) <0011| + |1100)(1100|. Using such a projection will tell with certainty that there are two errors. But there is no way of telling whether these errors occurred on the first two or the last two 24.1 Quantum repetition code Probability p Figure 24.2 Fidelity of 4-qubit repetition coding in bit-flip channel for no error correction (F), and up to one error correction (Fmin), showing improvement in the region p < Pmax = 0.23. qubits, these two events being equiprobable. The same ambiguity prevails with the two other possible projectors, P = |0101)(0101 + 1010) (1010| and P = |0110) (0110| + |1001) (1001|. The detection of three errors would take the same projectors as Pi=