irst and third qubits are the same, which confirms the previous diagnostic. Table 24.3 summarizes the different diagnostic and corrective action possibilities in the general case. Clearly, the complete diagnostic and error-correction procedure requires one to perform the two comparative measurements Z12,Z23 on each of the three blocks A, B, C, representing altogether six measurements and 4 superscript(3) = 64 possible bit-flip error diagnostics. It is also clear that a bit flip error occurring on any block U of [0) replicates on the same block U of [ (observing from Eq. (24.23) that these two blocks only differ by their relative phase factor). The above description concerned the detection and correction of either phase-flip or bit-flip errors occurring in the encoded 9-qubits [Ô) and [i). It is clear that since the syndrome measurements leave the qubit invariant, both error types can be corrected in Quantum error correction any order. What about error correction on encoded 9-qubits of the form (24.31) corresponding to an originator message q) = a/0) + 1 B|1)? To answer this question, we may first define the channel output qubit as (24.32) As we have seen earlier, the phase-flip syndrome diagnosis consists of making two successive measurements of the type (ijklmn = 123456, 123789, 456789): (24.33) In the above result, we first used the property that the measurement Xijklmn leaves the state invariant, therefore, the nondiagonal elements in the right-hand side in Eq. (24.33) are zero. Since the same phase-flip error must be found in the two qubits () and [i'), the two nonzero matrix elements must be equal, and considering the property |a|2 +181==1, = we finally have (Xijklmn) = +1, corresponding to no (+1) or one (-1) phase-flip error between the blocks A, B. Thus, syndrome diagnostic and error correction can be implemented on the encoded qubit = + BII) in exactly the same way as described earlier for the qubits (ô) and [i'). Clearly, the same conclusion applies to the case of bit-flip errors with two successive mea