and conceive that it may also correspond to that of a quantum memory. Whether information is transmitted or stored, the principle remains the same: an originator must encode the information and input the result into a physical system, and a recipient must do the reverse operation. We may, thus, equivalently refer to either transmission or storage "fidelity" to qualify the quantum channel or system into or through which information is being passed. Such a notion will be developed later. As an example of a noisy quantum system, consider the bit-flip channel described in Chapter 23. To recall, the bit-flip channel converts the qubits |0) and |1) into each other with probability p. Hence, the original qubit |q) = a|0) + B|1) has probability p of being transformed by the quantum channel into |q) = a|1) + B|0). Based on the principle of the four-bit repetition code, it is then sensible to encode the input qubit 1q) l9)=a0000)+B1111), (24.2) for instance (we could have chosen a 3-qubit repetition code as well). This is equivalent to encoding the basis states |0), |1) in the form (24.3) where |Ô), |I) have the meaning of the logical |0), |1) qubits. The quantum circuit for this repetition code, which is based on three CNOT gates (see Chapter 15), is shown in Fig. 24.1. After passing through the bit-flip channel, the 4-qubit basis states |0000), |1111) might have experienced zero, one, two, three, or four qubit flips or errors. We consider here the first two possibilities, whose detailed outcomes are summarized in Table 24.1. From the recipient's end, the detection and correction of errors is a matter of syndrome diagnosis. The notion of "syndrome" for linear block codes and cyclic codes was described in Chapter 11. Here, it is similarly applied to quantum error correction. The first step is to detect the existence of errors, namely of the occurrence of single qubit flips in the received codeword, which is achieved by means of an error syndrome measurement. Such a measurement consists of projecting the received codeword u