CNOT gates, entangling the system with two ancillary qubits initialized in the state . The encoded state is what is now passed through the noisy channel. The channel acts on by flipping some subset (possibly empty) of its qubits. No qubit is flipped with probability , a single qubit is flipped with probability , two qubits are flipped with probability , and all three qubits are flipped with probability . Note that a further assumption about the channel is made here: we assume that acts equally and independently on each of the three qubits in which the state is now encoded. The problem is now how to detect and correct such errors, while not corrupting the transmitted state. Let us assume for simplicity that is small enough that the probability of more than a single qubit being flipped is negligible. One can then detect whether a qubit was flipped, without also querying for the values being transmitted, by asking whether one of the qubits differs from the others. This amounts to performing a measurement with four different outcomes, corresponding to the following four projective measurements:This reveals which qubits are different from the others, without at the same time giving information about the state of the qubits themselves. If the outcome corresponding to is obtained, no correction is applied, while if the outcome corresponding to is observed, then the Pauli X gate is applied to the -th qubit. Formally, this correcting procedure corresponds to the application of the following map to the output of the channel: Note that, while this procedure perfectly corrects the output when zero or one flips are introduced by the channel, if more than one qubit is flipped then the output is not properly corrected. For example, if the first and second qubits are flipped, then the syndrome measurement gives the outcome , and the third qubit is flipped, instead of the first two. To assess the performance of this error correcting scheme for a general input we can study the fidelity between the input and the output . Being the output state correct when no more than one qubit is flipped, which happens with probability , we can write it as , where the dots denote components of resulting from errors not properly corrected by the protocol. It follows that This fidelity is to be compared with the corresponding fidelity obtained when no error correcting protocol