the above operators are in the same type and both of them involve patterns of open strings and loops. Therefore, the initial state goes to its excitations if the combination of $\hat{\mathcal{E}}(X)$ and $ \hat{\eta}(X)$ corresponds to a pattern that includes open strings and it remains in the subspace of $|\psi _+\ra$ and $|\psi_ -\ra$ if the above combination corresponds to a pattern of loops. Furthermore, here we can also consider coherence in the above subspace where loops that cross the string $\gamma$ for an odd number of times lead to decoherence in the above subspace. In order to characterize the coherence in the above subspace, we define the following parameter as a measure of coherence: $$\label{coh}
O[\boldsymbol\eta]=\frac{W_+[\boldsymbol\eta] -W_-[\boldsymbol\eta]}{W[\boldsymbol\eta]}.$$ Where $W_+[\boldsymbol\eta]$ is the fidelity of state $\Phi[\boldsymbol\eta]$ with $|\psi_+ \ra$ in the form of $W_+[\boldsymbol\eta] =\la \psi_+ |\Phi[\boldsymbol\eta] |\psi_+ \ra$. Similarly, $W_-[\boldsymbol\eta]$ is the fidelity of $\Phi[\boldsymbol\eta]$ and $|\psi_- \ra$ which is $W_-[\boldsymbol\eta] =\la \psi_- |\Phi[\boldsymbol\eta] |\psi_- \ra$ and $W[\boldsymbol\eta]=W_+[\boldsymbol\eta] + W_-[\boldsymbol\eta]$. By a geometrical description, $W_+[\boldsymbol\eta]$ is interpreted as a probability that the combination of errors $\hat{\mathcal{E}}(X)$ and $ \hat{\eta}(X)$ generates loop configurations which cross string $\gamma$ an even number of times and $W_-[\boldsymbol\eta]$ is the probability that the 