Eq.(\[deg2\]) is written in the following quantum form: $$\label{deg3}
m_n[\mathbf{J}]=\frac{\la \alpha[\mathbf{J}] |\prod_{e\in \gamma_n}Z_e |G\ra}{\la \alpha[\mathbf{J}] |G\ra},$$ where the operator $\Gamma_Z =\prod_{e\in \gamma_n} Z_e$ refers to a product of $Z$ operators corresponding to all edge qubits belonging to string $\gamma_n$. Furthermore, we note that since here we have an Ising model with open boundary, the state $|G\ra$ in the above relation is also the Toric code state defined on a lattice with open boundary condition. Thus, the face operators in the boundary should be a local operator in the form of a product of three $Z$ operators which is called a three-local operator, see Fig.(\[string\]). The operator $\Gamma_Z$ is in fact the quantum form of the product $(s_n s_{n-1})s_{n-1} ... s_1 (s_1 s_0)$ in Eq.(\[deg2\]). Now we have a quantum formalism for the order parameter of the random bond Ising model with an arbitrary quenched configuration of disorder. We should emphasize that it is necessary to take the mean value $\overline{m_n[\mathbf{J}]}$ for different quenched configurations. Therefore, the mean value $m=(\sum_n \overline{m_n[\mathbf{J}]})/N$ will be a function of $\beta$ and $p$. In the next section, we find a physical meaning for this quantity where we show that it is related to a decoherence 