Alice and Bob generate their resource state, i.e, an $(n+k)$-qubit stabilizer state with generators \begin{equation} \{ g_i, \, \bar{X}_j X_{n+j}, \, \bar{Z}_j Z_{n+j} \} \, . \end{equation}
Here, $g_i$ represents the code stabilizer generator, where $i = 1,2, \dots, n-k$. The code has supports on the first $n$ qubits. $\bar{X}_j$ and $\bar{Z}_j$ represent the Pauli operators of the $j$-th logical qubit, $X_{n+j}$ and $Z_{n+j}$ represent the Pauli operators of the $(n+j)$-th physical qubit, where $j = 1,2, \dots, k$. $X_j X_{n+j}$ and $\bar{Z}_j Z_{n+j}$ denote the tensor products $\bar{X}_j \otimes X_{n+j}$ and $\bar{Z}_j \otimes Z_{n+j}$, respectively.
The resource state forms $k$ logical-physical Bell states, represented by \begin{equation} \left|\mathcal{R}_D\right\rangle = \frac{1}{\sqrt{2^k}} \bigotimes_{j=1}^{k} \left( |\bar{0}_j\rangle |0_{n+j}\rangle + \|\bar{1}_j\rangle |1_{n+j}\rangle \right), \end{equation} where $|\bar{0}_j\rangle$ and $|\bar{1}_j\rangle$ represent the computational bases of the $j$-th logical qubit, and $|0_{n+j}\rangle$ and $|1_{n+j}\rangle$ represent the bases of the $(n+j)$-th physical qubit.
I am trying to generate a resource state informed by a $[[4,2,2]]$ CSS code. It should be 6-qubit stabilizer state with generators \begin{equation} \{ g_1 = X_1 X_3, \quad g_2 = Z_1 Z_3, \quad g_3 = X_2 X_4, \quad g_4 = Z_2 Z_4, \quad \bar{X}_1 X_5, \quad \bar{Z}_1 Z_5, \quad \bar{X}_2 X_6, \quad \bar{Z}_2 Z_6 \} \end{equation} where
\begin{array}{ll} \textbf{Logical X operators:} & \bar{X}_1 = X_1 X_3, \quad \bar{X}_2 = X_2 X_4 \\[6pt] \textbf{Logical Z operators:} & \bar{Z}_1 = Z_1 Z_2, \quad \bar{Z}_2 = Z_3 Z_4 \end{array}
The logical qubit encoding for the $[[4,2,2]]$ CSS code is:
\begin{equation} |\bar{0}_1 \bar{0}_2\rangle = \frac{1}{\sqrt{2}} (|0000\rangle + |1100\rangle), \end{equation}
\begin{equation} |\bar{0}_1 \bar{1}_2\rangle = \frac{1}{\sqrt{2}} (|0011\rangle + |1111\rangle), \end{equation}
\begin{equation} |\bar{1}_1 \bar{0}_2\rangle = \frac{1}{\sqrt{2}} (|0000\rangle + |0011\rangle), \end{equation}
\begin{equation} |\bar{1}_1 \bar{1}_2\rangle = \frac{1}{\sqrt{2}} (|1100\rangle + |1111\rangle). \end{equation}
Expanding the tensor product for the $[[4,2,2]]$ CSS code:
\begin{equation} |\mathcal{R}_D\rangle = \frac{1}{\sqrt{4}} \left( |\bar{0}_1\rangle |0_5\rangle + |\bar{1}_1\rangle |1_5\rangle \right) \otimes \left( |\bar{0}_2\rangle |0_6\rangle + |\bar{1}_2\rangle |1_6\rangle \right). \end{equation}
Expanding the tensor product:
\begin{equation} |\mathcal{R}_D\rangle = \frac{1}{2} \sum_{a,b \in \{0,1\}} \left( |\bar{a}_1\rangle |a_5\rangle \otimes |\bar{b}_2\rangle |b_6\rangle \right). \end{equation}
Explicitly,
\begin{equation} |\mathcal{R}_D\rangle = \frac{1}{2} \Big( |\bar{0}_1\rangle |0_5\rangle \otimes |\bar{0}_2\rangle |0_6\rangle + |\bar{0}_1\rangle |0_5\rangle \otimes |\bar{1}_2\rangle |1_6\rangle + \end{equation}
\begin{equation} |\bar{1}_1\rangle |1_5\rangle \otimes |\bar{0}_2\rangle |0_6\rangle + |\bar{1}_1\rangle |1_5\rangle \otimes |\bar{1}_2\rangle |1_6\rangle \Big). \end{equation}
Substituting the logical qubit encodings:
\begin{equation} |\mathcal{R}_D\rangle = \frac{1}{2\sqrt{2}} \Big( (|0000\rangle + |1100\rangle) |0_5\rangle |0_6\rangle + (|0011\rangle + |1111\rangle) |0_5\rangle |1_6\rangle + \end{equation}
\begin{equation} (|0000\rangle + |0011\rangle) |1_5\rangle |0_6\rangle + (|1100\rangle + |1111\rangle) |1_5\rangle |1_6\rangle \Big). \end{equation}
\begin{equation} |\mathcal{R}_D\rangle = \frac{1}{2\sqrt{2}} \Big( |000000\rangle + |110000\rangle + |001101\rangle + |111101\rangle + \end{equation}
\begin{equation} |000010\rangle + |001110\rangle + |110011\rangle + |111111\rangle \Big). \end{equation} Can somebody verify this calculation?
