In the image below for stabilizers for shors 9 qubit QECC, I understand the role of S1 to S8 operators but I am not getting role of X-bar and Z-bar operators. I know how to obtain them using standard form but not clear with its use and why do we need it
If you just use $S_1$ to $S_8$ on 9 qubits, you are left with a space of dimension 2. When you see something of dimension 2, you think "aha! an effective qubit". But to properly describe a qubit, you need two anti-commuting operators, conventionally $X$ and $Z$. Think: two axes of the Bloch sphere.
So, the $\bar X$ and $\bar Z$ are the operators that anti-commute with each other, giving the algebra of qubit, while operating entirely within the two-dimensional subspace (i.e. commuting with $S_1$ to $S_8$).
