While reading through some lecture notes on quantum error correction, I read the statement:
"In particular, the syndrome doesn’t depend on the specific codeword, only on the Pauli error."
I'm trying to come up with a proof as to why this is the case. I know that codewords are vectors in the shared +1 eigenvalue subspace among the Pauli errors in the stabilizer group, $S$.
I suppose the question is whether given a Pauli error (not in the stabilizer group), $P$, does $P$ have the same error syndrome for all codewords? This seems like it would not be true because we only know how Pauli errors in $S$ act on the codespace.