By definition, we define the logical Pauli operator as element of the centralizer of the stabilizer group $S$ in $G_n$ (the $n$-Pauli group).
It is a definition so why not.
My question is: mathematically speaking, is it really necessary ? Is this definition taken for mathematical simplicity (but we could imagine being less restrictive).
Indeed, what we need in the end is (i) to define a family of logical operator (ii) that verify the appropriate Pauli algebra.
Calling $U$ a logical operator, the minimal mathematical requirements we need are that it keeps any state $|\psi\rangle$ that is in the code space $C(S)$ inside of this code space. Thus, formally that:
$$\forall |\psi\rangle \in C(S): U |\psi\rangle \in C(S)$$
And this is all what we need conceptually. I guess that in the end we could find such operators $U$ not in the Pauli group that will respect the Pauli algebra. Are there hidden conditions I am not seing that would actually show that the logical Pauli operator must be elements of $G_n$ ?
