5 Logical operators, codes, and distance
The encoded information is acted on by the quotient \(\mathcal{C}(\mathcal{S})/\mathcal{S}\): operators that preserve the codespace, modulo those that act trivially on it.
A Pauli operator is *logical* when it lies in the centralizer definition 26, i.e. it commutes with every stabilizer and therefore maps the codespace to itself.
It suffices to test commutation against a generating set: \(g\) is logical iff it commutes with every generator of \(\mathcal{S}\).
Commuting with \(a\) and \(b\) implies commuting with \(ab\), so the set of elements commuting with \(g\) is a subgroup; if it contains the generators it contains their closure. This is what makes logicality a finite check for every concrete code in chapter 10.
A *nontrivial* logical operator is one that acts on the encoded information rather than fixing it. The predicate has two conditions: \(g\) is in the centralizer, and no element of \(\mathcal{S}\) has the same operator part as \(g\).
The second condition is stronger than the familiar \(g \notin \mathcal{S}\) (which it implies, at \(s = g\)) and is what makes the CSS bridge arguments work: with only the weaker condition, \(g\) could differ from a stabilizer by a phase, which is not a nontrivial action on the codespace.
The two conditions of definition 31 spelled out as a conjunction, which is the form the concrete distance proofs consume.
Definitional.
The second condition of definition 31 is deliberately stronger than the familiar \(g \notin \mathcal{S}\), which it implies by taking \(s = g\): no stabilizer may share the operator part of \(g\). With only the weaker condition a \(g\) differing from a stabilizer by a phase would count as acting nontrivially, and the CSS bridge argument of theorem 57 would fail.
Nontriviality only depends on the underlying subgroup, so it transfers between two stabilizer groups with the same carrier.
Both conditions of definition 31 are stated in terms of the subgroup alone. This lemma is the glue that lets a distance theorem proved against a hand-built stabilizer group be reused for the packaged definition 37, whose generator list is trimmed.
A bundled pair of logical \(\overline{X}\) and \(\overline{Z}\) operators for each of the \(k\) encoded qubits, packaged with the proofs that they are logical, are not stabilizers, and anticommute in the right pattern.
5.1 Packaging a code
A generator list is independent when no generator lies in the group generated by the others, so that \(|\mathcal{S}| = 2^{n-k}\) for a list of length \(n - k\).
If the rows of the check matrix are linearly independent over \(\mathbb {F}_2\) (definition 19) then the generators are independent in the sense of definition 35.
A relation among the generators pushes forward, under the homomorphism definition 14, to a linear relation among the rows of the check matrix. Linear independence of the rows therefore forces the relation to be trivial. This converts a group-theoretic side condition into a rank computation over \(\mathbb {F}_2\).
An \([[n,k]]\) stabilizer code: a list of \(n-k\) independent, pairwise commuting generators avoiding \(-I\) — and nothing else. The code *is* its stabilizer group; a choice of logical operators is derived data, bundled separately by definition 38. The type carries \(n\) and \(k\), so instantiating it *is* the theorem that a given family of operators encodes \(k\) qubits into \(n\).
Instantiating definition 37 for a concrete family is the theorem that the family encodes \(k\) qubits into \(n\): the type carries \(n\) and \(k\), and the obligations — generator count, independence, pairwise commutation, and exclusion of \(-I\) — must all be discharged to build an inhabitant. Nothing about logical operators enters the definition: a stabilizer code is its stabilizer group.
A definition 37 together with \(k\) pairs of logical operators (definition 34), one per encoded qubit, such that the pairs of distinct qubits commute. This is what the constructions that genuinely need a basis consume — logical Clifford actions, the encoding step of code concatenation — while distance (definition 39) lives on the bare code.
A chosen logical basis is layered on top. Its extra obligations are the centralizer and anticommutation conditions on the logical operators, and only the constructions that act on encoded qubits need it; the distance theory of the next section never looks at it.
5.2 Distance
A code *has distance \(d\)* when every nontrivial logical operator (definition 31) has weight at least \(d\), and some nontrivial logical operator has weight exactly \(d\). Both halves matter: the lower bound is the error-correction guarantee, the witness makes the value exact rather than merely a bound.
If a code has distance \(d\) then every nontrivial logical operator has weight at least \(d\); equivalently, no undetectable error of weight below \(d\) exists.
The first projection of definition 39.
An \([[n,k,d]]\) code: a definition 37 bundled with a proof of definition 39. The headline results of this development are inhabitants of this type.
Both halves of definition 39 matter. The lower bound is the error-correction guarantee; the witness of exact weight is what makes the distance a value rather than a bound, and it is usually the easier half to forget.