If every $\operatorname{wt}(E_a)<d/2$, then
$$
\operatorname{wt}(E_a^\dagger E_b)
\leq\operatorname{wt}(E_a)+\operatorname{wt}(E_b)<d.
$$
By the definition of the <distance of a stabilizer code>, no Pauli operator of weight below $d$ is a nontrivial logical operator. Part (b) therefore proves the <local correctability of a stabilizer code> for this error set.

The converse fails because pairwise products, rather than individual weights, control correctability. For example, let $P$ be a high-weight Pauli outside $C(\mathcal S)$ and take the error set $\{I,P\}$. If $P$ anticommutes with a stabilizer, then $\Pi P\Pi=0$, while $P^\dagger P=I$; the KL conditions hold even when $\operatorname{wt}(P)\geq d/2$. A still simpler singleton set containing any known unitary Pauli error is always reversible regardless of its weight.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 / 2 / c / Solution

If every pairwise product $E_a^\dagger E_b$ has weight below the code distance, it cannot be a nontrivial logical operator. The Knill--Laflamme conditions then hold for the whole error set.


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 Incoming links: Local correctability of a stabilizer code