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Minor corrections #46

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Oct 24, 2024
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Minor corrections #46

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tim-at-topos May 30, 2024
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Merge branch 'main' into typos
thosgood Oct 24, 2024
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4 changes: 2 additions & 2 deletions source/14-quantum-error-correction.Rmd
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Expand Up @@ -2285,11 +2285,11 @@ defines a CSS code with specification $[[n^2+m^2,(n-m)^2,\min(d,d^t)]]$.

### Error-correcting conditions, algebraically {#error-correcting-conditions-algebraically}

Let $\mathcal{S}\leq\mathcal{P}_n$ be a stabiliser group, and let $\mathcal{E}\subseteq\mathcal{P}_n$ be a set of physical errors.
Let $\mathcal{S}\leq\mathcal{P}_n$ be a stabiliser group, and let $\mathcal{E}\subseteq\mathcal{P}_n$ be a set such that $\id\in\mathcal{E}$.
Prove that the stabiliser code defined by $\mathcal{S}$ can perfectly correct for all errors in $\mathcal{E}$ if and only if
$$
E_1^\dagger E_2
\in N(\mathcal{S})\setminus\mathcal{S}
\not\in N(\mathcal{S})\setminus\mathcal{S}
$$
for all $E_1,E_2\in\mathcal{E}$.

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