Approximate operator-algebra error-correcting code[1,2] 

Given some algebra \(\mathcal{A}\), \(\mathcal{A}\) is \(\epsilon\)-correctable under noise channel \(\mathcal{N}\) if there exists some quantum channel \(\mathcal{R}\) such that \begin{align} ||(\mathcal{R}\circ\mathcal{N})-P_{\mathcal{A}}||_{\diamond}\leq\epsilon~, \tag*{(1)}\end{align} where \(P_{\mathcal{A}}\) is the projector onto algebra \(A\) and we use the diamond norm \(\diamond\) [3].

Let the minimal error for some algebra \(\mathcal{A}\) under noise channel \(\mathcal{N}\) be \begin{align} \epsilon_{\mathcal{A}}=\min_{\mathcal{R}} ||\mathcal{R}\circ\mathcal{N}-P_{\mathcal{A}}||_{\diamond}~. \tag*{(2)}\end{align} Let \(\delta_{\mathcal{A}}=||\mathcal{N}^C-\mathcal{N}^C\circ P_{\mathcal{A}'}||_{\diamond}\) for commutant \(\mathcal{A}'\) of algebra \(\mathcal{A}\) and complementary channel \(\mathcal{N}^C\) of noise channel \(\mathcal{N}\). Then [1], \begin{align} \delta_{\mathcal{A}}^2/4\leq \epsilon_{\mathcal{A}}\leq 2\delta_{\mathcal{A}}^{1/2}~. \tag*{(3)}\end{align}