Approximate operator-algebra QECC[1,2] 

Given an 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}

Complementary channel formulation

Given projector \(\mathcal{P}_{\mathcal{A}}\) onto algebra \(\mathcal{A}\) and noise channel \(\mathcal{N}\). Using worst-case entanglement fidelity \(F\), we can quantify approximate operator algebra error correction as \begin{align}F(R\mathcal{N},\mathcal{P}_{\mathcal{A}})\geq 1-\epsilon\tag*{(4)}\end{align} for some small \(\epsilon\) and recovery channel \(R\).

This is equivalent to considering the complementary channel \(\mathcal{N}^C\) with projector \(\mathcal{P}_{\mathcal{A}'}\) onto the commutant \(\mathcal{A}'\) of \(\mathcal{A}\) such that \begin{align} F(\mathcal{N},R'\mathcal{P}_{\mathcal{A}'})\geq 1-\epsilon \tag*{(5)}\end{align} for that same value of \(\epsilon\) and some channel \(R'\).

This formulation acts as a necessary and sufficient condition for approximate operator-algebra QECC and has been generalized to scenarios where there are restrictions on locality [4]. When \(\epsilon=0\) we can derive the standard correctability conditions for operator algebras [2].