Operator-algebra error-correcting code[1–6] 

Given an error operation \(\mathcal{E}\), one says that \(\mathcal{A}\) is correctable for \(\mathcal{E}\) if there exists a recovery operation \(\mathcal{R}\) such that \begin{align}P_{\mathcal{A}} (\mathcal{R} \circ \mathcal{E})^\dagger(X) P_{\mathcal{A}} = X\tag*{(3)}\end{align} for all \(X \in \mathcal{A}\), where \(P_{\mathcal{A}}\) is the unit projection onto \(\mathcal{A}\).

Equivalently, \(\mathcal{A}\) is correctable for \(\mathcal{E}\) if there exists a recovery operation \(\mathcal{R}\) such that for any \(\gamma\) and density operators \(\rho_\gamma,\sigma_\gamma\) supported on \(A_\gamma\) and \(B_\gamma\), respectively, there exists a state \(\tau_\gamma\) supported on \(A_\gamma\) such that \begin{align}(\mathcal{R} \circ \mathcal{E})(\rho_\gamma \otimes \sigma_\gamma) = \tau_\gamma \otimes \sigma_\gamma.\tag*{(4)}\end{align}

An algebraic condition for correctability can be given in terms of the Kraus operators \(E_j\) of \(\mathcal{E}\). Indeed, \(\mathcal{A}\) is correctable for \(\mathcal{E}\) if \begin{align}P_{\mathcal{A}} E_j^\dagger E_k P_{\mathcal{A}} \in \mathcal{A}'\tag*{(5)}\end{align} for all \(j,k\), where \(\mathcal{A}'\) is the commutant of \(\mathcal{A}\).