Finite-dimensional quantum error-correcting code 

Denoting Hilbert spaces by the letter \(\mathsf{H}\), a finite-dimensional quantum code \((U,\cal{E})\) is a partial isometry \(U:\mathsf{H}_{K}\to\mathsf{H}_{N}\) and a set of correctable errors \({\cal{E}}:\mathsf{H}_N\to\mathsf{H}_M\) with the following property: there exists a quantum operation \({\cal{D}}:\mathsf{H}_M\to \mathsf{H}_K\) such that for all \(E\in\cal{E}\) and states \(|\psi\rangle\in\mathsf{H}_{K}\), \begin{align} {\cal D}(EU|\psi\rangle\langle\psi|U^{\dagger}E^{\dagger})=c(E,|\psi\rangle)|\psi\rangle\langle\psi|\tag*{(1)}\end{align} for some constant \(c\) [1]. A code is said to protect against or correct the errors \(\mathcal{E}\).

Knill-Laflamme error correction conditions

Equivalently, correction capability is determined by of the quantum error-correction conditions, which may admit infinite terms due to non-normalizability of ideal code states. A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an approximate code.

The Knill-Laflamme conditions can alternatively be expressed in terms of the complementary channel, or in an information-theoretic way via a data processing inequality [6,7][8; Eq. (29)]. They have been extended to sequences of multiple errors and rounds of correction [9].

Correctability of quantum channels

From now on, we use \(\mathcal{E}\) to denote a noise channel constructed out of the set of errors \(E\) and let \(\mathcal{U}(\cdot)=U(\cdot)U^\dagger\) be the superoperator corresponding to the partial encdoing isometry \(U\). A noise channel is correctable if there exists a recovery channel \(\mathcal{D}\) such that \begin{align} \mathcal{D}\mathcal{E}\mathcal{U}(\rho)=\rho \tag*{(3)}\end{align} for all logical states \(\rho\).

The above is equivalent to the fidelity between \(\rho\) and \(\mathcal{D}\mathcal{E}\mathcal{U}(\rho)\) being one for any notion of distance between quantum states. In particular, we can consider a scenario where we send only one part of an entangled state through a channel and determine whether the entanglement has been preserved during transmission. Using the notion of entanglement fidelity, a quantum channel \(\mathcal{E}\) is exactly correctable iff there exists a quantum channel \(\mathcal{D}\) such that \begin{align} (\mathcal{D}\mathcal{E}\mathcal{U}\otimes\mathrm{id})(\ket{\psi}\bra{\psi})=\ket{\psi}\bra{\psi} \tag*{(4)}\end{align} for all states \(\rho\) and their corresponding purifications \(\ket{\psi}\) (i.e., states \(\ket{\psi}\) such that \(\text{Tr}_{2}(|\psi\rangle\langle\psi|)=\rho\)).

The above entanglement fidelity condition can be alternatively expressed using complementary channels.

A channel \(\mathcal{E}\) is correctable if \(\mathcal{E}^C(\rho)=\rho_0\mathrm{Tr}(\rho)\) for some constant state \(\rho_0\), which is equivalent to the Knill-Laflamme conditions [11].