Description
A code family that encompasses ordinary (i.e., subspace) codes, subsystem codes, classical-quantum codes, and hybrid codes using an operator-algebraic framework.
A simple example encompassing elements of all subfamilies encodes quantum information and a single classical bit into a direct sum of two subsystem codes. A quantum subsystem code \(\mathsf{A}_j\otimes\mathsf{B}_j\), with \(\mathsf{A}_j\) the logical factor associated with the quantum information, and \(\mathsf{B}_j\) the gauge factor, is associated with each of the two values \(j\in\{1,2\}\) of the classical bit. The corresponding decomposition of the Hilbert space \(\mathsf{H}\) is \begin{align} \mathsf{H}=(\mathsf{A}_{1}\otimes\mathsf{B}_{1})\oplus(\mathsf{A}_{2}\otimes\mathsf{B}_{2})\oplus\mathsf{C}^{\perp}~, \tag*{(1)}\end{align} where \(\mathsf{C}^\perp\) is the combined error space of both codes. The above code reduces to a subsystem code when \(\mathsf{A}_{2}\otimes\mathsf{B}_{2}\) is trivial, reduces to a classical-quantum code when \(\mathsf{A}_{1,2}\) are both trivial, reduces to a hybrid code when \(\mathsf{B}_{1,2}\) are both trivial, and reduces to an ordinary (i.e., subspace) code when \(\mathsf{B}_1\) and \(\mathsf{A}_{2}\otimes\mathsf{B}_{2}\) are both trivial.
In general, an OAQECC is determined by a finite dimensional \(C^*\) algebra \(\mathcal{A}\) of operators on \(\mathsf{H}\). This logical algebra induces a decomposition of the Hilbert space as \begin{align}\mathsf{H} = \bigoplus_\gamma \mathsf{A}_\gamma \otimes \mathsf{B}_\gamma,\tag*{(2)}\end{align} with respect to which \(\mathcal{A}\) takes the form \begin{align}\mathcal{A} = \bigoplus_\gamma I_\gamma \otimes \mathcal{L}(\mathsf{B}_\gamma),\tag*{(3)}\end{align} where \(\mathcal{L}(\mathsf{B}_\gamma)\) denotes the full set of linear maps on \(\mathsf{B}_\gamma\). The \(\mathsf{A}_{\gamma}\) factors can be used to store quantum information, \(\gamma\) indexes the block structure of the code, while \(\mathsf{B}_{\gamma}\) determine its gauge structure. Together, the above forms the most general form of an information preserving structure [1,4,11–14]. Logical operators form the commutant of \(\mathcal{A}\) as a result of the double commutant (a.k.a. double centralizer) theorem [15].
Protection
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}\Pi_{\mathcal{A}} (\mathcal{R} \circ \mathcal{E})^\dagger(X) \Pi_{\mathcal{A}} = X\tag*{(4)}\end{align} for all \(X \in \mathcal{A}\), where \(\Pi_{\mathcal{A}}\) 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 \(\mathsf{A}_\gamma\) and \(\mathsf{B}_\gamma\), respectively, there exists a state \(\tau_\gamma\) supported on \(\mathsf{A}_\gamma\) such that \begin{align}(\mathcal{R} \circ \mathcal{E})(\rho_\gamma \otimes \sigma_\gamma) = \tau_\gamma \otimes \sigma_\gamma.\tag*{(5)}\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}\Pi_{\mathcal{A}} E_j^\dagger E_k \Pi_{\mathcal{A}} \in \mathcal{A}'\tag*{(6)}\end{align} for all \(j,k\), where \(\mathcal{A}'\) is the commutant of \(\mathcal{A}\).
Conversely, a private algebra \(\mathcal{A}\) for a channel \(\mathcal{E}\) is one which is completely decohered by the channel [16]. In other words, no information about the algebra is retained after the action of the channel. Tradeoffs between error correction and privacy have been studied [17].
Cousins
- Approximate operator-algebra QECC— Approximate OAQECCs correcting a noise channel exactly reduce to OAQECCs.
- Entanglement-assisted operator-algebra QECC (EAOA QECC)— EAOA QECCs use pre-shared entangled ancillary subsystems, while OAQECCs recover the same operator-algebraic structures when those ancillary subsystems are instead treated as noiseless physical subsystems.