## Description

A code family that encompasses ordinary (i.e., subspace) codes, subsystem codes, classical-quantum codes, and hybrid codes using a unified 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 \(j\in\{0,1\}\) 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 code 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 \(\mathsf{A}_j\) factors can be used to store quantum information, \(\gamma\) indexes the block structure of the code, while \(\mathsf{B}_j\) determine its gauge structure. Together, the above forms the most general form of an information preserving structure [7,8].

## 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 \(A_\gamma\) 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 \(\mathcal{A}\).

Tradeoffs between error correction and privacy have been studied [9].

## Parent

- Approximate operator-algebra QECC — Approximate OAQECCs correcting a noise channel exactly reduce to OAQECCs.

## Children

- Classical-quantum (c-q) code — An OAQECC that retains its block structure for storing classical information but stores no quantum information and has no gauge degrees of freedom (e.g., gauge qubits) is a c-q code.
- Hybrid QECC — An OAQECC which has no gauge structure (e.g., gauge qubits) but has a block structure that corresponds to a classical code is a hybrid QECC.
- Subsystem QECC — An OAQECC which has gauge structure (e.g., gauge qubits) but no block structure is a subsystem QECC.
- Quantum error-correcting code (QECC) — An OAQECC which has no gauge structure (e.g., gauge qubits) and no block structure is a QECC.
- Operator-algebra (OA) qubit code

## References

- [1]
- C. Bény, A. Kempf, and D. W. Kribs, “Generalization of Quantum Error Correction via the Heisenberg Picture”, Physical Review Letters 98, (2007) arXiv:quant-ph/0608071 DOI
- [2]
- C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction of observables”, Physical Review A 76, (2007) arXiv:0705.1574 DOI
- [3]
- C. Bény, “Information flow at the quantum-classical boundary”, (2009) arXiv:0901.3629
- [4]
- G. Kuperberg and N. Weaver, “A von Neumann algebra approach to quantum metrics”, (2010) arXiv:1005.0353
- [5]
- C. BÉNY, D. W. KRIBS, and A. PASIEKA, “ALGEBRAIC FORMULATION OF QUANTUM ERROR CORRECTION”, International Journal of Quantum Information 06, 597 (2008) DOI
- [6]
- K. Furuya, N. Lashkari, and S. Ouseph, “Real-space RG, error correction and Petz map”, Journal of High Energy Physics 2022, (2022) arXiv:2012.14001 DOI
- [7]
- R. Blume-Kohout et al., “Characterizing the Structure of Preserved Information in Quantum Processes”, Physical Review Letters 100, (2008) arXiv:0705.4282 DOI
- [8]
- R. Blume-Kohout et al., “Information-preserving structures: A general framework for quantum zero-error information”, Physical Review A 82, (2010) arXiv:1006.1358 DOI
- [9]
- D. W. Kribs et al., “Quantum Complementarity and Operator Structures”, (2019) arXiv:1811.10425

## Page edit log

- Michael Liu (2023-17-06) — most recent
- Victor V. Albert (2021-11-24)

## Cite as:

“Operator-algebra QECC (OAQECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), NaN. https://errorcorrectionzoo.org/c/oaecc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/oaecc.yml.