Description
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}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 \(\mathcal{A}\).
Children
- Approximate operator-algebra error-correcting code
- Error-correcting code (ECC) — Any ECC can be embedded into a quantum Hilbert space, and thus passed through a quantum channel, by associating elements of the alphabet with basis vectors in a Hilbert space over the complex numbers. In other words, classical codewords are elements of an alphabet, while what codewords are functions on the alphabet. For example, a bit of information can be embedded into a two-dimensional vector space by associating the two bit values with two basis vectors for the space.
- Classical-quantum (c-q) code
- Entanglement-assisted hybrid classical-quantum (EACQ) code — EACQ codes utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to operator-algebra QECCs when said subsystems are interpreted as noiseless physical subsystems.
- Subsystem quantum error-correcting code
- Quantum error-correcting code (QECC)
- Operator-algebra qubit stabilizer code
References
- [1]
- G. Kuperberg, “The capacity of hybrid quantum memory”, (2003) arXiv:quant-ph/0203105
- [2]
- 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
- [3]
- C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction of observables”, Physical Review A 76, (2007) arXiv:0705.1574 DOI
- [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
Page edit log
- Michael Liu (2023-17-06) — most recent
- Victor V. Albert (2021-11-24)
Cite as:
“Operator-algebra error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), NaN. https://errorcorrectionzoo.org/c/oaecc
Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/oaecc.yml.