Operator-algebra error-correcting code[16] 

Description

Encodes quantum and/or classical information into a collection of subsystems of a Hilbert space \(\mathsf{H}\) that 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 A_\gamma \otimes B_\gamma,\tag*{(1)}\end{align} with respect to which \(\mathcal{A}\) takes the form \begin{align}\mathcal{A} = \bigoplus_\gamma I_\gamma \otimes \mathcal{L}(B_\gamma),\tag*{(2)}\end{align} where \(\mathcal{L}(B_\gamma)\) denotes the full set of linear maps on \(B_\gamma\).

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

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
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Zoo Code ID: oaecc

Cite as:
“Operator-algebra error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), NaN. https://errorcorrectionzoo.org/c/oaecc
BibTeX:
@incollection{eczoo_oaecc, title={Operator-algebra error-correcting code}, booktitle={The Error Correction Zoo}, year={NaN}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/oaecc} }
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“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/edit/main/codes/oaecc.yml.