Operator-algebra error-correcting code

Operator-algebra error-correcting code[1][2][3][4][5][6]

Stub.

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.
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. 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.
Quantum error-correcting code (QECC)
Subsystem quantum error-correcting code

Holographic code — Properties of holographic codes are often quantified in the Heisenberg picture, i.e., in terms of operator algebras [7][8].

[1]: Greg Kuperberg, “The capacity of hybrid quantum memory”. 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). DOI; quant-ph/0608071
[3]: C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction of observables”, Physical Review A 76, (2007). DOI; 0705.1574
[4]: Greg Kuperberg and Nik Weaver, “A von Neumann algebra approach to quantum metrics”. 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). DOI; 2012.14001
[7]: A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT”, Journal of High Energy Physics 2015, (2015). DOI; 1411.7041
[8]: F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017). DOI; 1612.00017

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“Operator-algebra error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/oaecc