Entanglement-assisted (EA) QECC

Entanglement-assisted (EA) QECC[1,2]

QECC whose encoding and decoding utilizes pre-shared entanglement between sender and receiver.

See Ref. [3] for an introduction to EA QEC.

Quantum error-correcting code (QECC) — EA QECCs utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to QECCs when said subsystems are interpreted as noiseless physical subsystems.
Entanglement-assisted hybrid classical-quantum (EACQ) code

Modular-qudit code — Pure modular-qudit codes can be used to make EA-QECCs with the same distance and dimension; see Thm. 10 of Ref. [4].
Linear binary code — Any linear binary code can be used to construct an EAQECC.
Linear \(q\)-ary code — Any linear \(q\)-ary code can be used to construct an EAQECC.
Error-corrected sensing code — Metrologically optimal codes can be thought of as being entanglement-assisted because they require error-free ancillas for optimal local parameter estimation, and the estimation procedure uses an entangling gate.

[1]: T. A. Brun, I. Devetak, and M.-H. Hsieh, “Catalytic Quantum Error Correction”, IEEE Transactions on Information Theory 60, 3073 (2014) arXiv:quant-ph/0608027 DOI
[2]: T. Brun, I. Devetak, and M.-H. Hsieh, “Correcting Quantum Errors with Entanglement”, Science 314, 436 (2006) arXiv:quant-ph/0610092 DOI
[3]: T. A. Brun and M.-H. Hsieh, “Entanglement-assisted quantum error-correcting codes”, Quantum Error Correction 181 (2013) DOI
[4]: M. Grassl, F. Huber, and A. Winter, “Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 68, 3942 (2022) arXiv:2010.07902 DOI

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“Entanglement-assisted (EA) QECC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/eaqecc