Entanglement-assisted (EA) hybrid quantum code

Entanglement-assisted (EA) hybrid quantum code[1–3]

Description

Code that encodes quantum and classical information and requires pre-shared entanglement for transmission.

EQ hybrid block quantum codes on \(n\) Galois qubits of dimensional \(q\) are denoted by \(((n,k:c,d;e))_q\), where \(k\) (\(c\)) is the number of encoded qubits (classical bits), where \(d\) is the distance, and where \(e\) is the required number of pre-shared ebits. Similarly, block codes on \(n\) modular qudits are denoted by \(((n,k:c,d;e))_{\mathbb{Z}_q}\).

In alternative conventions (not used here), EA hybrid codes are called entanglement-assisted classical-quantum (EACQ) codes. Here, we use the term classical-quantum for codes for transmitting classical information over quantum channels.

Protection

The EQ hybrid Singleton bound represents a triple trade-off region in the combined classical-bit, qubit, and e-bit space [4].

Rate

Tradeoff between classical communication, quantum communication, and entanglement distribution has been examined [5–7]; see also Ref. [8].

Child

Entanglement-assisted (EA) QECC

Cousins

Hybrid QECC — EQ hybrid codes utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to hybrid QECCs when said subsystems are interpreted as noiseless physical subsystems.
Entanglement-assisted (EA) QECC — An EA hybrid QECC storing no classical information reduces to an EA QECC. Conversely, any EA QECC can be converted into an EA hybrid QECC by using a portion of its logical subspace to store only classical information.

References

[1]: I. Kremsky, M.-H. Hsieh, and T. A. Brun, “Classical enhancement of quantum-error-correcting codes”, Physical Review A 78, (2008) arXiv:0802.2414 DOI
[2]: M.-H. Hsieh, “Entanglement-assisted Coding Theory”, (2008) arXiv:0807.2080
[3]: A. Nemec and A. Klappenecker, “Infinite Families of Quantum-Classical Hybrid Codes”, (2020) arXiv:1911.12260
[4]: M. Mamindlapally and A. Winter, “Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes”, (2023) arXiv:2202.02184
[5]: M.-H. Hsieh and M. M. Wilde, “Entanglement-Assisted Communication of Classical and Quantum Information”, IEEE Transactions on Information Theory 56, 4682 (2010) arXiv:0811.4227 DOI
[6]: Min-Hsiu Hsieh and M. M. Wilde, “Trading classical communication, quantum communication, and entanglement in quantum Shannon theory”, IEEE Transactions on Information Theory 56, 4705 (2010) arXiv:0901.3038 DOI
[7]: M.-H. Hsieh and M. M. Wilde, “Public and private communication with a quantum channel and a secret key”, Physical Review A 80, (2009) arXiv:0903.3920 DOI
[8]: J. Yard, P. Hayden, and I. Devetak, “Capacity theorems for quantum multiple-access channels: classical-quantum and quantum-quantum capacity regions”, IEEE Transactions on Information Theory 54, 3091 (2008) arXiv:quant-ph/0501045 DOI

Page edit log

Victor V. Albert (2022-05-12) — most recent

Cite as:

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

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