Entanglement-assisted (EA) hybrid QECC

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

Description

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

EA 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

If an EA hybrid code is viewed as transmitting \(C\) cbits, \(Q\) qubits, and consuming \(E\) ebits, then the EA hybrid Singleton bound is the set of triples \((C,Q,E)\) for which there exists a parameter \(t\in[0,\log q]\) such that \begin{align} C + 2Q & \leq (n-d+1)(\log q + t)~,\tag*{(1)}\\ Q - E & \leq (n-2d+2)t~,\tag*{(2)}\\ C + Q - E & \leq (n-d+1)\log q - (d-1)t~, \tag*{(3)}\end{align} for \(q\)-ary physical systems [4; Thm. 8].

Rate

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

Notes

Examples from the original paper include a \([[9,1:3,3;0]]\) code obtained from the Shor code, a \([[8,1:3,3;1]]\) code obtained from an \([[8,1,3;1]]\) EAQECC, and a \([[63,21:12,7;6]]\) code obtained from the \([[63,21,9;6]]\) EAQECC built from a classical \([63,39,9]\) BCH code [1].Inside the EAOAQEC stabilizer framework, hybrid stabilizer codes are a proper subclass of the broader EA hybrid subspace codes because the EACQ transversal operators obey additional constraints not required in general [9].

Cousins

Hybrid QECC— EA 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.
EAOA qubit stabilizer code— The original EACQ formalism describes a proper subclass of EA hybrid subspace codes inside the EAOAQEC stabilizer framework; EACQ representability imposes extra constraints on the transversal operators beyond belonging to distinct normalizer cosets [9].

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Entanglement-assisted operator-algebra QECC (EAOA QECC)

Parents

Entanglement-assisted operator-algebra QECC (EAOA QECC)

An EAOA QECC that has no gauge structure (e.g., gauge qubits), that has a block structure that corresponds to a classical code, and that utilizes pre-shared entanglement is an EA hybrid QECC.

Entanglement-assisted (EA) hybrid QECC

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”, IEEE Transactions on Information Theory 69, 5857 (2023) arXiv:2202.02184 DOI
[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
[9]: P. J. Nadkarni, S. Adonsou, G. Dauphinais, D. W. Kribs, and M. Vasmer, “Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction”, (2024) arXiv:2411.14389

Page edit log

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

Cite as:

“Entanglement-assisted (EA) hybrid QECC”, 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.