Classical-quantum (c-q) code 

Description

Code designed specifically for transmission of classical information through non-classical channels, e.g., quantum channels, hybrid quantum-classical channels, or channels with classical inputs and quantum outputs. Such codes include maps from a classical alphabet into a quantum Hilbert space.

Rate

The Holevo channel capacity, \begin{align} C=\lim_{n\to\infty}\frac{1}{n}\chi\left({\cal N}^{\otimes n}\right)~, \tag*{(1)}\end{align} where \(\chi\) is the Holevo information, is the highest rate of classical information transmission through a quantum channel with arbitrarily small error rate [1–3].

This capacity is equal to the single-letter Holevo information of a single copy of the channel, \(\chi(\cal{N})\), for all known deterministically constructed channels. However, it is known to be superadditive, i.e., not equal to the single-copy case, for particular random channels [4].

Corrections to the Holevo capacity and tradeoff between decoding error, code rate and code length are determined in quantum generalizations of small [5], moderate [6,7], and large [8] deviation analysis. Bounds also exist on the one-shot capacity, i.e., the achievability of classical codes given only one use of the quantum channel [9–14]; see [14; Table 2] for a summary.

Parent

Children

References

[1]
A. S. Holevo, “Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel”, Probl. Peredachi Inf., 9:3 (1973), 3–11; Problems Inform. Transmission, 9:3 (1973), 177–183
[2]
B. Schumacher and M. D. Westmoreland, “Sending classical information via noisy quantum channels”, Physical Review A 56, 131 (1997) DOI
[3]
A. S. Holevo, “The capacity of the quantum channel with general signal states”, IEEE Transactions on Information Theory 44, 269 (1998) DOI
[4]
M. B. Hastings, “Superadditivity of communication capacity using entangled inputs”, Nature Physics 5, 255 (2009) arXiv:0809.3972 DOI
[5]
M. Tomamichel and V. Y. F. Tan, “Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels”, Communications in Mathematical Physics 338, 103 (2015) arXiv:1308.6503 DOI
[6]
C. T. Chubb, V. Y. F. Tan, and M. Tomamichel, “Moderate Deviation Analysis for Classical Communication over Quantum Channels”, Communications in Mathematical Physics 355, 1283 (2017) arXiv:1701.03114 DOI
[7]
X. Wang, K. Fang, and M. Tomamichel, “On Converse Bounds for Classical Communication Over Quantum Channels”, IEEE Transactions on Information Theory 65, 4609 (2019) arXiv:1709.05258 DOI
[8]
M. Mosonyi and T. Ogawa, “Strong Converse Exponent for Classical-Quantum Channel Coding”, Communications in Mathematical Physics 355, 373 (2017) arXiv:1409.3562 DOI
[9]
M. V. Burnashev and A. S. Holevo, “On Reliability Function of Quantum Communication Channel”, (1998) arXiv:quant-ph/9703013
[10]
M. Hayashi and H. Nagaoka, “A general formula for the classical capacity of a general quantum channel”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0206186 DOI
[11]
M. Hayashi, “Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding”, Physical Review A 76, (2007) arXiv:quant-ph/0611013 DOI
[12]
L. Wang and R. Renner, “One-Shot Classical-Quantum Capacity and Hypothesis Testing”, Physical Review Letters 108, (2012) arXiv:1007.5456 DOI
[13]
S. Beigi and A. Gohari, “Quantum Achievability Proof via Collision Relative Entropy”, IEEE Transactions on Information Theory 60, 7980 (2014) arXiv:1312.3822 DOI
[14]
H.-C. Cheng, “A Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels”, (2022) arXiv:2208.02132
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Zoo Code ID: classical_into_quantum

Cite as:
“Classical-quantum (c-q) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/classical_into_quantum
BibTeX:
@incollection{eczoo_classical_into_quantum,
  title={Classical-quantum (c-q) code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/classical_into_quantum}
}
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Cite as:

“Classical-quantum (c-q) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/classical_into_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical_into_quantum/classical_into_quantum.yml.