## Description

## 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.

Ideal decoding error scales is suppressed exponentially with the number of subsystems \(n\) (for c-q block codes), and the exponent has been studied in Ref. [15].

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## 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, “Simple and Tighter Derivation of Achievability for Classical Communication Over Quantum Channels”, PRX Quantum 4, (2023) arXiv:2208.02132 DOI
- [15]
- S. Beigi and M. Tomamichel, “Lower Bounds on Error Exponents via a New Quantum Decoder”, (2023) arXiv:2310.09014

## Page edit log

- Victor V. Albert (2022-11-08) — most recent

## 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