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.

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].

Parent

  • Operator-algebra QECC (OAQECC) — OAQECC that retain its block structure for storing classical information but stores no quantum information is a c-q code.

Children

Cousins

  • Error-correcting code (ECC) — Any ECC can be embedded into a quantum Hilbert space, and thus passed through a quantum channel, by associating elements of the alphabet with basis vectors in a Hilbert space over the complex numbers. In other words, classical codewords are elements of an alphabet, while quantum codewords are functions on the alphabet. Classical codes can be unified with quantum codes using various algebraic frameworks [16,17].
  • Hybrid QECC — A hybrid QECC storing no quantum information reduces to a c-q code.

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
[16]
M. K. Patra and S. L. Braunstein, “An algebraic framework for information theory: Classical Information”, (2009) arXiv:0910.1536
[17]
G. Kuperberg and N. Weaver, “A von Neumann algebra approach to quantum metrics”, (2010) arXiv:1005.0353
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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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“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/edit/main/codes/classical_into_quantum/classical_into_quantum.yml.