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Classical-quantum (c-q) code

Root code for the Classical-quantum Domain
Codes for classical communication over quantum channels

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 exist on the one-shot capacity, i.e., the achievability of classical codes given only one use of the quantum channel. The ideal decoding error is suppressed exponentially with the number of subsystems \(n\) (for c-q block codes), and the achievable exponent has been studied in Refs. [9–17]; see [16; Table 2] for a summary. Achievable error exponents for communication are related to those for privacy amplification [18]. In the high-rate case, a lower [19] and upper [20] bound on the error exponent for general channels matches a conjecture by Holevo [10].

Unambiguous state discrimination (USD) can be used to achieve Holevo capacity on a general pure-state c-q channel [21].

Decoding

Unambiguous state discrimination (USD) [21].

Cousin

  • Hybrid QECC— A hybrid QECC storing no quantum information reduces to a c-q code.

Member of code lists

Primary Hierarchy

Classical-quantum Domain
Parents
Operator-algebra QECC (OAQECC)Quantum
An OAQECC that retains its block structure for storing classical information but stores no quantum information and has no gauge degrees of freedom (e.g., gauge qubits) is a c-q code.
Classical-quantum (c-q) code
Children
Error-correcting code (ECC)Modulation scheme Lattice-shell Constant-weight FP Distributed-storage LRC LCC LDC LTC Quantum-inspired classical block Small-distance block Skew-cyclic Constacyclic \(t\)-design Spherical design Universally optimal Sharp configuration Convolutional Generalized concatenated code (GCC) Concatenated Array MDS array STC MRD MSRD Balanced OA AG Editing Perfect Combinatorial design Nearly perfect Perfect binary Lattice-based Linear \(q\)-ary Tanner \(q\)-ary LDPC MDS Evaluation GRS GRM Projective geometry Divisible Self-dual additive Self-dual linear Linear code over \(\mathbb{Z}_q\) Gray Cyclic QR Polytope Polyhedron
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 [22,23].
Concatenated c-q code
Bosonic c-q codeCoherent-state c-q
Qubit 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]
A. S. Holevo, “Reliability function of general classical-quantum channel”, IEEE Transactions on Information Theory 46, 2256 (2000) arXiv:quant-ph/9907087 DOI
[11]
M. Hayashi and H. Nagaoka, “A general formula for the classical capacity of a general quantum channel”, Proceedings IEEE International Symposium on Information Theory, 71 arXiv:quant-ph/0206186 DOI
[12]
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
[13]
M. Hayashi, “Universal Coding for Classical-Quantum Channel”, Communications in Mathematical Physics 289, 1087 (2009) arXiv:0805.4092 DOI
[14]
L. Wang and R. Renner, “One-Shot Classical-Quantum Capacity and Hypothesis Testing”, Physical Review Letters 108, (2012) arXiv:1007.5456 DOI
[15]
S. Beigi and A. Gohari, “Quantum Achievability Proof via Collision Relative Entropy”, IEEE Transactions on Information Theory 60, 7980 (2014) arXiv:1312.3822 DOI
[16]
H.-C. Cheng, “Simple and Tighter Derivation of Achievability for Classical Communication Over Quantum Channels”, PRX Quantum 4, (2023) arXiv:2208.02132 DOI
[17]
S. Beigi and M. Tomamichel, “Lower Bounds on Error Exponents via a New Quantum Decoder”, (2023) arXiv:2310.09014
[18]
J. M. Renes, “Achievable error exponents of data compression with quantum side information and communication over symmetric classical-quantum channels”, 2023 IEEE Information Theory Workshop (ITW) 170 (2023) arXiv:2207.08899 DOI
[19]
M. Dalai, “Lower Bounds on the Probability of Error for Classical and Classical-Quantum Channels”, IEEE Transactions on Information Theory 59, 8027 (2013) arXiv:1201.5411 DOI
[20]
K. Li and D. Yang, “Reliability Function of Classical-Quantum Channels”, Physical Review Letters 134, (2025) arXiv:2407.12403 DOI
[21]
M. Takeoka, H. Krovi, and S. Guha, “Achieving the Holevo capacity of a pure state classical-quantum channel via unambiguous state discrimination”, 2013 IEEE International Symposium on Information Theory 166 (2013) DOI
[22]
M. K. Patra and S. L. Braunstein, “An algebraic framework for information theory: Classical Information”, (2009) arXiv:0910.1536
[23]
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.