Description
Bosonic c-q code whose codewords form a constellation of coherent states. Encodes real numbers into coherent states for transmission over a quantum channel and decoding with a quantum-enhanced receiver.
The codes consists of \(K\) coherent states on \(n\) modes, where the \(j\)th state, or codeword, is uniquely defined through the amplitude vector \(\boldsymbol{\alpha}^j=(\alpha_1^j, \alpha_2^j, \cdots, \alpha_n^j)\). The codebook, \begin{align} C=\left(\begin{array}{c} \boldsymbol{\alpha}^{1}\\ \vdots\\ \boldsymbol{\alpha}^{K} \end{array}\right)=\left(\begin{array}{cccc} \alpha_{1}^{1} & \alpha_{2}^{1} & \dots & \alpha_{n}^{1}\\ \vdots & \vdots & \ddots & \vdots\\ \alpha_{1}^{K} & \alpha_{2}^{K} & \dots & \alpha_{n}^{K} \end{array}\right)~, \tag*{(1)}\end{align} collects each codeword into the matrix \(C\) that characterizes the system of states to discriminate.
From the properties of \(C\), we can assess whether it is possible to optimally discriminate the codebook unambiguously. Optimal unambiguous state discrimination requires each vector to be linearly independent, which corresponds to a full-rank codebook. This is possible only if \(K \leq n\).
Rate
Random Gaussian-distributed coherent-state c-q codes achieve the capacity of the lossy bosonic channel [1].Decoding
Optimal receiver performance in ambiguous state discrimination is determined using the Yuen-Kennedy-Lax (YKL) conditions [2]. See review [3] for details on receivers used for coherent-state c-q codes.Joint-detection receiver that can attain channel capacity [4].Various near-optimal receiver designs that can handle arbitrary constellations of coherent states with possible degeneracies [5].The square-root measurement (a.k.a. pretty good measurement) [6–8] is optimal for geometrically uniform [9–12], direct sums of geometrically uniform [13], and compound geometrically uniform [14] constellations.Notes
See book [18].Cousins
- Coherent-state constellation code— Coherent-state c-q codes encode classical alphabets into constellations of coherent states, while coherent-state constellation codes encode quantum information into superpositions of coherent states.
- Modulation scheme— Coherent-state c-q codes are modulation schemes to transmission of classical information over quantum analog channels.
- Two-component cat code— Two-component cat codes can be thought of as coherent-state c-q codes because they protect against only one type of noise and thus only reliably store classical information.
Primary Hierarchy
References
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- I. A. Burenkov, M. V. Jabir, and S. V. Polyakov, “Practical quantum-enhanced receivers for classical communication”, AVS Quantum Science 3, (2021) DOI
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- M. Rosati, “Performance of Coherent Frequency-Shift Keying for Classical Communication on Quantum Channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2203.09822 DOI
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- J. Zhou, S. Chessa, E. Chitambar, and F. Leditzky, “On the distinguishability of geometrically uniform quantum states”, (2025) arXiv:2501.12376
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- N. Dalla Pozza and G. Pierobon, “Optimality of square-root measurements in quantum state discrimination”, Physical Review A 91, (2015) arXiv:1504.04908 DOI
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- H. Krovi, S. Guha, Z. Dutton, and M. P. da Silva, “Optimal measurements for symmetric quantum states with applications to optical communication”, Physical Review A 92, (2015) arXiv:1507.04737 DOI
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- F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States”, Physical Review Letters 88, (2002) arXiv:quant-ph/0109084 DOI
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- F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states”, Nature 421, 238 (2003) arXiv:quant-ph/0312016 DOI
- [18]
- J. Gazeau, Coherent States in Quantum Physics (Wiley, 2009) DOI
Page edit log
- Victor V. Albert (2023-05-31) — most recent
Cite as:
“Coherent-state c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/coherent_state_c-q