Coherent-state c-q code 


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 \(K \leq n\).


Random Gaussian-distributed coherent-state c-q codes achieve the capacity of the lossy bosonic channel [1].


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 is optimal for geometrically uniform constellations [68].


Continuous-variable quantum key distribution (CV-QKD) [911].


See book [12].




  • 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.


V. Giovannetti et al., “Classical Capacity of the Lossy Bosonic Channel: The Exact Solution”, Physical Review Letters 92, (2004) arXiv:quant-ph/0308012 DOI
H. Yuen, R. Kennedy, and M. Lax, “Optimum testing of multiple hypotheses in quantum detection theory”, IEEE Transactions on Information Theory 21, 125 (1975) DOI
I. A. Burenkov, M. V. Jabir, and S. V. Polyakov, “Practical quantum-enhanced receivers for classical communication”, AVS Quantum Science 3, (2021) DOI
S. Guha, “Structured Optical Receivers to Attain Superadditive Capacity and the Holevo Limit”, Physical Review Letters 106, (2011) arXiv:1101.1550 DOI
J. S. Sidhu et al., “Linear optics and photodetection achieve near-optimal unambiguous coherent state discrimination”, Quantum 7, 1025 (2023) arXiv:2109.00008 DOI
Y. C. Eldar and G. D. Forney Jr, “On Quantum Detection and the Square-Root Measurement”, (2000) arXiv:quant-ph/0005132
Y. C. Eldar, A. Megretski, and G. C. Verghese, “Optimal Detection of Symmetric Mixed Quantum States”, (2002) arXiv:quant-ph/0211111
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
P. van Loock and S. L. Braunstein, “Unconditional teleportation of continuous-variable entanglement”, Physical Review A 61, (1999) arXiv:quant-ph/9907073 DOI
F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States”, Physical Review Letters 88, (2002) arXiv:quant-ph/0109084 DOI
F. Grosshans et al., “Quantum key distribution using gaussian-modulated coherent states”, Nature 421, 238 (2003) arXiv:quant-ph/0312016 DOI
J. Gazeau, Coherent States in Quantum Physics (Wiley, 2009) DOI
Page edit log

Your contribution is welcome!

on (edit & pull request)— see instructions

edit on this site

Zoo Code ID: coherent_state_c-q

Cite as:
“Coherent-state c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_coherent_state_c-q, title={Coherent-state c-q code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Coherent-state c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.