Bosonic c-q code

This code defines the Bosonic/analog c-q Kingdom


Bosonic code designed for transmission of classical information through non-classical channels. 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 each state is uniquely defined through the amplitude vectors\(\boldsymbol{\alpha}^j=(\alpha_1^j, \alpha_2^j, …, \alpha_n^j)\} for \{j = {1, 2, \cdots, K}\) known as the codeword. 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 \(K \leq n\). See review [1] for details on receivers used for bosonic c-q codes.


The Holevo (also known as Gordon) capacity has been calculated for various bosonic quantum channels such as AGWN [2] (see Ref. [3] for a review).


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].




  • Bosonic code — Bosonic c-q codes are bosonic codes designed to transmit classical information.
  • Coherent-state constellation code — Typical bosonic c-q codes encode classical alphabets into constellations of coherent states.
  • Sphere packing — Bosonic c-q codes are extensions of sphere packings to transmission of classical information over quantum analog channels.
  • Two-component cat code — Two-component cat codes can be thought of as classical codes because they protect against only one type of noise.


I. A. Burenkov, M. V. Jabir, and S. V. Polyakov, “Practical quantum-enhanced receivers for classical communication”, AVS Quantum Science 3, 025301 (2021) DOI
V. Giovannetti et al., “Ultimate classical communication rates of quantum optical channels”, Nature Photonics 8, 796 (2014) arXiv:1312.6225 DOI
K. Banaszek et al., “Quantum Limits in Optical Communications”, Journal of Lightwave Technology 38, 2741 (2020) arXiv:2002.05766 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”, (2022) arXiv:2109.00008
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Zoo Code ID: bosonic_classical_into_quantum

Cite as:
“Bosonic c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_bosonic_classical_into_quantum, title={Bosonic c-q code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

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