Here is a list of classical-quantum (c-q) codes whose codewords are coherent states.

Code | Description |
---|---|

BPSK c-q code | Coherent-state c-q binary code encoding into two coherent states \(|\pm\alpha\rangle\) for complex \(\alpha\). A shifted version, with codewords \(\{|0\rangle,|\alpha\rangle\}\), is called binary amplitude modulation (BAM), The three-state subcode \(\{|\alpha,\alpha\rangle,|-\alpha,\alpha\rangle,|\alpha,-\alpha\rangle\}\) of two-mode BPSK is called the single-degeneracy code [1]. |

Coherent FSK (CFSK) c-q code | Coherent-state c-q code encoding into coherent states that are frequency-shifted with certain initial relative phase. |

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. |

Hadamard BPSK c-q code | Multimode coherent-state c-q code that is a concatenation of a Hadamard code with a BPSK c-q code. Its codewords are \(n\)-mode coherent states whose components \(\pm\alpha\) are arranged according to rows of a Hadamard matrix. |

Niset-Andersen-Cerf code | Coherent-state c-q code encoding two-mode coherent states \(\{|\alpha\rangle, |\beta\rangle\}\) into four modes such that the complex values \((\alpha,\beta)\) are recoverable after a single-mode erasure. There are two variations of the storage procedure: a deterministic protocol that offers recovery against a single mode erasure, and a probabalistic that can protect against multiple errors with post selection. This code is effectively protecting classical information stored in \((\alpha,\beta)\) using quantum operations. |

On-off keyed (OOK) c-q code | Coherent-state c-q binary code whose encoding is either in the vacuum \(|0\rangle\) or in a nonzero coherent state \(|\alpha\rangle\). |

PPM c-q code | A \(q\)-PPM c-q code is a coherent-state c-q code whose \(j\)th codeword corresponds to a tensor-product state of zero-amplitude coherent states at all modes except mode \(j\). For example, a 3-PPM encoding corresponds to the three-mode states \(|\alpha\rangle|0\rangle|0\rangle\), \(|0\rangle|\alpha\rangle|0\rangle\), and \(|0\rangle|0\rangle|\alpha\rangle\) for some complex \(\alpha\). The dual of a PPM code is obtained by the exchange \(0\leftrightarrow\alpha\). |

PSK c-q code | Coherent-state c-q \(q\)-ary code whose \(j\)th codeword corresponds to a coherent state whose phase is the \(j\)th multiple of \(2\pi/q\). These states are also called geometrically uniform states (GUS) [2]. |

## References

- [1]
- S. Guha, “Structured Optical Receivers to Attain Superadditive Capacity and the Holevo Limit”, Physical Review Letters 106, (2011) arXiv:1101.1550 DOI
- [2]
- Y. C. Eldar and G. D. Forney, “On quantum detection and the square-root measurement”, IEEE Transactions on Information Theory 47, 858 (2001) DOI