Welcome to the Bosonic/analog c-q Kingdom.

Bosonic c-q code
Bosonic code designed for transmission of classical information through non-classical channels. Typically, such codes encode real numbers into coherent states for transmission over a quantum channel and decoding with a quantum-enhanced receiver.
Parents:
Classical-quantum (c-q) code.
Parent of:
Coherent FSK (CFSK) c-q code, Niset-Andersen-Cerf code, On-off keyed (OOK) c-q code, PPM c-q code.
Cousins:
Bosonic code, Coherent-state constellation code, Sphere packing.

Coherent FSK (CFSK) c-q code[1][2]
Bosonic c-q code encoding into coherent states that are frequency-shifted with certain initial relative phase. Codewords are coherent states \(|\alpha_m\rangle\), where \begin{align} \alpha_m = \alpha e^{i(\omega_0+[m-1]\Delta\omega)t+i(m-1)\Delta\theta} \end{align} for common frequency \(\omega_0\), frequency shift \(\Delta\omega < 2\pi/T\), total time \(T\), and phase shift \(\Delta\theta\).
Parents:
Bosonic c-q code.
Parent of:
PSK c-q code.
Cousins:
Frequency-shift keyring (FSK) code.

Niset-Andersen-Cerf code[3]
Bosonic 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.
Protection: The deterministic protocol protects against a single erasure error on a known mode. This recovers one state perfectly and the other state with fidelity \(F = \frac{1}{1 + e^{-2 r}}\) for an initial EPR pair squeezed with variance \(e^{-2r}\). The probabalistic protocol utilizes post-selection to protect against multiple erasures with state-dependent fidelity.
Parents:
Bosonic c-q code.
Cousins:
Quadrature-amplitude modulation (QAM) code.
Cousin of:
Homological bosonic code.

On-off keyed (OOK) c-q code[4]
Bosonic c-q binary code whose encoding is either in the vacuum \(|0\rangle\) or in a nonzero coherent state \(|\alpha\rangle\).
Parents:
Bosonic c-q code.
Cousins:
BPSK c-q code.

PPM c-q code[5]
A \(q\)-PPM c-q code is a bosonic 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 \(\alpha\).
Parents:
Bosonic c-q code.
Cousins:
Pulse-position modulation (PPM) code.

PSK c-q code[6]
Bosonic c-q \(q\)-ary code whose \(j\)th codeword corresponds to a coherent state whose phase is the \(j\)th multiple of \(2\pi/q\).
Parents:
Coherent FSK (CFSK) c-q code.
Parent of:
BPSK c-q code.
Cousins:
Phase-shift keyring (PSK) code, Cat code.

BPSK c-q code
Bosonic c-q binary code encoding one bit of information into coherent states \(|\pm\alpha\rangle\) for complex \(\alpha\).
Parents:
PSK c-q code.
Cousins:
Binary PSK (BPSK) code, Two-component cat code.
Cousin of:
On-off keyed (OOK) c-q code.

## References

- [1]
- I. A. Burenkov, O. V. Tikhonova, and S. V. Polyakov, “Quantum receiver for large alphabet communication”, Optica 5, 227 (2018). DOI; 1802.08287
- [2]
- I. A. Burenkov et al., “Time-Resolving Quantum Measurement Enables Energy-Efficient, Large-Alphabet Communication”, PRX Quantum 1, (2020). DOI
- [3]
- J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally Feasible Quantum Erasure-Correcting Code for Continuous Variables”, Physical Review Letters 101, (2008). DOI; 0710.4858
- [4]
- R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement”, Nature 446, 774 (2007). DOI
- [5]
- J. Chen et al., “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver”, Nature Photonics 6, 374 (2012). DOI; 1111.4017
- [6]
- F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications”, Nature Photonics 9, 48 (2014). DOI