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.
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.
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. Cousin of: Homological bosonic code.
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.
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.
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. 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