## Description

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].

## Rate

## Decoding

Linear-optical quantum receiver [3].Kennedy receiver [4,5].Photon-number resolving detector [6].Non-Gaussian near-optimal receiver [5].Multi-stage quantum receiver [7].

## Realizations

Linear-optical quantum receiver [3].Homodyne receiver [8].Kennedy receiver [8,9].Photon-number resolving detector [6].Communication over dephasing [10], time-varying phase-noise [11], and thermal-noise [12] channels.Adaptive decoder using displacements and photon detection [13].Superconducting circuits: a bit-flip time of 1s has been achieved for the two-legged cat code in the classical-bit regime [14].BPQM detector on a BPSK-modulated tree code [15].

## Parents

## Cousins

- Binary PSK (BPSK) code — BPSK (BPSK c-q) codes are used to transmit classical information using antipodal coherent states over classical (quantum) channels.
- Two-component cat code — BPSK c-q (two-component cat) codes are used to transmit classical (quantum) information using (superpositions of) antipodal coherent states over quantum channels.
- On-off keyed (OOK) c-q code — OOK c-q codewords are related to BPSK c-q codewords by a displacement in phase space.

## 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]
- J. R. Buck, S. J. van Enk, and C. A. Fuchs, “Experimental proposal for achieving superadditive communication capacities with a binary quantum alphabet”, Physical Review A 61, (2000) DOI
- [3]
- K. Tsujino et al., “Quantum Receiver beyond the Standard Quantum Limit of Coherent Optical Communication”, Physical Review Letters 106, (2011) arXiv:1103.5592 DOI
- [4]
- Kennedy, Robert S. "A near-optimum receiver for the binary coherent state quantum channel." Quarterly Progress Report 108 (1973): 219-225.
- [5]
- M. Takeoka and M. Sasaki, “Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers”, Physical Review A 78, (2008) arXiv:0706.1038 DOI
- [6]
- M. T. DiMario and F. E. Becerra, “Robust Measurement for the Discrimination of Binary Coherent States”, Physical Review Letters 121, (2018) arXiv:1807.05199 DOI
- [7]
- D. Sych and G. Leuchs, “Practical Receiver for Optimal Discrimination of Binary Coherent Signals”, Physical Review Letters 117, (2016) arXiv:1404.5033 DOI
- [8]
- C. Wittmann et al., “Demonstration of Near-Optimal Discrimination of Optical Coherent States”, Physical Review Letters 101, (2008) arXiv:0809.4953 DOI
- [9]
- M. L. Shcherbatenko et al., “Sub-shot-noise-limited fiber-optic quantum receiver”, Physical Review A 101, (2020) arXiv:1911.08932 DOI
- [10]
- M. T. DiMario et al., “Optimized communication strategies with binary coherent states over phase noise channels”, npj Quantum Information 5, (2019) arXiv:1907.12515 DOI
- [11]
- M. T. DiMario and F. E. Becerra, “Phase tracking for sub-shot-noise-limited receivers”, Physical Review Research 2, (2020) DOI
- [12]
- R. Yuan et al., “Optimally Displaced Threshold Detection for Discriminating Binary Coherent States Using Imperfect Devices”, (2020) arXiv:2007.11109
- [13]
- M. T. DiMario and F. E. Becerra, “Demonstration of optimal non-projective measurement of binary coherent states with photon counting”, npj Quantum Information 8, (2022) arXiv:2207.12234 DOI
- [14]
- C. Berdou et al., “One Hundred Second Bit-Flip Time in a Two-Photon Dissipative Oscillator”, PRX Quantum 4, (2023) arXiv:2204.09128 DOI
- [15]
- C. Delaney et al., “Demonstration of a quantum advantage by a joint detection receiver for optical communication using quantum belief propagation on a trapped-ion device”, Physical Review A 106, (2022) arXiv:2102.13052 DOI

## Page edit log

- Victor V. Albert (2023-01-13) — most recent

## Cite as:

“BPSK c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_bpsk