# PSK c-q code[1]

## Description

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].

## Protection

The error probability for \(q=4\) under an optimal quantum detector is worked out in [3; Sec. IV.3]; see also [2,4,5].

## Decoding

## Realizations

Unambiguous state discrimination using displacement-based receiver for 4-PSK [13].Multi-stage quantum receivers [1,14–16].Bayesian inference [17].Time resolving quantum receiver opertaing in the telecom C band [18].Displacements and photon detection [19].Adaptive decoder using linear-optical elements and photon detection [20].

## Parent

- Coherent FSK (CFSK) c-q code — The CFSK c-q code reduces to the \(q\)-ary PSK c-q code when \(\Delta\omega = 0\) and \(\Delta\theta = 2\pi/q\).

## Child

## Cousins

- Phase-shift keying (PSK) code — PSK (PSK c-q) codes are used to transmit classical information using single-mode coherent states distributed on a circle over classical (quantum) channels.
- Cat code — PSK c-q (cat) codes are used to transmit classical (quantum) information using (superpositions of) single-mode coherent states distributed on a circle over quantum channels.
- Polygon code — The PSK coherent-state constellation forms the vertices of a \(q\)-gon.

## References

- [1]
- 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
- [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
- [3]
- Carl W. Helstrom. Quantum Detection and Estimation Theory. Elsevier, 1976.
- [4]
- K. Kato et al., “Quantum detection and mutual information for QAM and PSK signals”, IEEE Transactions on Communications 47, 248 (1999) DOI
- [5]
- G. Cariolaro, R. Corvaja, and G. Pierobon, “Gaussian states and geometrically uniform symmetry”, Physical Review A 90, (2014) arXiv:1410.5282 DOI
- [6]
- M. Takeoka et al., “Implementation of projective measurements with linear optics and continuous photon counting”, Physical Review A 71, (2005) arXiv:quant-ph/0410133 DOI
- [7]
- F. E. Becerra et al., “M-ary-state phase-shift-keying discrimination below the homodyne limit”, Physical Review A 84, (2011) DOI
- [8]
- C. Wittmann, U. L. Andersen, and G. Leuchs, “Discrimination of optical coherent states using a photon number resolving detector”, Journal of Modern Optics 57, 213 (2010) arXiv:0905.2496 DOI
- [9]
- S. Izumi et al., “Displacement receiver for phase-shift-keyed coherent states”, Physical Review A 86, (2012) arXiv:1208.1815 DOI
- [10]
- S. Izumi et al., “Quantum receivers with squeezing and photon-number-resolving detectors forM-ary coherent state discrimination”, Physical Review A 87, (2013) arXiv:1302.2691 DOI
- [11]
- K. Li, Y. Zuo, and B. Zhu, “Suppressing the Errors Due to Mode Mismatch for \(M\)-Ary PSK Quantum Receivers Using Photon-Number-Resolving Detector”, IEEE Photonics Technology Letters 25, 2182 (2013) arXiv:1304.7316 DOI
- [12]
- I. A. Burenkov, O. V. Tikhonova, and S. V. Polyakov, “Quantum receiver for large alphabet communication”, Optica 5, 227 (2018) arXiv:1802.08287 DOI
- [13]
- F. E. Becerra, J. Fan, and A. Migdall, “Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states”, Nature Communications 4, (2013) DOI
- [14]
- F. E. Becerra et al., “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination”, Nature Photonics 7, 147 (2013) DOI
- [15]
- S. Izumi et al., “Experimental Demonstration of a Quantum Receiver Beating the Standard Quantum Limit at Telecom Wavelength”, Physical Review Applied 13, (2020) arXiv:2001.05902 DOI
- [16]
- A. R. Ferdinand, M. T. DiMario, and F. E. Becerra, “Multi-state discrimination below the quantum noise limit at the single-photon level”, npj Quantum Information 3, (2017) arXiv:1711.00074 DOI
- [17]
- I. A. Burenkov et al., “Experimental demonstration of time resolving quantum receiver for bandwidth and power efficient communications”, Conference on Lasers and Electro-Optics (2020) DOI
- [18]
- M. V. Jabir et al., “Versatile quantum-enabled telecom receiver”, AVS Quantum Science 5, 015001 (2023) DOI
- [19]
- S. Izumi, J. S. Neergaard-Nielsen, and U. L. Andersen, “Adaptive Generalized Measurement for Unambiguous State Discrimination of Quaternary Phase-Shift-Keying Coherent States”, PRX Quantum 2, (2021) arXiv:2009.02558 DOI
- [20]
- 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

## Page edit log

- Victor V. Albert (2022-12-04) — most recent

## Cite as:

“PSK c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_psk