PSK c-q code[1] 


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].


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


Multi-stage quantum receivers [611].Bayesian inference [12].


Unambiguous state discrimination using displacement-based receiver for 4-PSK [13].Multi-stage quantum receivers [1,1416].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].


  • 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 \(\Delta\theta = 2\pi/q\).



  • 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.


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
Y. C. Eldar and G. D. Forney, “On quantum detection and the square-root measurement”, IEEE Transactions on Information Theory 47, 858 (2001) DOI
Carl W. Helstrom. Quantum Detection and Estimation Theory. Elsevier, 1976.
K. Kato et al., “Quantum detection and mutual information for QAM and PSK signals”, IEEE Transactions on Communications 47, 248 (1999) DOI
G. Cariolaro, R. Corvaja, and G. Pierobon, “Gaussian states and geometrically uniform symmetry”, Physical Review A 90, (2014) arXiv:1410.5282 DOI
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
F. E. Becerra et al., “M-ary-state phase-shift-keying discrimination below the homodyne limit”, Physical Review A 84, (2011) DOI
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
S. Izumi et al., “Displacement receiver for phase-shift-keyed coherent states”, Physical Review A 86, (2012) arXiv:1208.1815 DOI
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
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
I. A. Burenkov, O. V. Tikhonova, and S. V. Polyakov, “Quantum receiver for large alphabet communication”, Optica 5, 227 (2018) arXiv:1802.08287 DOI
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
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
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
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
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
M. V. Jabir et al., “Versatile quantum-enabled telecom receiver”, AVS Quantum Science 5, 015001 (2023) DOI
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
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
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Zoo Code ID: quantum_psk

Cite as:
“PSK c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={PSK c-q code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“PSK c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.