Abstract

We experimentally investigate a strategy to discriminate between quaternary phase-shift keyed coherent states based on single-shot measurements that is compatible with high-bandwidth communications. We extend previous theoretical work in single-shot measurements to include critical experimental parameters affecting the performance of practical implementations. Specifically, we investigate how the visibility of the optical displacement operations required in the strategy impacts the achievable discrimination error probability, and identify the experimental requirements to outperform an ideal heterodyne measurement. Our experimental implementation is optimized based on the experimental parameters and allows for the investigation of realistic single-shot measurements for multistate discrimination.

© 2018 Optical Society of America

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References

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2017 (2)

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 Inf. 3, 43 (2017).
[Crossref]

M. Bina, A. Allevi, M. Bondani, and S. Olivares, “Homodyne-like detection for coherent state-discrimination in the presence of phase noise,” Opt. Express 25, 10685–10692 (2017).
[Crossref]

2016 (3)

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

J. M. Arrazola, P. Wallden, and E. Andersson, “Multiparty quantum signature schemes,” Quantum Inf. Comput. 16, 435–464 (2016).

J. M. Arrazola, M. Karasamanis, and N. Lütkenhaus, “Practical quantum retrieval games,” Phys. Rev. A 93, 062311 (2016).
[Crossref]

2015 (2)

F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications,” Nat. Photonics 9, 48–53 (2015).
[Crossref]

C. R. Müller and C. Marquardt, “A robust quantum receiver for phase shift keyed signals,” New J. Phys. 17, 032003 (2015).
[Crossref]

2014 (1)

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Phys. Rev. A 89, 032318 (2014).
[Crossref]

2013 (3)

S. Izumi, M. Takeoka, K. Ema, and M. Sasaki, “Quantum receivers with squeezing and photon-number-resolving detectors for M-ary coherent state discrimination,” Phys. Rev. A 87, 042328 (2013).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nat. Photonics 7, 147–152 (2013).
[Crossref]

F. E. Becerra, J. Fan, and A. Migdall, “Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states,” Nat. Commun. 4, 2028 (2013).
[Crossref]

2012 (4)

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6, 374–379 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

P. J. Winzer, “High-spectral-efficiency optical modulation formats,” J. Lightwave Technol. 30, 3824–3835 (2012).
[Crossref]

2011 (2)

2010 (2)

D. Sych and G. Leuchs, “Coherent state quantum key distribution with multi letter phase-shift keying,” New J. Phys. 12, 053019 (2010).
[Crossref]

C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Demonstration of coherent-state discrimination using a displacement-controlled photon-number-resolving detector,” Phys. Rev. Lett. 104, 100505 (2010).
[Crossref]

2009 (1)

2008 (2)

M. Takeoka and M. Sasaki, “Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers,” Phys. Rev. A 78, 022320 (2008).
[Crossref]

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[Crossref]

2007 (1)

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446, 774–777 (2007).
[Crossref]

2004 (1)

J. A. Bergou, U. Herzog, and M. Hillery, “Discrimination of quantum states,” Lect. Notes Phys. 649, 417–465 (2004).
[Crossref]

2002 (2)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
[Crossref]

1996 (1)

1993 (1)

1969 (1)

Allevi, A.

Amemiya, K.

Andersen, U. L.

C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Demonstration of coherent-state discrimination using a displacement-controlled photon-number-resolving detector,” Phys. Rev. Lett. 104, 100505 (2010).
[Crossref]

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[Crossref]

Andersson, E.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

J. M. Arrazola, P. Wallden, and E. Andersson, “Multiparty quantum signature schemes,” Quantum Inf. Comput. 16, 435–464 (2016).

Arrazola, J. M.

J. M. Arrazola, M. Karasamanis, and N. Lütkenhaus, “Practical quantum retrieval games,” Phys. Rev. A 93, 062311 (2016).
[Crossref]

J. M. Arrazola, P. Wallden, and E. Andersson, “Multiparty quantum signature schemes,” Quantum Inf. Comput. 16, 435–464 (2016).

Assalini, A.

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

Barnett, S. M.

Baumgartner, G.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nat. Photonics 7, 147–152 (2013).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[Crossref]

Becerra, F. E.

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 Inf. 3, 43 (2017).
[Crossref]

F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications,” Nat. Photonics 9, 48–53 (2015).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nat. Photonics 7, 147–152 (2013).
[Crossref]

F. E. Becerra, J. Fan, and A. Migdall, “Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states,” Nat. Commun. 4, 2028 (2013).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[Crossref]

Bennet, C. H.

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Malvern Physics Series, Bangalore, India, 1984, p. 175.

Bennett, J. M.

J. M. Bennett, (Polarizers) Handbook of Optics, M. Bass, ed., 2nd ed. (McGraw-Hill, 1995), Vol. II.

Bergou, J. A.

J. A. Bergou, U. Herzog, and M. Hillery, “Discrimination of quantum states,” Lect. Notes Phys. 649, 417–465 (2004).
[Crossref]

Betti, S.

S. Betti, G. De Marchis, and E. Lannone, Coherent Optical Communications Systems (Wiley, 2000).

Bina, M.

Bondani, M.

Bondurant, R. S.

Brassard, G.

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Malvern Physics Series, Bangalore, India, 1984, p. 175.

Cassemiro, K. N.

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[Crossref]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Chen, J.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6, 374–379 (2012).
[Crossref]

Cook, R. L.

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446, 774–777 (2007).
[Crossref]

Croal, C.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

Croke, S.

Cromer, C. L.

De Marchis, G.

S. Betti, G. De Marchis, and E. Lannone, Coherent Optical Communications Systems (Wiley, 2000).

DiMario, M. T.

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 Inf. 3, 43 (2017).
[Crossref]

Dolinar, S. J.

S. J. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” (Research Laboratory of Electronics, MIT, 1973), p. 115.

Dutton, Z.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6, 374–379 (2012).
[Crossref]

Ema, K.

S. Izumi, M. Takeoka, K. Ema, and M. Sasaki, “Quantum receivers with squeezing and photon-number-resolving detectors for M-ary coherent state discrimination,” Phys. Rev. A 87, 042328 (2013).
[Crossref]

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

Fan, J.

F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications,” Nat. Photonics 9, 48–53 (2015).
[Crossref]

F. E. Becerra, J. Fan, and A. Migdall, “Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states,” Nat. Commun. 4, 2028 (2013).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nat. Photonics 7, 147–152 (2013).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[Crossref]

Ferdinand, A. R.

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 Inf. 3, 43 (2017).
[Crossref]

Fujii, G.

Fujino, H.

Fujiwara, M.

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

Fukuda, D.

García-Patrón, R.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Gentile, T. R.

Geremia, J. M.

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446, 774–777 (2007).
[Crossref]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Goldhar, J.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nat. Photonics 7, 147–152 (2013).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[Crossref]

Grangier, P.

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
[Crossref]

Grieser, D. R.

Grosshans, F.

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
[Crossref]

Guha, S.

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Phys. Rev. A 89, 032318 (2014).
[Crossref]

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6, 374–379 (2012).
[Crossref]

Habif, J. L.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6, 374–379 (2012).
[Crossref]

Heim, B.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory, Mathematics in Science and Engineering (Academic, 1976), Vol. 123.

Herzog, U.

J. A. Bergou, U. Herzog, and M. Hillery, “Discrimination of quantum states,” Lect. Notes Phys. 649, 417–465 (2004).
[Crossref]

Hillery, M.

J. A. Bergou, U. Herzog, and M. Hillery, “Discrimination of quantum states,” Lect. Notes Phys. 649, 417–465 (2004).
[Crossref]

Houston, J. M.

Inoue, S.

Ishii, H.

Itatani, T.

Izumi, S.

S. Izumi, M. Takeoka, K. Ema, and M. Sasaki, “Quantum receivers with squeezing and photon-number-resolving detectors for M-ary coherent state discrimination,” Phys. Rev. A 87, 042328 (2013).
[Crossref]

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

Karasamanis, M.

J. M. Arrazola, M. Karasamanis, and N. Lütkenhaus, “Practical quantum retrieval games,” Phys. Rev. A 93, 062311 (2016).
[Crossref]

Kennedy, R. S.

R. S. Kennedy, “A near-optimum receiver for the binary coherent state quantum channel,” (Research Laboratory of Electronics, MIT, 1972), unpublished.

Khan, I.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

Korolkova, N.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

Kosloski, J. T.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nat. Photonics 7, 147–152 (2013).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[Crossref]

J. T. Kosloski, “A Kennedy receiver for optical quadrature phase shift keying,” Ph.D. thesis (Johns Hopkins University, 2012), 257 pp.

Lannone, E.

S. Betti, G. De Marchis, and E. Lannone, Coherent Optical Communications Systems (Wiley, 2000).

Lazarus, R.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6, 374–379 (2012).
[Crossref]

Leuchs, G.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Demonstration of coherent-state discrimination using a displacement-controlled photon-number-resolving detector,” Phys. Rev. Lett. 104, 100505 (2010).
[Crossref]

D. Sych and G. Leuchs, “Coherent state quantum key distribution with multi letter phase-shift keying,” New J. Phys. 12, 053019 (2010).
[Crossref]

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[Crossref]

Lloyd, S.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Lütkenhaus, N.

J. M. Arrazola, M. Karasamanis, and N. Lütkenhaus, “Practical quantum retrieval games,” Phys. Rev. A 93, 062311 (2016).
[Crossref]

Marquardt, C.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

C. R. Müller and C. Marquardt, “A robust quantum receiver for phase shift keyed signals,” New J. Phys. 17, 032003 (2015).
[Crossref]

Martin, P. J.

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446, 774–777 (2007).
[Crossref]

Migdall, A.

F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications,” Nat. Photonics 9, 48–53 (2015).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nat. Photonics 7, 147–152 (2013).
[Crossref]

F. E. Becerra, J. Fan, and A. Migdall, “Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states,” Nat. Commun. 4, 2028 (2013).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[Crossref]

Moeller, C. E.

Müller, C. R.

C. R. Müller and C. Marquardt, “A robust quantum receiver for phase shift keyed signals,” New J. Phys. 17, 032003 (2015).
[Crossref]

Nair, R.

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Phys. Rev. A 89, 032318 (2014).
[Crossref]

Numata, T.

Olivares, S.

Peuntinger, C.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

Pirandola, S.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Polyakov, S. V.

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[Crossref]

Pozza, N. D.

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

Ralph, T. C.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Sasaki, M.

S. Izumi, M. Takeoka, K. Ema, and M. Sasaki, “Quantum receivers with squeezing and photon-number-resolving detectors for M-ary coherent state discrimination,” Phys. Rev. A 87, 042328 (2013).
[Crossref]

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[Crossref]

M. Takeoka and M. Sasaki, “Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers,” Phys. Rev. A 78, 022320 (2008).
[Crossref]

Shapiro, J. H.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Sych, D.

C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Demonstration of coherent-state discrimination using a displacement-controlled photon-number-resolving detector,” Phys. Rev. Lett. 104, 100505 (2010).
[Crossref]

D. Sych and G. Leuchs, “Coherent state quantum key distribution with multi letter phase-shift keying,” New J. Phys. 12, 053019 (2010).
[Crossref]

Takeoka, M.

S. Izumi, M. Takeoka, K. Ema, and M. Sasaki, “Quantum receivers with squeezing and photon-number-resolving detectors for M-ary coherent state discrimination,” Phys. Rev. A 87, 042328 (2013).
[Crossref]

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Demonstration of coherent-state discrimination using a displacement-controlled photon-number-resolving detector,” Phys. Rev. Lett. 104, 100505 (2010).
[Crossref]

M. Takeoka and M. Sasaki, “Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers,” Phys. Rev. A 78, 022320 (2008).
[Crossref]

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[Crossref]

Tan, S.-H.

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Phys. Rev. A 89, 032318 (2014).
[Crossref]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Tsuchida, H.

Wallden, P.

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

J. M. Arrazola, P. Wallden, and E. Andersson, “Multiparty quantum signature schemes,” Quantum Inf. Comput. 16, 435–464 (2016).

Weedbrook, C.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Winzer, P. J.

Wittmann, C.

C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Demonstration of coherent-state discrimination using a displacement-controlled photon-number-resolving detector,” Phys. Rev. Lett. 104, 100505 (2010).
[Crossref]

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[Crossref]

Yoshizawa, A.

Zama, T.

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Opt. (2)

J. Lightwave Technol. (1)

Lect. Notes Phys. (1)

J. A. Bergou, U. Herzog, and M. Hillery, “Discrimination of quantum states,” Lect. Notes Phys. 649, 417–465 (2004).
[Crossref]

Nat. Commun. (1)

F. E. Becerra, J. Fan, and A. Migdall, “Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states,” Nat. Commun. 4, 2028 (2013).
[Crossref]

Nat. Photonics (3)

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nat. Photonics 7, 147–152 (2013).
[Crossref]

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6, 374–379 (2012).
[Crossref]

F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications,” Nat. Photonics 9, 48–53 (2015).
[Crossref]

Nature (1)

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446, 774–777 (2007).
[Crossref]

New J. Phys. (2)

D. Sych and G. Leuchs, “Coherent state quantum key distribution with multi letter phase-shift keying,” New J. Phys. 12, 053019 (2010).
[Crossref]

C. R. Müller and C. Marquardt, “A robust quantum receiver for phase shift keyed signals,” New J. Phys. 17, 032003 (2015).
[Crossref]

npj Quantum Inf. (1)

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 Inf. 3, 43 (2017).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (6)

S. Izumi, M. Takeoka, K. Ema, and M. Sasaki, “Quantum receivers with squeezing and photon-number-resolving detectors for M-ary coherent state discrimination,” Phys. Rev. A 87, 042328 (2013).
[Crossref]

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Phys. Rev. A 89, 032318 (2014).
[Crossref]

F. E. Becerra, J. Fan, G. Baumgartner, S. V. Polyakov, J. Goldhar, J. T. Kosloski, and A. Migdall, “M-ary-state phase-shift-keying discrimination below the homodyne limit,” Phys. Rev. A 84, 062324 (2011).
[Crossref]

S. Izumi, M. Takeoka, M. Fujiwara, N. D. Pozza, A. Assalini, K. Ema, and M. Sasaki, “Displacement receiver for phase-shift-keyed coherent states,” Phys. Rev. A 86, 042328 (2012).
[Crossref]

J. M. Arrazola, M. Karasamanis, and N. Lütkenhaus, “Practical quantum retrieval games,” Phys. Rev. A 93, 062311 (2016).
[Crossref]

M. Takeoka and M. Sasaki, “Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers,” Phys. Rev. A 78, 022320 (2008).
[Crossref]

Phys. Rev. Lett. (4)

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
[Crossref]

C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, “Demonstration of near-optimal discrimination of optical coherent states,” Phys. Rev. Lett. 101, 210501 (2008).
[Crossref]

C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Demonstration of coherent-state discrimination using a displacement-controlled photon-number-resolving detector,” Phys. Rev. Lett. 104, 100505 (2010).
[Crossref]

C. Croal, C. Peuntinger, B. Heim, I. Khan, C. Marquardt, G. Leuchs, P. Wallden, E. Andersson, and N. Korolkova, “Free-space quantum signatures using heterodyne measurements,” Phys. Rev. Lett. 117, 100503 (2016).
[Crossref]

Quantum Inf. Comput. (1)

J. M. Arrazola, P. Wallden, and E. Andersson, “Multiparty quantum signature schemes,” Quantum Inf. Comput. 16, 435–464 (2016).

Rev. Mod. Phys. (2)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Other (7)

S. J. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” (Research Laboratory of Electronics, MIT, 1973), p. 115.

R. S. Kennedy, “A near-optimum receiver for the binary coherent state quantum channel,” (Research Laboratory of Electronics, MIT, 1972), unpublished.

S. Betti, G. De Marchis, and E. Lannone, Coherent Optical Communications Systems (Wiley, 2000).

C. W. Helstrom, Quantum Detection and Estimation Theory, Mathematics in Science and Engineering (Academic, 1976), Vol. 123.

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Malvern Physics Series, Bangalore, India, 1984, p. 175.

J. M. Bennett, (Polarizers) Handbook of Optics, M. Bass, ed., 2nd ed. (McGraw-Hill, 1995), Vol. II.

J. T. Kosloski, “A Kennedy receiver for optical quadrature phase shift keying,” Ph.D. thesis (Johns Hopkins University, 2012), 257 pp.

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Figures (6)

Fig. 1.
Fig. 1. Theoretical scheme. (a) The input states are prepared in one of four possible states | α k { | α , | i α , | α , | i α } . (b) The power of the input state is split into three arms with ratios R 1 , R 2 , and R 3 , and each arm is designed to test a different hypothesis of the input state by displacement operations and photon counting. Arms 1, 2, and 3 are set to test states | α , | i α , and | α , respectively, with optimized displacements D ^ ( β i ) that minimize the probability of error. The detection outcomes of three SPDs for zero ( d i = 0 ) or non-zero ( d i = 1 ) photons provide information about the possible input state. These outcomes are used to obtain a decision rule based on maximum a posteriori probability criterion, which is shown in (c).
Fig. 2.
Fig. 2. Theoretical optimal parameters and error probabilities. (a) The optimal displacement ratios ( | β opt , i | 2 / R i | α | 2 ) for the three detection arms that minimize the probability of error. Arms 1 and 3 are in a red solid line and arm 2 is in a blue dashed line. The inset shows the optimal power ratios R 1 , R 2 , and R 3 in arms 1, 2, and 3, respectively. (b) The error probabilities for different values of detection efficiency ( η ), dark counts ( ν ), and visibility ( ξ ) of the displacement operations for fixed splitting ratios of { R 1 , R 2 , R 3 } = { 0.40 , 0.20 , 0.40 } in arms 1, 2, and 3, respectively. Cases shown: detection efficiency of η = 1.0 , dark count rate of ν = 10 6 , and visibility of ξ = 0.995 (red solid line); η = 0.80 , ν = 10 6 , with ξ = 1.0 (green solid line); and η = 1.0 , ν = 10 6 , with ξ = 1.0 (blue solid line). Thin dotted lines of the same color represent zero dark counts ν = 0 . The Helstrom bound, corresponding to the ultimate quantum limit [1], and heterodyne limit (QNL) are also plotted in thick dashed lines.
Fig. 3.
Fig. 3. Experimental configuration. The coherent state pulses are generated by a 633 nm He–Ne laser and an acousto-optic modulator (AOM) and then coupled into the experimental setup using a single-mode fiber (SMF). A variable attenuator (VA) defines the mean photon number of the input state. A half-wave plate ( HWP 0 ) and quarter-wave plate ( QWP 0 ) define the four possible states in the vertical polarization that are then split into the three arms with a splitting ratio of {0.40, 0.20, 0.40} with two beam splitters. The input state in the vertical polarization co-propagates with the strong displacement field (LO) in the horizontal polarization. The displacements of the input state in each arm are performed by a HWP and a polarizing beam splitter (PBS) and the photons in the displaced state D ^ ( β i ) | α k are counted with SPD’s. The detection record d is collected by a field-programmable gate array (FPGA) that transfers the data to the computer for analysis of the discrimination strategy. The extinction ratio (ER) for each PBS i is shown and for a power ratio of the input state to LO of 1/100 this leads to the observed visibilities of {0.991(1), 0.990(1), 0.993(1)} in arms 1, 2, and 3, respectively.
Fig. 4.
Fig. 4. Interference in arm 1. The normalized intensity after the PBS 1 as a function of the HWP 1 angle in arm 1 testing the hypothesis that the input state is | α for the four possible input states { | α , | i α , | α , | i α } with phases ϕ = { 0 , π / 2 , π , 3 π / 2 } , respectively. Point A corresponds to the maximum nulling at the nulling angle ψ 1 and point D corresponds to the intensity at the nulling angle ψ 1 when sending the state with a π phase shift ( ϕ = π ). These two points are the minimum (A) and the maximum (D) of an interference fringe and are used to estimate the visibility ξ of the displacement operation in arm 1. The inset (i) shows the expected normalized interference as a function of input phase ϕ , with minimum and maximum corresponding to points A and D, respectively. Points B and C correspond to the HWP 1 being set to the nulling angle ψ 1 and sending the states ϕ = 3 π / 2 and ϕ = π / 2 . These two points are used to check the quality of state preparation. Also shown (green squares) are examples of the angles that correspond to the optimal displacement ratios S i ( n ) for n = 1 , 2 , 3 . The angle δ 1 corresponds to the HWP 1 angle that allows only light with a vertical polarization to be transmitted through the PBS 1 .
Fig. 5.
Fig. 5. Experimental results. The experimental results for the error probability as a function of mean photon number of the input state n = | α | 2 . Included are the Helstrom bound (green) and QNL (black) as well as the QNL adjusted for the overall experimental detection efficiency of 77.8% (gray). The effect of non-ideal visibility can be seen by the floor that is imposed by the reduced visibility resulting in an achieved P E of 3.6 × 10 2 at a mean photon number of | α | 2 10 . The experimental results are in very good agreement with theoretical predictions (red). Our numerical study can provide a guide for future implementations of these single-shot strategies regarding the required experimental parameters needed to outperform a heterodyne (QNL) measurement.
Fig. 6.
Fig. 6. Error probability for non-ideal parameters. A comparison of the error probability P E for the non-Gaussian measurement with the heterodyne limit P het in a logarithmic scale log 10 ( P E / P het ) as a function of the visibility and mean photon number, with (a) detection efficiency η = 1 and (b)  η = 90 for ν = 10 6 . Negative values of log 10 ( P E / P het ) indicate improvement over heterodyne, i.e., P E / P het < 1 . Empty areas above red boundaries correspond to values of log 10 ( P E / P het ) > 1 . (c) The ratio P E / P het in the parameter space for visibility and detection efficiency for n = 6 . Note that it is possible to attain improvement over a heterodyne, P E / P het < 1 , with non-ideal detection efficiency and visibility. The black circle corresponds to a receiver with η = 98 % and ξ = 99.98 % .

Equations (6)

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P ( α k | d ) = P ( α k | d 1 , β 1 ) P ( α k | d 2 , β 2 ) P ( α k | d 3 , β 3 ) ,
P ( d i | α k , β i ) = { e | α k β i | 2 , d i = 0 , 1 e | α k β i | 2 , d i = 1 .
P E = 1 1 4 k = 1 4 P ( α k | α k , β 1 , β 2 , β 3 ) ,
I i ( α , β , ϕ , Δ ) = R i | α | 2 cos 2 ( 2 Δ ) + | β i | 2 sin 2 ( 2 Δ ) 2 R i | α | | β i | sin ( 2 Δ ) cos ( 2 Δ ) cos ( γ ) .
I i norm ( α , β , ϕ , Δ ) = cos ( 2 Δ ) 2 + f i 2 sin ( 2 Δ ) 2 2 f i sin ( 2 Δ ) cos ( 2 Δ ) cos ( γ ) ,
S i ( n ) = | β opt , i ( θ , δ i ) | | α i ( θ , δ i ) | = | β i | sin [ 2 ( θ i ( n ) δ i ) ] R i | α | cos [ 2 ( θ i ( n ) δ i ) ] = f i tan [ 2 ( θ i ( n ) δ i ) ] .

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