Abstract

We study the problem of determining the photon number statistics of an unknown quantum state using conjugate optical homodyne detection. We quantify the information gain in a single-shot measurement and show that the photon number statistics can be recovered in repeated measurements on an ensemble of identical input states without scanning the phase of the input state or randomizing the phase of the local oscillator used in homodyne detection. We demonstrate how the expectation maximization algorithm and Bayesian inference can be utilized to facilitate the reconstruction and illustrate our approach by conducting experiments to study the photon number distributions of a weak coherent state and a thermal state source.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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  33. We could remove the constant 1 in Eq. (9) by redefining Eq. (1) as $Z=X_{3}^{2}+P_{4}^{2}-1$Z=X32+P42−1. However, such a definition may result a negative measurement result of Z in the case of single-shot measurement.
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  43. M. Hillery, “Quantum cryptography with squeezed states,” Phys. Rev. A 61(2), 022309 (2000).
    [Crossref]
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    [Crossref]
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    [Crossref]
  46. B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the Local Oscillator “Locally" in Continuous-Variable Quantum Key Distribution Based on Coherent Detection,” Phys. Rev. X 5(4), 041009 (2015).
    [Crossref]
  47. H. Qin, R. Kumar, and R. Alléaume, “Quantum hacking: Saturation attack on practical continuous-variable quantum key distribution,” Phys. Rev. A 94(1), 012325 (2016).
    [Crossref]
  48. H. Qin, R. Kumar, V. Makarov, and R. Alléaume, “Homodyne-detector-blinding attack in continuous-variable quantum key distribution,” Phys. Rev. A 98(1), 012312 (2018).
    [Crossref]
  49. S. Ghorai, P. Grangier, E. Diamanti, and A. Leverrier, “Asymptotic security of continuous-variable quantum key distribution with a discrete modulation,” Phys. Rev. X 9(2), 021059 (2019).
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  53. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible,” Phys. Rev. Lett. 89(13), 137903 (2002).
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  54. J. Fiurášek, “Gaussian Transformations and Distillation of Entangled Gaussian States,” Phys. Rev. Lett. 89(13), 137904 (2002).
    [Crossref]
  55. G. Giedke and I. J. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66(3), 032316 (2002).
    [Crossref]
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    [Crossref]
  57. S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient Classical Simulation of Continuous Variable Quantum Information Processes,” Phys. Rev. Lett. 88(9), 097904 (2002).
    [Crossref]
  58. S. Lloyd and S. L. Braunstein, “Quantum Computation over Continuous Variables,” Phys. Rev. Lett. 82(8), 1784–1787 (1999).
    [Crossref]
  59. U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48(6), 4598–4604 (1993).
    [Crossref]

2019 (2)

T. A. Eriksson, T. Hirano, B. J. Puttnam, G. Rademacher, R. S. Luís, M. Fujiwara, R. Namiki, Y. Awaji, M. Takeoka, N. Wada, and M. Sasaki, “Wavelength division multiplexing of continuous variable quantum key distribution and 18.3 Tbit/s data channels,” Commun. Phys. 2(1), 9 (2019).
[Crossref]

S. Ghorai, P. Grangier, E. Diamanti, and A. Leverrier, “Asymptotic security of continuous-variable quantum key distribution with a discrete modulation,” Phys. Rev. X 9(2), 021059 (2019).
[Crossref]

2018 (3)

H. Qin, R. Kumar, V. Makarov, and R. Alléaume, “Homodyne-detector-blinding attack in continuous-variable quantum key distribution,” Phys. Rev. A 98(1), 012312 (2018).
[Crossref]

C. Lüders, J. Thewes, and M. Assmann, “Real time g(2) monitoring with 100 kHz sampling rate,” Opt. Express 26(19), 24854–24863 (2018).
[Crossref]

F. Raffaelli, G. Ferranti, D. H. Mahler, P. Sibson, J. E. Kennard, A. Santamato, G. Sinclair, D. Bonneau, M. G. Thompson, and J. C. F. Matthews, “A homodyne detector integrated onto a photonic chip for measuring quantum states and generating random numbers,” Quantum Sci. Technol. 3(2), 025003 (2018).
[Crossref]

2017 (1)

2016 (2)

E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
[Crossref]

H. Qin, R. Kumar, and R. Alléaume, “Quantum hacking: Saturation attack on practical continuous-variable quantum key distribution,” Phys. Rev. A 94(1), 012325 (2016).
[Crossref]

2015 (2)

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the Local Oscillator “Locally" in Continuous-Variable Quantum Key Distribution Based on Coherent Detection,” Phys. Rev. X 5(4), 041009 (2015).
[Crossref]

R. Kumar, H. Qin, and R. Alléaume, “Coexistence of continuous variable QKD with intense DWDM classical channels,” New J. Phys. 17(4), 043027 (2015).
[Crossref]

2014 (3)

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

R. Namiki, O. Gittsovich, S. Guha, and N. Lütkenhaus, “Gaussian-only regenerative stations cannot act as quantum repeaters,” Phys. Rev. A 90(6), 062316 (2014).
[Crossref]

2013 (2)

H. M. Chrzanowski, S. M. Assad, J. Bernu, B. Hage, A. P. Lund, T. C. Ralph, P. K. Lam, and T. Symul, “Reconstruction of photon number conditioned states using phase randomized homodyne measurements,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104009 (2013).
[Crossref]

G. Roumpos and S. T. Cundiff, “Photon number distributions from a diode laser,” Opt. Lett. 38(2), 139 (2013).
[Crossref]

2012 (1)

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(2), 621–669 (2012).
[Crossref]

2011 (1)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-Photon Sources and Detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[Crossref]

2010 (1)

B. Qi, W. Zhu, L. Qian, and H.-K. Lo, “Feasibility of quantum key distribution through a dense wavelength division multiplexing network,” New J. Phys. 12(10), 103042 (2010).
[Crossref]

2009 (3)

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3(12), 696–705 (2009).
[Crossref]

2007 (1)

Y. Zhao, B. Qi, and H.-K. Lo, “Experimental quantum key distribution with active phase randomization,” Appl. Phys. Lett. 90(4), 044106 (2007).
[Crossref]

2005 (2)

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95(6), 063602 (2005).
[Crossref]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71(6), 061803 (2005).
[Crossref]

2004 (1)

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum Cryptography Without Switching,” Phys. Rev. Lett. 93(17), 170504 (2004).
[Crossref]

2003 (4)

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and Ph. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref]

E. Waks, K. Inoue, W. D. Oliver, E. Diamanti, and Y. Yamamoto, “High-efficiency photon-number detection for quantum information processing,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1502–1511 (2003).
[Crossref]

K. Banaszek and I. A. Walmsley, “Photon counting with a loop detector,” Opt. Lett. 28(1), 52 (2003).
[Crossref]

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

2002 (5)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible,” Phys. Rev. Lett. 89(13), 137903 (2002).
[Crossref]

J. Fiurášek, “Gaussian Transformations and Distillation of Entangled Gaussian States,” Phys. Rev. Lett. 89(13), 137904 (2002).
[Crossref]

G. Giedke and I. J. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66(3), 032316 (2002).
[Crossref]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient Classical Simulation of Continuous Variable Quantum Information Processes,” Phys. Rev. Lett. 88(9), 097904 (2002).
[Crossref]

2000 (1)

M. Hillery, “Quantum cryptography with squeezed states,” Phys. Rev. A 61(2), 022309 (2000).
[Crossref]

1999 (2)

T. C. Ralph, “Continuous variable quantum cryptography,” Phys. Rev. A 61(1), 010303 (1999).
[Crossref]

S. Lloyd and S. L. Braunstein, “Quantum Computation over Continuous Variables,” Phys. Rev. Lett. 82(8), 1784–1787 (1999).
[Crossref]

1998 (2)

Th. Richter, “Determination of photon statistics and density matrix from double homodyne detection measurements,” J. Mod. Opt. 45(8), 1735–1749 (1998).
[Crossref]

K. Banaszek, “Maximum-likelihood estimation of photon-number distribution from homodyne statistics,” Phys. Rev. A 57(6), 5013–5015 (1998).
[Crossref]

1996 (3)

W. Grice and I. A. Walmsley, “Homodyne Detection in a Photon Counting Application,” J. Mod. Opt. 43(4), 795–805 (1996).
[Crossref]

S. Schiller, G. Breitenbach, S. F. Pereira, T. Müller, and J. Mlynek, “Quantum Statistics of the Squeezed Vacuum by Measurement of the Density Matrix in the Number State Representation,” Phys. Rev. Lett. 77(14), 2933–2936 (1996).
[Crossref]

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127(1-3), 144–160 (1996).
[Crossref]

1995 (1)

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon-number statistics from the phase-averaged quadrature-field distribution: Theory and ultrafast measurement,” Phys. Rev. A 52(2), R924–R927 (1995).
[Crossref]

1993 (2)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner Distribution and the Density Matrix of a Light Mode Using Optical Homodyne Tomography: Application to Squeezed States and the Vacuum,” Phys. Rev. Lett. 70(9), 1244–1247 (1993).
[Crossref]

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48(6), 4598–4604 (1993).
[Crossref]

1991 (1)

J. W. Noh, A. Fougères, and L. Mandel, “Measurement of the Quantum Phase by Photon Counting,” Phys. Rev. Lett. 67(11), 1426–1429 (1991).
[Crossref]

1989 (2)

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40(3), 1371–1384 (1989).
[Crossref]

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40(5), 2847–2849 (1989).
[Crossref]

1986 (1)

N. G. Walker and J. E. Carroll, “Multiport homodyne detection near the quantum noise limit,” Opt. Quantum Electron. 18(5), 355–363 (1986).
[Crossref]

1983 (2)

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum Likelihood from Incomplete Data via the EM Algorithm,” J. R. Stat. Soc. Ser. B (Methodol.) 39(1), 1–22 (1977).
[Crossref]

1963 (1)

R. J. Glauber, “The Quantum Theory of Optical Coherence,” Phys. Rev. 130(6), 2529–2539 (1963).
[Crossref]

Abbas, G. L.

Alléaume, R.

H. Qin, R. Kumar, V. Makarov, and R. Alléaume, “Homodyne-detector-blinding attack in continuous-variable quantum key distribution,” Phys. Rev. A 98(1), 012312 (2018).
[Crossref]

H. Qin, R. Kumar, and R. Alléaume, “Quantum hacking: Saturation attack on practical continuous-variable quantum key distribution,” Phys. Rev. A 94(1), 012325 (2016).
[Crossref]

R. Kumar, H. Qin, and R. Alléaume, “Coexistence of continuous variable QKD with intense DWDM classical channels,” New J. Phys. 17(4), 043027 (2015).
[Crossref]

Anderson, M. E.

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon-number statistics from the phase-averaged quadrature-field distribution: Theory and ultrafast measurement,” Phys. Rev. A 52(2), R924–R927 (1995).
[Crossref]

Andreoni, A.

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95(6), 063602 (2005).
[Crossref]

Assad, S. M.

H. M. Chrzanowski, S. M. Assad, J. Bernu, B. Hage, A. P. Lund, T. C. Ralph, P. K. Lam, and T. Symul, “Reconstruction of photon number conditioned states using phase randomized homodyne measurements,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104009 (2013).
[Crossref]

Assche, G. V.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and Ph. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref]

Assmann, M.

Awaji, Y.

T. A. Eriksson, T. Hirano, B. J. Puttnam, G. Rademacher, R. S. Luís, M. Fujiwara, R. Namiki, Y. Awaji, M. Takeoka, N. Wada, and M. Sasaki, “Wavelength division multiplexing of continuous variable quantum key distribution and 18.3 Tbit/s data channels,” Commun. Phys. 2(1), 9 (2019).
[Crossref]

Banaszek, K.

K. Banaszek and I. A. Walmsley, “Photon counting with a loop detector,” Opt. Lett. 28(1), 52 (2003).
[Crossref]

K. Banaszek, “Maximum-likelihood estimation of photon-number distribution from homodyne statistics,” Phys. Rev. A 57(6), 5013–5015 (1998).
[Crossref]

Bartlett, S. D.

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G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95(6), 063602 (2005).
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T. A. Eriksson, T. Hirano, B. J. Puttnam, G. Rademacher, R. S. Luís, M. Fujiwara, R. Namiki, Y. Awaji, M. Takeoka, N. Wada, and M. Sasaki, “Wavelength division multiplexing of continuous variable quantum key distribution and 18.3 Tbit/s data channels,” Commun. Phys. 2(1), 9 (2019).
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F. Raffaelli, G. Ferranti, D. H. Mahler, P. Sibson, J. E. Kennard, A. Santamato, G. Sinclair, D. Bonneau, M. G. Thompson, and J. C. F. Matthews, “A homodyne detector integrated onto a photonic chip for measuring quantum states and generating random numbers,” Quantum Sci. Technol. 3(2), 025003 (2018).
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D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner Distribution and the Density Matrix of a Light Mode Using Optical Homodyne Tomography: Application to Squeezed States and the Vacuum,” Phys. Rev. Lett. 70(9), 1244–1247 (1993).
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H. M. Chrzanowski, S. M. Assad, J. Bernu, B. Hage, A. P. Lund, T. C. Ralph, P. K. Lam, and T. Symul, “Reconstruction of photon number conditioned states using phase randomized homodyne measurements,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104009 (2013).
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T. A. Eriksson, T. Hirano, B. J. Puttnam, G. Rademacher, R. S. Luís, M. Fujiwara, R. Namiki, Y. Awaji, M. Takeoka, N. Wada, and M. Sasaki, “Wavelength division multiplexing of continuous variable quantum key distribution and 18.3 Tbit/s data channels,” Commun. Phys. 2(1), 9 (2019).
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B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
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E. Waks, K. Inoue, W. D. Oliver, E. Diamanti, and Y. Yamamoto, “High-efficiency photon-number detection for quantum information processing,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1502–1511 (2003).
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G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95(6), 063602 (2005).
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Y. Zhao, B. Qi, and H.-K. Lo, “Experimental quantum key distribution with active phase randomization,” Appl. Phys. Lett. 90(4), 044106 (2007).
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B. Qi, W. Zhu, L. Qian, and H.-K. Lo, “Feasibility of quantum key distribution through a dense wavelength division multiplexing network,” New J. Phys. 12(10), 103042 (2010).
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Appl. Phys. Lett. (1)

Y. Zhao, B. Qi, and H.-K. Lo, “Experimental quantum key distribution with active phase randomization,” Appl. Phys. Lett. 90(4), 044106 (2007).
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T. A. Eriksson, T. Hirano, B. J. Puttnam, G. Rademacher, R. S. Luís, M. Fujiwara, R. Namiki, Y. Awaji, M. Takeoka, N. Wada, and M. Sasaki, “Wavelength division multiplexing of continuous variable quantum key distribution and 18.3 Tbit/s data channels,” Commun. Phys. 2(1), 9 (2019).
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H. M. Chrzanowski, S. M. Assad, J. Bernu, B. Hage, A. P. Lund, T. C. Ralph, P. K. Lam, and T. Symul, “Reconstruction of photon number conditioned states using phase randomized homodyne measurements,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104009 (2013).
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Nature (1)

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and Ph. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
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B. Qi, W. Zhu, L. Qian, and H.-K. Lo, “Feasibility of quantum key distribution through a dense wavelength division multiplexing network,” New J. Phys. 12(10), 103042 (2010).
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B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
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E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
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Opt. Commun. (1)

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127(1-3), 144–160 (1996).
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Opt. Express (1)

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Opt. Quantum Electron. (1)

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Other (6)

See, for example, http://www.optoplex.com/

R. Loudon, The Quantum Theory of Light (Oxford University, 2000).

We could remove the constant 1 in Eq. (9) by redefining Eq. (1) as $Z=X_{3}^{2}+P_{4}^{2}-1$Z=X32+P42−1. However, such a definition may result a negative measurement result of Z in the case of single-shot measurement.

J. Lin, T. Upadhyaya, and N. Lütkenhaus, “Asymptotic security analysis of discrete-modulated continuous-variable quantum key distribution,” arXiv:1905.10896 (2019).

E. Kaur, S. Guha, and M. M. Wilde, “Asymptotic security of discrete-modulation protocols for continuous-variable quantum key distribution,” arXiv:1901.10099 (2019).

H. P. Yuen and J. H. Shapiro, “Quantum Statistics of Homodyne and Heterodyne Detection,” in Coherence and Quantum Optics IV, edited by L. Mandel and E. Wolf, eds. (Plenum, 1978), p. 719–727.

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Figures (8)

Fig. 1.
Fig. 1. Conjugate optical homodyne detection. $BS_{1-3}$ : symmetric beam spliter; PD: photo detector; $LO_\theta$ ( $LO_{\theta +\pi /2}$ ): local oscillator with phase $\theta$ ( $\theta +\pi /2$ ).
Fig. 2.
Fig. 2. Simulation results of detection efficiency $\eta$ (Dash-dot line), dark count probability $D$ (Dashed line), and the ratio $R=\eta /D$ (Solid line).
Fig. 3.
Fig. 3. EM reconstruction of photon number statistics from a sequence of simulated homodyne measurements for a coherent state. The blue histogram bars represent a true photon distribution. The gray histogram bars correspond to a distribution reconstructed by using the EM algorithm.
Fig. 4.
Fig. 4. Models of realistic photo-detector with detection efficiency $\eta$ . (a) The actual setup. (b) An equivalent model of (a). See details in Appendix A.
Fig. 5.
Fig. 5. Experimental setup. S-photon source under test; LO-local oscillator; PC-polarization controller; Att-variable optical attenuator; ADC-data acquisition board.
Fig. 6.
Fig. 6. Histograms of the reconstructed photon number distributions for a coherent state $|\alpha \rangle$ : Simulated photon number statistics for $|\alpha |^2=5$ assuming noiseless detectors (blue bar); Simulated photon number statistics for $|\alpha |^2=5$ with noisy detectors (red bar); Photon number statistics reconstructed from experimental data (green bar).
Fig. 7.
Fig. 7. Histograms of the reconstructed photon number distributions for a thermal state with the mean photon number $=15.3$ : Simulated photon number statistics with noisy detectors (red bar); Photon number statistics reconstructed from experimental data (blue bar); Thermal distribution with the mean photon number $=15.3$ (black line).
Fig. 8.
Fig. 8. Two equivalent models. (a) A symmetric beam splitter followed by two virtual beam splitters. (b) A virtual beam splitter placed in front of the symmetric beam splitter.

Equations (33)

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Z = X 3 2 + P 4 2 .
X ^ 3 = 1 2 [ a ^ 3 exp ( i θ ) + a ^ 3 exp ( i θ ) ] ,
P ^ 4 = i 2 [ a ^ 4 exp ( i θ ) a ^ 4 exp ( i θ ) ] .
Z ^ = X ^ 3 2 + P ^ 4 2 .
a ^ 3 = 1 2 ( a ^ 1 + a ^ 2 ) ,
a ^ 4 = 1 2 ( a ^ 1 a ^ 2 ) ,
[ a ^ j , a ^ j ] = 1 , j { 1 , 2 , 3 , 4 }
Z ^ = n ^ 1 + n ^ 2 + a ^ 1 a ^ 2 e i 2 θ + a ^ 1 a ^ 2 e i 2 θ + 1
Z ^ = ψ 1 0 2 | Z ^ | ψ 1 0 2 = n 1 + 1 ,
Δ Z 2 = ( Z ^ Z ^ ) 2 = Z ^ 2 Z ^ 2 .
Z ^ 2 = n ^ 1 2 + 3 n ^ 1 + 2.
Δ Z 2 = Δ n 1 2 + n 1 + 1 ,
g ( 2 ) ( 0 ) = a ^ 1 a ^ 1 a ^ 1 a ^ 1 a ^ 1 a ^ 1 2 .
a ^ 1 a ^ 1 a ^ 1 a ^ 1 = Z ^ 2 4 Z ^ + 2.
g ( 2 ) ( 0 ) = Z ^ 2 4 Z ^ + 2 ( Z ^ 1 ) 2 .
P X 3 , P 4 ( x 3 , p 4 ) = 1 π m , n = 0 ρ m n exp [ i ( n m ) θ ] ( m ! n ! ) 1 / 2 ( x 3 i p 4 ) m ( x 3 + i p 4 ) n exp [ ( x 3 2 + p 4 2 ) ] .
P R ( r ) = 0 2 π P X 3 , P 4 ( r cos ϕ , r sin ϕ ) d ϕ .
P R ( r ) = 2 exp ( r 2 ) n = 0 ρ n n n ! r 2 n + 1 .
P Z ( z ) = exp ( z ) n = 0 ρ n n n ! z n .
P ( Z = z | n ) = exp ( z ) z n n ! .
P ( N = n | z ) = P ( Z = z | n ) P N ( n ) P Z ( z ) = exp ( z ) z n n ! .
σ = Δ n 2 = z .
η = T P Z ( z | 1 ) d z
D = T P Z ( z | 0 ) d z
p ( Z k = z k | { p ( n ) } , n k ) = p ( n k ) exp ( z k ) z k n k n k ! .
L c = k = 1 M p ( Z k = z k | { p ( n ) } , n k ) .
L = k = 1 M n k = 0 N m a x p ( Z k = z k | { p ( n ) } , n k ) ,
k = 1 M n k = 0 N m a x = n 1 = 0 N m a x n M = 0 N m a x .
p 0 ( n ) = 1 N m a x + 1
X ^ 3 = 1 2 cos γ X ^ 1 + 1 2 cos γ X ^ 2 + sin γ X ^ 5 = 1 2 cos γ X ^ 1 + 1 2 1 + sin 2 γ X ^ V 1
P ^ 4 = 1 2 cos γ P ^ 1 1 2 cos γ P ^ 2 + sin γ P ^ 6 = 1 2 cos γ P ^ 1 + 1 2 1 + sin 2 γ P ^ V 2
X ^ 3 = 1 2 cos γ X ^ 1 + 1 2 sin γ X ^ 5 + 1 2 X ^ 2 = 1 2 cos γ X ^ 1 + 1 2 1 + sin 2 γ X ^ V 3
P ^ 4 = 1 2 cos γ P ^ 1 + 1 2 sin γ P ^ 5 1 2 P ^ 2 = 1 2 cos γ P ^ 1 + 1 2 1 + sin 2 γ P ^ V 4

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