Abstract

We experimentally investigate the non-Gaussian features of the phase-randomized coherent states, a class of states exploited in communication channels and in decoy state-based quantum key distribution protocols. In particular, we reconstruct their phase-insensitive Wigner functions and quantify their non-Gaussianity. The measurements are performed in the mesoscopic photon-number domain by means of a direct detection scheme involving linear detectors.

© 2012 OSA

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  1. H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett.94, 230504 (2005).
    [CrossRef] [PubMed]
  2. Y. Zhao, B. Qi, and H.-K. Lo, “Experimental quantum key distribution with active phase randomization,” Appl. Phys. Lett.90, 044106 (2007).
    [CrossRef]
  3. H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D41, 599–627 (2007).
    [CrossRef]
  4. H.-K. Lo and J. Preskill, “Phase randomization improves the security of quantum key distribution,” CALT-68-2556 (2005), arXiv:quant-ph/0504209v1.
  5. M. Curty, X. Ma, B. Qi, and T. Moroder, “Passive decoy-state quantum key distribution with practical light sources,” Phys. Rev. A81, 022310 (2010).
    [CrossRef]
  6. S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Special Topics203, 3–24 (2012).
    [CrossRef]
  7. M. Curty, T. Moroder, X. Ma, and N. Lütkenhaus, “Non-Poissonian statistics from Poissonian light sources with application to passive decoy state quantum key distribution,” Opt. Lett.34, 3238–3240 (2009).
    [CrossRef] [PubMed]
  8. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys.81, 299–332 (2009).
    [CrossRef]
  9. K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev.177, 1882–1902 (1969).
    [CrossRef]
  10. S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A53, 4528–4533 (1996).
    [CrossRef] [PubMed]
  11. K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett.76, 4344–4347 (1996).
    [CrossRef] [PubMed]
  12. G. Zambra, A. Allevi, M. Bondani, A. Andreoni, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters,” Int. J. Quantum Inf.5, 305–309 (2007).
    [CrossRef]
  13. M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010).
    [CrossRef]
  14. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303(R) (2008).
    [CrossRef]
  15. M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett.34, 1444–1446 (2009).
    [CrossRef] [PubMed]
  16. M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt.56, 226–231 (2009).
    [CrossRef]
  17. A. Andreoni and M. Bondani, “Photon statistics in the macroscopic realm measured without photon counters,” Phys. Rev. A80, 013819 (2009).
    [CrossRef]
  18. A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni, and S. Olivares, “Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction,” Phys. Rev. A82, 013816 (2010).
    [CrossRef]

2012 (1)

S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Special Topics203, 3–24 (2012).
[CrossRef]

2010 (3)

M. Curty, X. Ma, B. Qi, and T. Moroder, “Passive decoy-state quantum key distribution with practical light sources,” Phys. Rev. A81, 022310 (2010).
[CrossRef]

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010).
[CrossRef]

A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni, and S. Olivares, “Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction,” Phys. Rev. A82, 013816 (2010).
[CrossRef]

2009 (5)

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett.34, 1444–1446 (2009).
[CrossRef] [PubMed]

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt.56, 226–231 (2009).
[CrossRef]

A. Andreoni and M. Bondani, “Photon statistics in the macroscopic realm measured without photon counters,” Phys. Rev. A80, 013819 (2009).
[CrossRef]

M. Curty, T. Moroder, X. Ma, and N. Lütkenhaus, “Non-Poissonian statistics from Poissonian light sources with application to passive decoy state quantum key distribution,” Opt. Lett.34, 3238–3240 (2009).
[CrossRef] [PubMed]

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

2008 (1)

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303(R) (2008).
[CrossRef]

2007 (3)

G. Zambra, A. Allevi, M. Bondani, A. Andreoni, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters,” Int. J. Quantum Inf.5, 305–309 (2007).
[CrossRef]

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

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D41, 599–627 (2007).
[CrossRef]

2005 (1)

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett.94, 230504 (2005).
[CrossRef] [PubMed]

1996 (2)

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A53, 4528–4533 (1996).
[CrossRef] [PubMed]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett.76, 4344–4347 (1996).
[CrossRef] [PubMed]

1969 (1)

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev.177, 1882–1902 (1969).
[CrossRef]

Agliati, A.

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt.56, 226–231 (2009).
[CrossRef]

Allevi, A.

A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni, and S. Olivares, “Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction,” Phys. Rev. A82, 013816 (2010).
[CrossRef]

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett.34, 1444–1446 (2009).
[CrossRef] [PubMed]

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt.56, 226–231 (2009).
[CrossRef]

G. Zambra, A. Allevi, M. Bondani, A. Andreoni, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters,” Int. J. Quantum Inf.5, 305–309 (2007).
[CrossRef]

Andreoni, A.

A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni, and S. Olivares, “Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction,” Phys. Rev. A82, 013816 (2010).
[CrossRef]

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt.56, 226–231 (2009).
[CrossRef]

A. Andreoni and M. Bondani, “Photon statistics in the macroscopic realm measured without photon counters,” Phys. Rev. A80, 013819 (2009).
[CrossRef]

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett.34, 1444–1446 (2009).
[CrossRef] [PubMed]

G. Zambra, A. Allevi, M. Bondani, A. Andreoni, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters,” Int. J. Quantum Inf.5, 305–309 (2007).
[CrossRef]

Banaszek, K.

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303(R) (2008).
[CrossRef]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett.76, 4344–4347 (1996).
[CrossRef] [PubMed]

Bondani, M.

A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni, and S. Olivares, “Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction,” Phys. Rev. A82, 013816 (2010).
[CrossRef]

A. Andreoni and M. Bondani, “Photon statistics in the macroscopic realm measured without photon counters,” Phys. Rev. A80, 013819 (2009).
[CrossRef]

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett.34, 1444–1446 (2009).
[CrossRef] [PubMed]

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt.56, 226–231 (2009).
[CrossRef]

G. Zambra, A. Allevi, M. Bondani, A. Andreoni, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters,” Int. J. Quantum Inf.5, 305–309 (2007).
[CrossRef]

Cahill, K. E.

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev.177, 1882–1902 (1969).
[CrossRef]

Chen, K.

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett.94, 230504 (2005).
[CrossRef] [PubMed]

Curty, M.

Genoni, M. G.

A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni, and S. Olivares, “Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction,” Phys. Rev. A82, 013816 (2010).
[CrossRef]

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303(R) (2008).
[CrossRef]

Glauber, R. J.

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev.177, 1882–1902 (1969).
[CrossRef]

Inamori, H.

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D41, 599–627 (2007).
[CrossRef]

Lo, H.-K.

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

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett.94, 230504 (2005).
[CrossRef] [PubMed]

H.-K. Lo and J. Preskill, “Phase randomization improves the security of quantum key distribution,” CALT-68-2556 (2005), arXiv:quant-ph/0504209v1.

Lütkenhaus, N.

Lvovsky, A. I.

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

Ma, X.

M. Curty, X. Ma, B. Qi, and T. Moroder, “Passive decoy-state quantum key distribution with practical light sources,” Phys. Rev. A81, 022310 (2010).
[CrossRef]

M. Curty, T. Moroder, X. Ma, and N. Lütkenhaus, “Non-Poissonian statistics from Poissonian light sources with application to passive decoy state quantum key distribution,” Opt. Lett.34, 3238–3240 (2009).
[CrossRef] [PubMed]

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett.94, 230504 (2005).
[CrossRef] [PubMed]

Mayers, D.

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D41, 599–627 (2007).
[CrossRef]

Moroder, T.

Olivares, S.

S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Special Topics203, 3–24 (2012).
[CrossRef]

A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni, and S. Olivares, “Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction,” Phys. Rev. A82, 013816 (2010).
[CrossRef]

Paris, M. G. A.

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303(R) (2008).
[CrossRef]

G. Zambra, A. Allevi, M. Bondani, A. Andreoni, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters,” Int. J. Quantum Inf.5, 305–309 (2007).
[CrossRef]

Preskill, J.

H.-K. Lo and J. Preskill, “Phase randomization improves the security of quantum key distribution,” CALT-68-2556 (2005), arXiv:quant-ph/0504209v1.

Qi, B.

M. Curty, X. Ma, B. Qi, and T. Moroder, “Passive decoy-state quantum key distribution with practical light sources,” Phys. Rev. A81, 022310 (2010).
[CrossRef]

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

Raymer, M. G.

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

Vogel, W.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A53, 4528–4533 (1996).
[CrossRef] [PubMed]

Wallentowitz, S.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A53, 4528–4533 (1996).
[CrossRef] [PubMed]

Wódkiewicz, K.

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett.76, 4344–4347 (1996).
[CrossRef] [PubMed]

Zambra, G.

G. Zambra, A. Allevi, M. Bondani, A. Andreoni, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters,” Int. J. Quantum Inf.5, 305–309 (2007).
[CrossRef]

Zhao, Y.

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

Appl. Phys. Lett. (1)

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

Eur. Phys. J. D (1)

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D41, 599–627 (2007).
[CrossRef]

Eur. Phys. J. Special Topics (1)

S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Special Topics203, 3–24 (2012).
[CrossRef]

Int. J. Quantum Inf. (1)

G. Zambra, A. Allevi, M. Bondani, A. Andreoni, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters,” Int. J. Quantum Inf.5, 305–309 (2007).
[CrossRef]

J. Mod. Opt. (1)

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt.56, 226–231 (2009).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev.177, 1882–1902 (1969).
[CrossRef]

Phys. Rev. A (6)

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A53, 4528–4533 (1996).
[CrossRef] [PubMed]

M. Curty, X. Ma, B. Qi, and T. Moroder, “Passive decoy-state quantum key distribution with practical light sources,” Phys. Rev. A81, 022310 (2010).
[CrossRef]

A. Andreoni and M. Bondani, “Photon statistics in the macroscopic realm measured without photon counters,” Phys. Rev. A80, 013819 (2009).
[CrossRef]

A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni, and S. Olivares, “Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction,” Phys. Rev. A82, 013816 (2010).
[CrossRef]

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303(R) (2008).
[CrossRef]

Phys. Rev. Lett. (2)

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett.94, 230504 (2005).
[CrossRef] [PubMed]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett.76, 4344–4347 (1996).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

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

Other (1)

H.-K. Lo and J. Preskill, “Phase randomization improves the security of quantum key distribution,” CALT-68-2556 (2005), arXiv:quant-ph/0504209v1.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. F: variable neutral density filter; BS: 50/50 beam splitter; Pz: piezoelectric movement; MF: multimode fiber (600 μm core).

Fig. 2
Fig. 2

Experimental reconstruction of a section of the phase-insensitive Wigner function of a PHAV (left panel), with |β|2 = 1.97, and of a 2-PHAV (right panel), with |β|2 = 1.03 and |β̃|2 = 0.91. Black dots: experimental data; red line: theoretical expectations given in Eqs. (6) and (7), respectively.

Fig. 3
Fig. 3

Experimental reconstruction of a section of the phase-insensitive Wigner function of a 2-PHAV at different mean values MT (see the text for details). Black dots: experimental data; red line: theoretical expectations given in Eq. (7).

Fig. 4
Fig. 4

Left panel: nonG measure ε for the detected number of photons as a function of the mean number of photons of balanced 2-PHAVs (see the text for details). Right panel: ε as a function of the balancing between the two components of the 2-PHAV, at fixed mean number of detected photons MT = M1 + M2 of the overall state. Black dots: experimental data; red squares: theoretical expectation.

Equations (7)

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ρ β = 0 2 π d ϕ 2 π | β β | = e | β | 2 n = 0 | β | 2 n n ! | n n | ,
W PHAV ( α ; β ) = 0 2 π d ϕ 2 π e | α β 2 | = 2 π I 0 ( 4 | α | | β | ) exp [ 2 ( | α | 2 + | β | 2 ) ] ,
δ [ ρ β ] = S ( σ ) S ( ρ β ) ,
δ [ ρ β ] = ( N + 1 ) ln ( N + 1 ) N ln N + n p n ln p n ,
W 2 PHAV ( α ; β , β ˜ , τ ) = 0 2 π d ϕ ˜ 2 π W PHAV ( α β ˜ 1 τ ; β τ )
W ˜ PHAV ( ξ α ) = W PHAV ( ξ α ) e 1 ξ ( | α | + | β | ) ,
W ˜ 2 PHAV ( ξ P α ) = W 2 PHAV ( ξ P α ) e [ 1 ξ P | α | + 1 ξ S ( | β | + | β ˜ | ) ] ,

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