Abstract

We present the full characterization of phase-randomized or phase-averaged coherent states, a class of states exploited in communication channels and in decoy state-based quantum key distribution protocols. We report on the suitable formalism to analytically describe the main features of these states and on their experimental investigation, that results in agreement with theory. In particular, we consider a recently proposed non-Gaussianity measure based on the quantum fidelity, that we compare with previous ones, and we use the mutual information to investigate the amount of correlations one can produce by manipulating this class of states.

© 2013 Optical Society of America

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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]
  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. D 41, 599–627 (2007).
    [CrossRef]
  4. S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Spec. Top. 203, 3–24 (2012).
    [CrossRef]
  5. 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]
  6. M. Bondani, A. Allevi, and A. Andreoni, “Light statistics by non-calibrated linear photodetectors,” Adv. Sci. Lett. 2, 463–468 (2009).
    [CrossRef]
  7. M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt. 56, 226–231 (2009).
    [CrossRef]
  8. A. Andreoni and M. Bondani, “Photon statistics in the macroscopic realm measured without photon counters,” Phys. Rev. A 80, 013819 (2009).
    [CrossRef]
  9. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 1 & 2.
  10. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1944).
  11. K. E. Cahill and R. J. Glauber, “Ordered expansions in Bosons amplitude operators,” Phys. Rev. 177, 1857–1881 (1969).
    [CrossRef]
  12. K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
    [CrossRef]
  13. V. Veitch, C. Ferrie, D. Gross, and J. Emerson, “Negative quasi-probability as a resource for quantum computation,” New J. Phys. 14, 113011 (2012).
    [CrossRef]
  14. V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, “Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation,” New J. Phys. 15, 013037 (2013).
    [CrossRef]
  15. A. Mari and J. Eisert, “Positive Wigner functions render classical simulation of quantum computation efficient,” Phys. Rev. Lett. 109, 230503 (2012).
    [CrossRef]
  16. M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. 34, 1444–1446 (2009).
    [CrossRef]
  17. S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
    [CrossRef]
  18. K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996).
    [CrossRef]
  19. A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
    [CrossRef]
  20. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
    [CrossRef]
  21. 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. A 78, 060303(R) (2008).
    [CrossRef]
  22. M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 052341 (2010).
    [CrossRef]
  23. I. Ghiu, P. Marian, and T. A. Marian, “Measures of non-Gaussianity for one-mode field states,” Phys. Scr. T153, 014028 (2013).
    [CrossRef]
  24. A. Uhlmann, “The ‘transition probability’ in the state space of a *-algebra,” Rep. Math. Phys. 9, 273–279 (1976).
    [CrossRef]
  25. A. Uhlmann, “Parallel transport and the ‘quantum holonomy’ along density operators,” Rep. Math. Phys. 24, 229–240 (1986).
    [CrossRef]
  26. A. Allevi, S. Olivares, and M. Bondani, “Manipulating the non-Gaussianity of phase-randomized coherent states,” Opt. Express 20, 24850–24855 (2012).
    [CrossRef]
  27. A. Allevi, S. Olivares, and M. Bondani, “Experimental quantification of non-Gaussianity of phase-randomized coherent states,” Int. J. Quantum Inform. 10, 1241006 (2012).
    [CrossRef]
  28. This results follows from the form of the joint photon statistics of the two outgoing modes that is factorized and reads p(n,m)=P(n;τ|β|2)P(m;(1−τ)|β|2), where P(n;N)=exp(−N)Nn(n!)−1 is the Poisson distribution.
  29. A. Allevi, M. Bondani, and A. Andreoni, “Photon-number correlations by photon-number resolving detectors,” Opt. Lett. 35, 1707–1709 (2010).
    [CrossRef]
  30. 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 Inform. 5, 305–309 (2007).
    [CrossRef]

2013

V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, “Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation,” New J. Phys. 15, 013037 (2013).
[CrossRef]

I. Ghiu, P. Marian, and T. A. Marian, “Measures of non-Gaussianity for one-mode field states,” Phys. Scr. T153, 014028 (2013).
[CrossRef]

2012

A. Allevi, S. Olivares, and M. Bondani, “Manipulating the non-Gaussianity of phase-randomized coherent states,” Opt. Express 20, 24850–24855 (2012).
[CrossRef]

A. Allevi, S. Olivares, and M. Bondani, “Experimental quantification of non-Gaussianity of phase-randomized coherent states,” Int. J. Quantum Inform. 10, 1241006 (2012).
[CrossRef]

A. Mari and J. Eisert, “Positive Wigner functions render classical simulation of quantum computation efficient,” Phys. Rev. Lett. 109, 230503 (2012).
[CrossRef]

S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Spec. Top. 203, 3–24 (2012).
[CrossRef]

V. Veitch, C. Ferrie, D. Gross, and J. Emerson, “Negative quasi-probability as a resource for quantum computation,” New J. Phys. 14, 113011 (2012).
[CrossRef]

2010

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 052341 (2010).
[CrossRef]

A. Allevi, M. Bondani, and A. Andreoni, “Photon-number correlations by photon-number resolving detectors,” Opt. Lett. 35, 1707–1709 (2010).
[CrossRef]

2009

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

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

M. Bondani, A. Allevi, and A. Andreoni, “Light statistics by non-calibrated linear photodetectors,” Adv. Sci. Lett. 2, 463–468 (2009).
[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. A 80, 013819 (2009).
[CrossRef]

2008

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. A 78, 060303(R) (2008).
[CrossRef]

2007

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (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. D 41, 599–627 (2007).
[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 Inform. 5, 305–309 (2007).
[CrossRef]

2005

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

1996

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[CrossRef]

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

1986

A. Uhlmann, “Parallel transport and the ‘quantum holonomy’ along density operators,” Rep. Math. Phys. 24, 229–240 (1986).
[CrossRef]

1976

A. Uhlmann, “The ‘transition probability’ in the state space of a *-algebra,” Rep. Math. Phys. 9, 273–279 (1976).
[CrossRef]

1969

K. E. Cahill and R. J. Glauber, “Ordered expansions in Bosons amplitude operators,” Phys. Rev. 177, 1857–1881 (1969).
[CrossRef]

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, S. Olivares, and M. Bondani, “Manipulating the non-Gaussianity of phase-randomized coherent states,” Opt. Express 20, 24850–24855 (2012).
[CrossRef]

A. Allevi, S. Olivares, and M. Bondani, “Experimental quantification of non-Gaussianity of phase-randomized coherent states,” Int. J. Quantum Inform. 10, 1241006 (2012).
[CrossRef]

A. Allevi, M. Bondani, and A. Andreoni, “Photon-number correlations by photon-number resolving detectors,” Opt. Lett. 35, 1707–1709 (2010).
[CrossRef]

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

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

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

M. Bondani, A. Allevi, and A. Andreoni, “Light statistics by non-calibrated linear photodetectors,” Adv. Sci. Lett. 2, 463–468 (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 Inform. 5, 305–309 (2007).
[CrossRef]

Andreoni, A.

A. Allevi, M. Bondani, and A. Andreoni, “Photon-number correlations by photon-number resolving detectors,” Opt. Lett. 35, 1707–1709 (2010).
[CrossRef]

M. Bondani, A. Allevi, and A. Andreoni, “Light statistics by non-calibrated linear photodetectors,” Adv. Sci. Lett. 2, 463–468 (2009).
[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. A 80, 013819 (2009).
[CrossRef]

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. 34, 1444–1446 (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 Inform. 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. A 78, 060303(R) (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

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

Bondani, A.

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

Bondani, M.

A. Allevi, S. Olivares, and M. Bondani, “Experimental quantification of non-Gaussianity of phase-randomized coherent states,” Int. J. Quantum Inform. 10, 1241006 (2012).
[CrossRef]

A. Allevi, S. Olivares, and M. Bondani, “Manipulating the non-Gaussianity of phase-randomized coherent states,” Opt. Express 20, 24850–24855 (2012).
[CrossRef]

A. Allevi, M. Bondani, and A. Andreoni, “Photon-number correlations by photon-number resolving detectors,” Opt. Lett. 35, 1707–1709 (2010).
[CrossRef]

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

A. Andreoni and M. Bondani, “Photon statistics in the macroscopic realm measured without photon counters,” Phys. Rev. A 80, 013819 (2009).
[CrossRef]

M. Bondani, A. Allevi, and A. Andreoni, “Light statistics by non-calibrated linear photodetectors,” Adv. Sci. Lett. 2, 463–468 (2009).
[CrossRef]

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 Inform. 5, 305–309 (2007).
[CrossRef]

Brida, G.

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

Cahill, K. E.

K. E. Cahill and R. J. Glauber, “Ordered expansions in Bosons amplitude operators,” Phys. Rev. 177, 1857–1881 (1969).
[CrossRef]

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]

Curty, M.

Eisert, J.

A. Mari and J. Eisert, “Positive Wigner functions render classical simulation of quantum computation efficient,” Phys. Rev. Lett. 109, 230503 (2012).
[CrossRef]

Emerson, J.

V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, “Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation,” New J. Phys. 15, 013037 (2013).
[CrossRef]

V. Veitch, C. Ferrie, D. Gross, and J. Emerson, “Negative quasi-probability as a resource for quantum computation,” New J. Phys. 14, 113011 (2012).
[CrossRef]

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 1 & 2.

Ferrie, C.

V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, “Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation,” New J. Phys. 15, 013037 (2013).
[CrossRef]

V. Veitch, C. Ferrie, D. Gross, and J. Emerson, “Negative quasi-probability as a resource for quantum computation,” New J. Phys. 14, 113011 (2012).
[CrossRef]

Genoni, M. G.

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 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. A 78, 060303(R) (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

Genovese, M.

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

Ghiu, I.

I. Ghiu, P. Marian, and T. A. Marian, “Measures of non-Gaussianity for one-mode field states,” Phys. Scr. T153, 014028 (2013).
[CrossRef]

Glauber, R. J.

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

K. E. Cahill and R. J. Glauber, “Ordered expansions in Bosons amplitude operators,” Phys. Rev. 177, 1857–1881 (1969).
[CrossRef]

Gramegna, M.

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

Gross, D.

V. Veitch, C. Ferrie, D. Gross, and J. Emerson, “Negative quasi-probability as a resource for quantum computation,” New J. Phys. 14, 113011 (2012).
[CrossRef]

Inamori, H.

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D 41, 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]

Lütkenhaus, N.

Ma, X.

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 1 & 2.

Mari, A.

A. Mari and J. Eisert, “Positive Wigner functions render classical simulation of quantum computation efficient,” Phys. Rev. Lett. 109, 230503 (2012).
[CrossRef]

Marian, P.

I. Ghiu, P. Marian, and T. A. Marian, “Measures of non-Gaussianity for one-mode field states,” Phys. Scr. T153, 014028 (2013).
[CrossRef]

Marian, T. A.

I. Ghiu, P. Marian, and T. A. Marian, “Measures of non-Gaussianity for one-mode field states,” Phys. Scr. T153, 014028 (2013).
[CrossRef]

Mayers, D.

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D 41, 599–627 (2007).
[CrossRef]

Moroder, T.

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 1 & 2.

Olivares, S.

S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Spec. Top. 203, 3–24 (2012).
[CrossRef]

A. Allevi, S. Olivares, and M. Bondani, “Manipulating the non-Gaussianity of phase-randomized coherent states,” Opt. Express 20, 24850–24855 (2012).
[CrossRef]

A. Allevi, S. Olivares, and M. Bondani, “Experimental quantification of non-Gaussianity of phase-randomized coherent states,” Int. J. Quantum Inform. 10, 1241006 (2012).
[CrossRef]

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

Paris, M. G. A.

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 052341 (2010).
[CrossRef]

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[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. A 78, 060303(R) (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[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 Inform. 5, 305–309 (2007).
[CrossRef]

Qi, B.

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

Traina, P.

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[CrossRef]

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 1 & 2.

Uhlmann, A.

A. Uhlmann, “Parallel transport and the ‘quantum holonomy’ along density operators,” Rep. Math. Phys. 24, 229–240 (1986).
[CrossRef]

A. Uhlmann, “The ‘transition probability’ in the state space of a *-algebra,” Rep. Math. Phys. 9, 273–279 (1976).
[CrossRef]

Veitch, V.

V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, “Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation,” New J. Phys. 15, 013037 (2013).
[CrossRef]

V. Veitch, C. Ferrie, D. Gross, and J. Emerson, “Negative quasi-probability as a resource for quantum computation,” New J. Phys. 14, 113011 (2012).
[CrossRef]

Vogel, W.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[CrossRef]

Wallentowitz, S.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1944).

Wiebe, N.

V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, “Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation,” New J. Phys. 15, 013037 (2013).
[CrossRef]

Wódkiewicz, K.

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

Zambra, G.

A. Allevi, A. Andreoni, A. Bondani, G. Brida, M. Genovese, M. Gramegna, S. Olivares, M. G. A. Paris, P. Traina, and G. Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (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 Inform. 5, 305–309 (2007).
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This results follows from the form of the joint photon statistics of the two outgoing modes that is factorized and reads p(n,m)=P(n;τ|β|2)P(m;(1−τ)|β|2), where P(n;N)=exp(−N)Nn(n!)−1 is the Poisson distribution.

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Figures (11)

Fig. 1.
Fig. 1.

Experimental setup. Fj, variable neutral density filter; BS, 5050 beam splitter; Pz, piezoelectric movement; MF, multimode fiber (600 μm core); and HPD, hybrid photodetector.

Fig. 2.
Fig. 2.

Detected-photon distribution of a PHAV for three different mean values. Colored dots: experimental data, lines: theoretical expectations. The purity is μ[ϱ^]=0.13 (red plot, dashed line), μ[ϱ^]=0.20 (blue plot, solid line), and μ[ϱ^]=0.39 (black plot, dotted line).

Fig. 3.
Fig. 3.

Experimental reconstruction of a section of the Wigner function of a PHAV with |β|2=1.97 and ξ=0.999. Blue dots: experimental data, orange mesh: theoretical expectation.

Fig. 4.
Fig. 4.

Comparison among the three measures of non-Gaussianity in the case of PHAVs as functions of the mean number of photons. Open squared symbols: experimental data, dots: theoretical expectations.

Fig. 5.
Fig. 5.

Mutual information between the two outputs of the BS at which a PHAV with energy MT is divided. Open squared symbols: experimental data, red dots: theory. The error bars are smaller than the symbol size.

Fig. 6.
Fig. 6.

Detected-photon distribution of a balanced 2-PHAV state for three different mean values. Colored dots: experimental data, lines: theoretical expectations (31). The purity is μ[ϱ^]=0.09 (red plot, dashed line), μ[ϱ^]=0.13 (blue plot, solid line) and μ[ϱ^]=0.21 (black plot, dotted line).

Fig. 7.
Fig. 7.

Left: experimental reconstruction of a section of the Wigner function of a PHAV with |β|2=1.97 and ξ=0.999. Blue dots: experimental data, orange mesh: theoretical expectation. Right: experimental reconstruction of a section of the Wigner function of a balanced 2-PHAV with |β1|2=1.03, |β2|2=0.91, ξP=0.95, and ξS=1. Red dots: experimental data, blue mesh: theoretical expectation.

Fig. 8.
Fig. 8.

Comparison among the three measures of non-Gaussianity in the case of balanced 2-PHAVs as functions of the mean number of photons. Open squared symbols: experimental data, dots: theoretical expectations.

Fig. 9.
Fig. 9.

Comparison among the three measures of non-Gaussianity in the case of 2-PHAVs at fixed total energy as functions of the balancing between the two single PHAVs. Open squared symbols: experimental data, dots: theoretical expectations.

Fig. 10.
Fig. 10.

Mutual information between the two outputs of the BS at which a balanced 2-PHAV is divided as a function of the total input energy MT. Open squared symbols: experimental data, red dots: theory. The error bars are smaller than the symbol size.

Fig. 11.
Fig. 11.

Mutual information between the two outputs of the BS at which a 2-PHAV at fixed input energy MT is divided as a function of the ratio R=|β1|/|β2| between the energies of the two PHAVs generating the 2-PHAV state. Open squared symbols: experimental data, red dots: theory. The error bars are smaller than the symbol size.

Equations (34)

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|β=exp(12|β|2)n=0|β|neinϕn!|n,
ϱ^=02πdϕ2π|ββ|=n=0ϱnn|nn|,
ϱnn=exp(|β|2)|β|2nn!,
μ[ϱ^]=n=0ϱnn2=exp(2|β|2)I0(2|β|2),
χ(λ;s)exp(s2|λ|2)Tr[ϱ^D^(λ)],(1s1)
χ(λ;s)=exp(1s2|λ|2)n=0ϱnnLn(|λ|2).
n|D^(λ)|n=exp(12|λ|2)Ln(|λ|2),
n=0Ln(z)xnn!=exJ0(2xz),
χ(λ;s)=exp(1s2|λ|2)J0(2|β||λ|),
W(z;s)=21sexp[2(|β|2+|z|2)1s]I0(4|z||β|1s),
Q(z)=1πW(z;1)=1πexp[(|β|2+|z|2)]I0(2|z||β|).
P(z)=W(z;1)=1πδ(|z|2|β|2)=12π|β|δ(|z||β|).
W(z)=2exp[2(|β|2+|z|2)]I0(4|z||β|),
W˜(ξα)=W(ξα)exp[(|α|+|β|)1ξ],
εA[ϱ^]DHS2[ϱ^,σ^]μ[ϱ^]=μ[ϱ^]+μ[σ^]2κ[ϱ^,σ^]2μ[ϱ^],
κ[ϱ^,σ^]=1|β|2+1exp(|β|2|β|2+1).
εB[ϱ^]S(σ^)S(ϱ^),
εC[ϱ^]1F(ϱ^,σ^),
F(ϱ^,σ^)={Tr[ϱ^σ^ϱ^]}2
MI(ϱ^12)=S(ϱ^1)+S(ϱ^2)S(ϱ^12),
χ(λ;s)=χ01(λτ;s)χ02(λ1τ;s),
χ(λ;s)=exp(1s2|λ|2)×J0(2|β1||λ|τ)J0(2|β2||λ|1τ),
ϱ^=1πd2d2λχ(λ;0)D^(λ).
ϱlm=2δlm0d|λ||λ|exp(12|λ|2)χ(|λ|)Lm(|λ|2),
ϱnn=1n!k=n(1)kn(kn)!(a^)ka^k.
(a^)ka^k=[|β1|2τ+|β2|2(1τ)]kukPk(1u),
u:=||β1|2τ|β2|2(1τ)||β1|2τ+|β2|2(1τ).
g(k)(0)=(a^)ka^ka^a^k=ukPk(1u).
ϱnn=1n!k=n(1)kn(kn)![|β1|2τ+|β2|2(1τ)]kukPk(1u).
g(k)(0)=(2k1)!!k!>1,(k>1),
ϱnn=exp(2|β|2)(2n1)!!(n!)2|β|2nF11(12;n+1;2|β|2),
W(α;s)=21sexp(2|α|21s)×k=0(1)kk!(2[|β1|2τ+|β2|2(1τ)]1s)k×ukPk(1u)Lk(2|α|21s),
W(α;s)=21sexp(2|α|21s)k=0(2k1)!!(k!)2×(2|β|21s)kLk(2|α|21s).
W˜2PHAV(ξPα)=W2PHAV(ξPα)×exp[|α|1ξP(|β1|+|β2|)1ξS],

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