## Manipulating the non-Gaussianity of phase-randomized coherent states |

Optics Express, Vol. 20, Issue 22, pp. 24850-24855 (2012)

http://dx.doi.org/10.1364/OE.20.024850

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### 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

## 1. Introduction

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

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

5. M. Curty, X. Ma, B. Qi, and T. Moroder, “Passive decoy-state quantum key distribution with practical light sources,” Phys. Rev. A **81**, 022310 (2010). [CrossRef]

6. S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Special Topics **203**, 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. A **53**, 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. A **82**, 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. A **78**, 060303(R) (2008). [CrossRef]

## 2. Theory

*ρ*of a single-mode PHAV of amplitude

_{β}*β*is given by: where |

*β*〉 is a coherent state,

*β*= |

*β*|

*e*, and {|

^{iϕ}*n*〉},

*n*= 0, 1, 2,..., is the photon number basis. The corresponding Wigner function is [15

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

*I*

_{0}(

*z*) being the modified Bessel function. The representation in terms of Wigner function emphasizes both the non-Gaussian nature of PHAV and its phase-insensitive nature (

*i.e.*rotational invariance about the origin of the phase space).

*ρ*, we consider the following measure [13

_{β}13. M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A **82**, 052341 (2010). [CrossRef]

*S*(

*ρ*) = −Tr[

*ρ*ln

*ρ*] is the von Neumann entropy of the state

*ρ*and

*σ*is a reference Gaussian state with the same covariance matrix as the state

*ρ*. As the PHAV is a diagonal state in the photon number basis, its reference state is a thermal equilibrium state, with the same mean number of photons

*N*= |

*β*|

^{2}[13

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

*p*= exp (−|

_{n}*β*|

^{2}) |

*β*|

^{2n}/

*n*!.

*ρ*and

_{β}*ρ*

_{β}_{˜}at a beam splitter (BS) with transmissivity

*τ*, which can be useful for application to passive decoy state QKD [7

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]

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

*β̃*= |

*β̃*|

*e*and the function in the integral is given by Eq. (2). Obviously, the Wigner function of the other outgoing mode can be obtained by replacing

^{iϕ̃}*τ*with the reflectivity (1 −

*τ*). In the following we will refer to this state as 2-PHAV and, since it is diagonal, its nonG is still given by Eq. (4), where now

*p*is the 2-PHAV non-trivial photon statistics reported in [12

_{n}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]

## 3. Experimental results

*μ*m span. The 2-PHAV was produced by mixing at a BS two single PHAVs obtained by duplicating the phase-randomization system. A filter inserted in the path of one of the two PHAVs allowed us to change the balancing between the two fields. We implemented a direct detection scheme involving a photon-counting detector, namely a hybrid photodetector (HPD, R10467U-40, maximum quantum efficiency ∼0.5 at 500 nm, 1.4 ns response time, Hamamatsu), that is a detector not only endowed with a partial photon-counting capability, but also characterized by a linear response up to 100 photons. Thanks to its features, the HPD can actually operate in the mesoscopic domain, where the states are robust with respect to losses. The output of the detector was amplified (preamplifier A250 plus amplifier A275, Amptek), synchronously integrated (SGI, SR250, Stanford) and digitized (AT-MIO-16E-1, National Instruments). The estimated overall quantum efficiency of the detection chain is equal to 0.45. By using the strategy described in Refs. [16

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. A **80**, 013819 (2009). [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]

*M*

_{T}= 1.97 and

*M*

_{T}= 1.94, respectively. In each panel we also show the theoretical expectations (red line) for the PHAV and 2-PHAV, respectively [15

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

*ξ*being the overall (spatial and temporal) overlap between the probe and the PHAV, and: where

*ξ*

_{P}describes the overall overlap between the probe and the 2-PHAV and

*ξ*

_{S}the overall overlap between the two components of the 2-PHAV. In both the Eqs. (6) and (7), |

*β*|

^{2}and |

*β*̃|

^{2}are now the mean numbers of photons we measured (see Fig. 2), thus including the quantum efficiency. In fact, it is worth noting that for classical states the functional form of the Wigner function is preserved also in the presence of losses and its expression, given in terms of detected photons, reads

**34**, 1444–1446 (2009). [CrossRef] [PubMed]

*ξ*,

*ξ*

_{P}and

*ξ*

_{S}reported in the panels of Fig. 2, we actually achieved a very good superposition in aligning the system. In our opinion, having overcome this criticality can be considered a demonstration of the robustness of our setup and also a fundamental step on the way to the exploitation of these states for possible applications.

*ε*, for detected photons, which represents a lower bound to nonG [13

**82**, 052341 (2010). [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. A **82**, 013816 (2010). [CrossRef]

*ε*

_{PHAV}= 0.207 ± 0.004 and

*ε*

_{2–PHAV}= 0.036 ± 0.005. From this comparison it emerges that a PHAV is more non-Gaussian than the 2-PHAV, thus demonstrating that a Wigner function exhibiting a dip in the origin of the phase space is more non-Gaussian than one characterized by a peak in the origin followed by a “shoulder”. The results prove that combining two non-Gaussian states does not necessarily lead to an increase of the overall nonG. Indeed, to better understand the nature of the “shoulder”, we generated balanced 2-PHAVs at different mean values and reconstructed their Wigner function. The experimental data are shown in Fig. 3 together with the theoretical curve. We can observe that the “shoulder” is more and more evident at increasing the mean number of photons of the balanced 2-PHAV. Indeed, the calculation of the nonG amount is in accordance with this result, as it is an increasing function of the mean number of photons (left panel

*ε*

_{2–PHAV}= 0.031 ± 0.005,

*M*

_{T}= 1.61, and right panel

*ε*

_{2–PHAV}= 0.061 ± 0.006,

*M*

_{T}= 2.92). For the sake of clarity, in the left panel of Fig. 4 we show the behavior of

*ε*as a function of the mean total energy of the 2-PHAV. To achieve a more complete characterization of this class of states, we also studied the behavior of the nonG amount by varying the balancing between the two components of the 2-PHAV and keeping the mean number of detected photons of the overall state fixed. The lower bound

*ε*of the nonG amount as a function of the ratio between the two components is plotted in the right panel of Fig. 4: as one may expect, it monotonically decreases at increasing the balancing. In fact, the most unbalanced condition reduces to the case in which there is only a single PHAV, whereas the most balanced one corresponds to have a balanced 2-PHAV. In Fig. 4 we note that the experimental values of

*ε*are larger than those expected from theory and the difference increases at high mean numbers (left panel) or at high unbalancing (right panel). We ascribe this behavior to small errors (few percent) in the determination of the mean values of detected photons that can occur at relatively high intensities due to some saturation of the detector. Indeed, the sensitivity of

*ε*to the mean number of photons is testified by the error bars associated to the experimental data. This critical character suggests the exploitation of such nonG measure to test decoy states based on 2-PHAVs [7

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]

## 4. Conclusion

## Acknowledgments

## References and links

1. | H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. |

2. | Y. Zhao, B. Qi, and H.-K. Lo, “Experimental quantum key distribution with active phase randomization,” Appl. Phys. Lett. |

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

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. A |

6. | S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Special Topics |

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. |

8. | A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. |

9. | K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. |

10. | S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A |

11. | K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. |

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. |

13. | M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A |

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. A |

15. | M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. |

16. | M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics,” J. Mod. Opt. |

17. | A. Andreoni and M. Bondani, “Photon statistics in the macroscopic realm measured without photon counters,” Phys. Rev. A |

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. A |

**OCIS Codes**

(230.5160) Optical devices : Photodetectors

(270.0270) Quantum optics : Quantum optics

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: July 20, 2012

Revised Manuscript: September 4, 2012

Manuscript Accepted: September 6, 2012

Published: October 16, 2012

**Citation**

Alessia Allevi, Stefano Olivares, and Maria Bondani, "Manipulating the non-Gaussianity of phase-randomized coherent states," Opt. Express **20**, 24850-24855 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24850

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### References

- H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett.94, 230504 (2005). [CrossRef] [PubMed]
- 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]
- H.-K. Lo and J. Preskill, “Phase randomization improves the security of quantum key distribution,” CALT-68-2556 (2005), arXiv:quant-ph/0504209v1.
- 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]
- S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J. Special Topics203, 3–24 (2012). [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]
- K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev.177, 1882–1902 (1969). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]

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