## Generating arbitrary photon-number entangled states for continuous-variable quantum informatics |

Optics Express, Vol. 20, Issue 13, pp. 14221-14233 (2012)

http://dx.doi.org/10.1364/OE.20.014221

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### Abstract

We propose two experimental schemes that can produce an arbitrary photon-number entangled state (PNES) in a finite dimension. This class of entangled states naturally includes non-Gaussian continuous-variable (CV) states that may provide some practical advantages over the Gaussian counterparts (two-mode squeezed states). We particularly compare the entanglement characteristics of the Gaussian and the non-Gaussian states in view of the degree of entanglement and the Einstein-Podolsky-Rosen correlation, and further discuss their applications to the CV teleportation and the nonlocality test. The experimental imperfection due to the on-off photodetectors with nonideal efficiency is also considered in our analysis to show the feasibility of our schemes within existing technologies.

© 2012 OSA

## 1. Introduction

11. T. Kiesel, W. Vogel, and B. Hage, “Entangled qubits in a non-Gaussian quantum state,” Phys. Rev. A **83**, 062319 (2011). [CrossRef]

12. E. S. Gomez, W. A. T. Nogueira, C. H. Monken, and G. Lima, “Quantifying the non-Gaussianity of the state of spatially correlated down-converted photons,” Opt. Express **20**, 3753–3772 (2012). [CrossRef] [PubMed]

13. T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A **61**, 032302 (2000). [CrossRef]

21. A. Kitagawa, M. Takeoka, K. Wakui, and M. Sasaki, “Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to the dense coding scheme,” Phys. Rev. A **72**, 022334 (2005). [CrossRef]

22. S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. **82**, 1784– 1787 (1999). [CrossRef]

25. R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. **93**, 130409 (2004). [CrossRef] [PubMed]

26. M. Allegra, P. Giorda, and M. G. A. Paris, “Role of initial entanglement and non-Gaussianity in the decoherence of photon-number entangled states evolving in a noisy channel,” Phys. Rev. Lett. **105**, 100503 (2010). [CrossRef] [PubMed]

27. M. Allegra, P. Giorda, and M. G. A. Paris, “Decoherence of Gaussian and nonGaussian photon-number entangled states in a noisy channel,” Int. J. Quant. Inf. **9**, 27–38 (2011). [CrossRef]

28. K. K. Sabapathy, J. S. Ivan, and R. Simon, “Robustness of non-Gaussian entanglement against noisy amplifier and attenuator environments,” Phys. Rev. Lett. **107**, 130501 (2011). [CrossRef] [PubMed]

30. H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett. **108**, 030503 (2012). [CrossRef] [PubMed]

31. H. Nha, G.-J. Milburn, and H. J. Carmichael, “Linear amplification and quantum cloning for non-Gaussian continuous variables,” New J. Phys. **12**, 103010 (2010). [CrossRef]

*n*〉 denotes a Fock state basis. One particular example is the TMSS with the coefficients

*C*=

_{n}*λ*(1 −

^{n}*λ*

^{2})

^{1/2}(

*λ*: squeezing parameter,

*N*→ ∞), which is the only Gaussian state among the photon-number entangled states. Another example is the pair-coherent state given by

*C*∼

_{n}*ζ*/

^{n}*n*! [32

32. G. S. Agarwal, “Generation of pair coherent states and squeezing via the competition of four-wave mixing and amplified spontaneous emission,” Phys. Rev. Lett. **57**, 827–830 (1986). [CrossRef] [PubMed]

33. C. C. Gerry, J. Mimih, and R. Birrittella, “State-projective scheme for generating pair coherent states in traveling-wave optical fields,” Phys. Rev. A **84**, 023810 (2011). [CrossRef]

34. A. Gábris and G. S. Agarwal, “Quantuem teleportation with pair-coherent states,” Int. J. Quantum Inf. **5**, 305–309 (2007). [CrossRef]

35. C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A **82**, 013831 (2010). [CrossRef]

36. A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett. **80**, 3169–3172 (1998). [CrossRef]

37. S. Daffer and P. L. Knight, “Generating optimal states for a homodyne Bell test,” Phys. Rev. A **72**, 034101 (2005). [CrossRef]

39. J. Wenger, M. Hafezi, F. Grosshans, R. Tualle-Brouri, and P. Grangier, “Maximal violation of Bell inequalities using continuous-variable measurements,” Phys. Rev. A **67**, 012105 (2003). [CrossRef]

*C*can be controlled with beam splitting and squeezing parameters. Both of our proposed schemes make use of coherent superposition operations in single-photon interferometic settings that erase the which-path information on the realized photonic operations. The first scheme employs the second-order superposition operation

_{n}*tââ*

^{†}+

*râ*

^{†}

*â*, which has been recently proposed and experimentally implemented in the context of proving bosonic commutation relation [

*â*,

*â*

^{†}]= 1 [40

40. M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. **101**, 260401 (2008). [CrossRef]

43. H.-J. Kim, J. Park, and H.-W. Lee, “Cavity-QED based scheme for realization of photon annihilation and creation operations and their combinations,” J. Opt. Soc. Am. B **27**, 464–475 (2010). [CrossRef]

*tââ*

^{†}+

*râ*

^{†}

*â*was also discussed in the context of noiseless quantum amplifier [44

44. A. Zavatta, J. Fiurasek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics **5**, 52 (2011) [CrossRef]

*tâ*+

*rb*̂

^{†}. Its single-mode version

*tâ*+

*râ*

^{†}was recently proposed for a quantum state engineering [45

45. S.-Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A **82**, 053812 (2010). [CrossRef]

20. S.-Y. Lee, S.-W. Ji, H.-J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A **84**, 012302 (2011). [CrossRef]

46. J. Park, S.-Y. Lee, H.-W. Lee, and H. Nha, “Enhanced Bell violation by a coherent superposition of photon subtraction and addition,” J. Opt. Soc. Am. B **29**, 906–911 (2012). [CrossRef]

8. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. **80**, 869– 872 (1998). [CrossRef]

47. K. Banaszek and K. Wódkiewicz, “Nonlocality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A **58**, 4345–4347 (1998). [CrossRef]

48. K. Banaszek and K. Wódkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. **82**, 2009–2013 (1999). [CrossRef]

*C*

_{0}|0〉

*|0〉*

_{a}*+*

_{b}*C*

_{1}|1〉

*|1〉*

_{a}*, can survive longer under noisy environments than the TMSS with the same degree of entanglement or energy [28*

_{b}28. K. K. Sabapathy, J. S. Ivan, and R. Simon, “Robustness of non-Gaussian entanglement against noisy amplifier and attenuator environments,” Phys. Rev. Lett. **107**, 130501 (2011). [CrossRef] [PubMed]

29. J. Lee, M. S. Kim, and H. Nha, “Comment on “Role of initial entanglement and non-Gaussianity in the decoherence of photon-number entangled states evolving in a noisy channel”,” Phys. Rev. Lett. **107**, 238901 (2011). [CrossRef] [PubMed]

*C*

_{0}|0〉

*|0〉*

_{a}*+*

_{b}*C*

_{1}|1〉

*|1〉*

_{a}*+*

_{b}*C*

_{2}|2〉

*|2〉*

_{a}*, considering realistic experimental conditions. In Sec. 6, our results are summarized.*

_{b}## 2. Entanglement and EPR correlation

*, the degree of entanglement can be quantified by the von Neumann entropy*

_{AB}*E*(

*ρ*)= −Tr

_{A}*[*

_{A}*ρ*log

_{A}_{2}

*ρ*] for the reduced density operator

_{A}*ρ*= Tr

_{A}*[|Ψ〉*

_{B}*〈Ψ|*

_{AB}*]. For the class of photon-number entangled states*

_{AB}*C*are identical. Thus, for the case of TMSS with

_{n}*C*=

_{n}*λ*(1 −

^{n}*λ*

^{2})

^{1/2}, the state can have an infinite degree of entanglement with infinite squeezing, i.e.

*λ*= 1, which is practically impossible to achieve. On the other hand, the finite-dimensional PNES can match or even surpass a finitely-squeezed TMSS as an entangled resource. In Fig. 1(a), we plot the degree of entanglement for the TMSS (blue solid) as a function of the squeezing parameter

*s*= tanh

^{−1}

*λ*. This is compared with the maximal possible entanglement for the PNES with equal coefficients (

*C*

_{1}=

*C*

_{2}= ⋯ =

*C*) of dimensions

_{N}*N*= 1 (red dotted), 2 (red dashed), and 10 (red dot-dashed). The degrees of entanglement for the PNESs are given by 1, 1.585 and 3.459, respectively. To achieve such degrees of entanglement, the squeezing of the TMSS should be

*s*= 0.5185 (4.506

*dB*),

*s*= 0.7335 (6.374

*dB*) and

*s*= 1.391 (12.09

*dB*), respectively. In the pulsed-regime generation of squeezed states, the level of squeezing currently available from an optical parametric amplifier is

*s*= 0.403 (3.5

*dB*) [49

49. A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. **98**, 030502 (2007). [CrossRef] [PubMed]

50. A recent experiment achieved a higher-squeezing level ∼6.8dB of a pulsed light at the wavelength *λ* =1500nm in optical fiber [51]. For a long-distance quantum communication, however, one may require a quantum memory to store the quantum state of light. For this purpose, alkali atoms have been employed with the wavelength range *λ* ∼800nm, e.g. [52]. Furthermore, the thermal photon noise that can be detrimental to the quantum nature of light usually increases with the wavelength, so we here compare the PNES with the pulsed squeezed light of *λ* =850nm reported in [49].

*N*= 1 can already surpass the entanglement of the TMSS.

^{2}(

*x*̂

*−*

_{A}*x*̂

*)+Δ*

_{B}^{2}(

*p*̂

*+*

_{A}*p*̂

*). Here*

_{B}*j*=

*A*,

*B*) are the quadrature amplitudes of the field that can be measured in homodyne detection. The value of EPR below 2 represents the quantum correlation between the quadrature amplitudes of two modes. In Fig. 1(b), the EPR correlations of the PNESs for the dimensions

*N*= 1, 2, and 10 are 1.172, 0.8315, and 0.2516, respectively. The corresponding levels of squeezing for the TMSS are given by

*s*= 0.2674 (2.324

*dB*),

*s*= 0.4388 (3.813

*dB*) and

*s*= 1.037 (9.008

*dB*). Thus the PNES with

*N*≥ 2 can surpass the currently available TMSS (

*s*= 0.403) in view of the EPR correlation.

## 3. Applications: CV teleportation and nonlocality test

8. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. **80**, 869– 872 (1998). [CrossRef]

47. K. Banaszek and K. Wódkiewicz, “Nonlocality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A **58**, 4345–4347 (1998). [CrossRef]

48. K. Banaszek and K. Wódkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. **82**, 2009–2013 (1999). [CrossRef]

**(i)**The teleportation fidelity in the Braunstein-Kimble (BK) scheme [8

8. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. **80**, 869– 872 (1998). [CrossRef]

*C*

_{out}(

*λ*)=

*C*

_{in}(

*λ*)

*C*(

_{E}*λ*

^{*},

*λ*) [53

53. P. Marian and T. A. Marian, “Continuous-variable teleportation in the characteristic-function description,” Phys. Rev. A **74**, 042306 (2006). [CrossRef]

*C*(

_{E}*λ*

^{*},

*λ*) is the characteristic function of a two-mode entangled state. We consider the finite-dimensional PNES

*N*= 1,2 and 3 as an entangled resource. For instance, the characteristic function of the PNES for

*N*= 2 is given by

*C*

_{0}|

^{2}+ |

*C*

_{1}|

^{2}+ |

*C*

_{2}|

^{2}= 1. For the case of teleporting an arbitrary coherent-state input, we find, by optimizing the fidelity (1) using Eq. (2), that the average fidelity can be achieved up to

*F*= 0.7334 at the choice of

*C*

_{0}≈ 0.765,

*C*

_{1}≈ 0.535 and

*C*

_{2}≈ 0.359.

*N*= 1, 2 and 3 corresponds to the fidelity via the TMSS with the squeezing parameters

*s*= 0.320 (2.776

*dB*),

*s*= 0.506 (4.397

*dB*), and

*s*= 0.638 (5.548

*dB*), respectively. Thus, the PNES at

*N*= 2 can surpass the fidelity via the TMSS with the currently available squeezing in the pulsed regime, i.e.,

*s*≈ 0.403 (3.5

*dB*) [53

53. P. Marian and T. A. Marian, “Continuous-variable teleportation in the characteristic-function description,” Phys. Rev. A **74**, 042306 (2006). [CrossRef]

*N*= 2) for CV teleportation.

**(ii)**We next consider the nonlocality test by Banaszek and Wódkiewicz that is addressed in phase space using the two-mode Wigner function [47

47. K. Banaszek and K. Wódkiewicz, “Nonlocality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A **58**, 4345–4347 (1998). [CrossRef]

48. K. Banaszek and K. Wódkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. **82**, 2009–2013 (1999). [CrossRef]

*W*(

*α*,

*β*) is the two-mode Wigner function. We find that for the PNES,

*N*= 2, the Bell inequality can be violated up to

*B*

_{BW}= 2.32088 with the coefficients

*C*

_{0}≈ 0.589,

*C*

_{1}≈ 0.700 and

*C*

_{2}≈ 0.404. In Fig. 2(b), we see that this degree of nonlocality almost reaches the level for the TMSS with infinite squeezing [54

54. H. Jeong, W. Son, M. S. Kim, D. Ahn, and C. Brukner, “Quantum nonlocality test for continuous-variable states with dichotomic observables,” Phys. Rev. A **67**, 012106 (2003). [CrossRef]

*N*= 3 surpasses the value

*B*

_{BW}of the TMSS in the entire region of squeezing. We can again achieve such degree of Bell violation using the weak squeezing regime in our schemes (Sec. 4).

## 4. Experimental schemes

**(i)**We first consider the operation

*tââ*

^{†}+

*râ*

^{†}

*â*acting on a single-mode

*a*, which is the coherent superposition of two product operations—photon addition followed by subtraction (

*ââ*

^{†}) and photon subtraction followed by addition (

*â*

^{†}

*â*). This coherent operation was experimentally implemented to prove the bosonic commutation relation [

*â*,

*â*

^{†}]= 1 [40

40. M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. **101**, 260401 (2008). [CrossRef]

41. A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. **103**, 140406 (2009). [CrossRef] [PubMed]

*ââ*

^{†}−

*â*

^{†}

*â*, we adopt an arbitrarily weighted superposition of the two operations, i.e.,

*tââ*

^{†}+

*râ*

^{†}

*â*. In particular, we show that the single-mode operation

*tââ*

^{†}+

*râ*

^{†}

*â*together with two-mode squeezing operations

*Ŝ*(

_{ab}*ξ*)= exp(−

*ξâ*

^{†}

*b*̂

^{†}+

*ξ*

^{*}

*b*̂

*â*) can constitute an essential building block to generate an arbitrary PNES.

*Ŝ*(

_{ab}*ξ*), the coherent operation

*tââ*

^{†}+

*râ*

^{†}

*â*and the inverse squeezing

*t*|

_{n}^{2}+ |

*r*|

_{n}^{2}= 1. In the above Eq. (4), the identity

*ξ*≡

*s*exp(

*iφ*) [55].

*|0〉*

_{a}*is injected as an input,*

_{b}*Ô*

_{1}yields a superposition of number states as

*Ô*

_{1}|0〉

*|0〉*

_{a}*= cosh*

_{b}^{2}

*s*

_{1}[(

*t*

_{1}+

*r*

_{1}tanh

^{2}

*s*

_{1})|0〉

*|0〉*

_{a}*− exp(*

_{b}*iφ*

_{1})(

*t*

_{1}+

*r*

_{1})tanh

*s*

_{1}|1〉

*|1〉*

_{a}*]. In principle, a succession of*

_{b}*Ô*applied on the vacuum states,

_{n}*s*,

_{n}*r*,

_{n}*t*and

_{n}*φ*. For instance, the state

_{n}*Ô*

_{1}|0〉

*|0〉*

_{a}*∼*

_{b}*C*

_{0}|0〉

*|0〉*

_{a}*+*

_{b}*C*

_{1}|1〉

*|1〉*

_{a}*can have a larger proportion of |1〉*

_{b}*|1〉*

_{a}*, i.e. |*

_{b}*C*

_{0}| < |

*C*

_{1}|, under the condition

*r*

_{1}tanh

*s*

_{1}>

*t*

_{1}. For comparison, if one instead applies the original quantum scissor scheme on the TMSS that projects an input onto the subspace spanned by |0〉 and |1〉 [56

56. D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical state truncation by projection synthesis,” Phys. Rev. Lett. **81**, 1604–1606 (1998). [CrossRef]

*|0〉*

_{a}*+*

_{b}*λ*|1〉

*|1〉*

_{a}*. That is, the vacuum state |0〉*

_{b}*|0〉*

_{a}*is always more weighted than the single-photon state |1〉*

_{b}*|1〉*

_{a}*. On the other hand, a generalized scissor scheme proposed for the noiseless quantum amplifier [57*

_{b}57. G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics **4**, 316–319 (2010). [CrossRef]

*Ô*can be experimentally realized as depicted in Fig. 3(a). An input state |

_{n}*ψ*〉

*is first injected into a nondegenerate parametric amplifier (NDPA) with coupling parameter*

_{ab}*ξ*, and then into the beam splitter BS1 (transmittance:

_{n}*T*

_{1}≈ 1) with the other input mode

*c*in a vacuum. This can be described by Next the mode

*a*is further injected into another NDPA with small coupling

*s*≪ 1 and the output is kept only under the condition of single-photon detection at PD0. That is, which is then injected into the BS2 (

*T*

_{2}≈ 1),

*= |0〉*

_{cd}*|0〉*

_{c}*. The next beam splitter BS3 making the transformations*

_{d}*ĉ*→

*t*+

_{n}ĉ*r*̂ and

_{n}d*d*̂ →

*t*̂ −

_{n}d*r*gives On detecting a single photon at PD1 (PD2) and no photon at PD2 (PD1), the state is projected to (

_{n}ĉ*tââ*

^{†}+

*râ*

^{†}

*â*)

*Ŝ*(

_{ab}*ξ*)|

_{n}*ψ*〉

*, where*

_{ab}*ξ*yields

_{n}**(ii)**Second, we show that the sequence of two first-order coherent superposition operations, (t

_{2n}

*â*+ r

_{2n}

*b*̂

^{†})(t

_{2n−1}

*b*̂ + r

_{2n−1}

*â*

^{†}), can also yield an operation similar to

*Ô*in Eq. (4). A similar type of coherent operation was previously investigated in a form acting on a single-mode,

_{n}*tâ*+

*râ*

^{†}, which is the superposition of photon subtraction and addition [45

45. S.-Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A **82**, 053812 (2010). [CrossRef]

*nonlocal*coherent superposition acting on two modes,

*tâ*+

*rb*̂

^{†}(

*tb*̂ +

*râ*

^{†}).

_{2n−1}|

^{2}+|r

_{2n−1}|

^{2}= 1 and |t

_{2n}|

^{2}+|r

_{2n}|

^{2}= 1. Given a vacuum state as an input,

*Ô*′

_{1}yields a superposition of number states as

*Ô*′

_{1}|0〉

*|0〉*

_{a}*= r*

_{b}_{1}(t

_{2}|0〉

*|0〉*

_{a}*+ r*

_{b}_{2}|1〉

*|1〉*

_{a}*). Furthermore, a succession of*

_{b}*Ô*′

*, i.e.,*

_{n}_{2n−1}, t

_{2n−1}, r

_{2n}and t

_{2n}. Here the coefficients can be readily controlled only by the beam-splitter parameters as shown below.

*Ô*′

*can be implemented as depicted in Fig. 4. First, an arbitrary two-mode state |*

_{n}*ψ*〉

*is injected into an NDPA with small coupling s*

_{ab}_{1}≪ 1 and a BS1 with high transmissivity T

_{1}≈ 1, with mode

*a*(

*b*) into NDPA (BS1). The other input modes to the NDPAs and the BSs are all in vacuum states. Then, the BS3 (transmissivity: t

_{2n−1}) yielding the transformations

*ĉ*

^{†}→ t

_{2n−1}

*ĉ*

^{†}+ r

_{2n−1}

*d*̂

^{†}and

*d*̂

^{†}

*→*t

_{2n−1}

*d*̂

^{†}− r

_{2n−1}

*ĉ*

^{†}gives the output With the detection of single photon at PD1 (PD2) and no photon at PD2 (PD1), we see from Eq. (11) that the state is projected to |Φ〉

_{ab}*≡*(t′

_{2n−1}

*b*̂ + r′

_{2n−1}

*â*

^{†})|

*ψ*〉

*, where*

_{ab}_{2n−1}∼−s

_{1}r

_{2n−1}(−s

_{1}t

_{2n−1}). Next, the output state is further injected into another NDPA with small coupling s

_{2}≪ 1 and a BS2 with high transmissivity T

_{2}≈ 1, with mode a (b) into BS2 (NDPA). Finally, a beam splitter BS4 (transmissivity: t

_{2n}) yielding the transformations

*ê*

^{†}→ t

_{2n}

*ê*

^{†}+ r

_{2n}

*f*̂

^{†}and

*f*̂

^{†}→ t

_{2n}

*f*̂

^{†}− r

_{2n}

*ê*

^{†}gives Once again, with the detection of single photon at PD3 (PD4) and no photon at PD4 (PD3), we see from Eq. (12) that the state is projected to (t′

_{2n}

*â*+ r′

_{2n}

*b*̂

^{†})(t′

_{2n−1}

*b*̂ + r′

_{2n−1}

*â*

^{†})|

*ψ*〉

*, where*

_{ab}_{2n}∼−s

_{2}t

_{2n}(s

_{2}r

_{2n}).

**(iii)**In Sec. 3, we have seen that the optimal PNES

*N*= 2 for CV teleportation has the coefficients

*C*

_{0}≈ 0.765,

*C*

_{1}≈ 0.535, and

*C*

_{2}≈ 0.359. Under our first scheme, these coefficients can be obtained using the experimental parameters, e.g.

*s*

_{1}=

*s*

_{2}= 0.1,

*ϕ*

_{1}= 0,

*ϕ*

_{2}=

*π*,

*r*

_{1}≈ 0.4589, and

*r*

_{2}≈ 0.9984, with

_{2}≈ 0.3, r

_{3}≈ 0.3863 and r

_{4}≈ 0.6193, with

*s*= 0.1 (0.869

*dB*). Furthermore, in the second scheme, we can obtain the same PNES only by adjusting the beam-splitter parameters, therefore, a high-level of squeezing is not necessary in our schemes. This is also true for the case of nonlocality test shown in Sec. 3. Under the first scheme, the optimized coefficients of the PNES for the nonlocality test are obtained using the parameters

*s*

_{1}=

*s*

_{2}= 0.1,

*ϕ*

_{1}= 0,

*ϕ*

_{2}=

*π*,

*r*

_{1}≈ 0.38 and

*r*

_{2}≈ 0.999, with

_{2}≈ 0.3, r

_{3}≈ 0.391 and r

_{4}≈ 0.221, with

## 5. Experimental feasibility

*C*

_{0}|0〉

*|0〉*

_{a}*+*

_{b}*C*

_{1}|1〉

*|1〉*

_{a}*, or two-photon correlation,*

_{b}*C*

_{0}|0〉

*|0〉*

_{a}*+*

_{b}*C*

_{1}|1〉

*|1〉*

_{a}*+*

_{b}*C*

_{2}|2〉

*|2〉*

_{a}*, as examples. In the two schemes of Figs. 3 and 4, we particularly consider each photodetector as an on-off detector that only distingushes two events, detection and non-detection, with efficiency*

_{b}*η*. The photodetection can then be characterized by a positive operator-valued measure (POVM) [58

58. D. Mogilevtsev, “Diagonal element inference by direct detection,” Opt. Commun. **156**, 307–310 (1998). [CrossRef]

60. A. R. Rossi, S. Olivares, and M. G. A. Paris, “Photon statistics without counting photons,” Phys. Rev. A **70**, 055801 (2004). [CrossRef]

_{1}=

*Î*− Π̂

_{0}(click).

*tââ*

^{†}+

*râ*

^{†}

*â*heralded by nonideal on-off detectors, and the inverse two-mode squeezing. This sequence yields an output state where

*ρ*

_{in}≡ |

*ψ*〉〈

*ψ*|

*⊗ |0〉〈0|*

_{ab}*and*

_{cde}*N*= 1,2) and the corresponding output state,

*N*= 1,2). In Fig. 5, we first show the case of PNES up to one-photon correlation, where the fidelity

*F*

_{1}(blue dot) is plotted as a function of |

*C*

_{0}|

^{2}with the detector efficiency

*η*= 0.66. With the parameters

*s*

_{1}= 0.1 and

*C*

_{0}|

^{2}, with the detector efficiency

*η*= 0.66 currently available [61

61. D. Achilles, C. Silberhorn, C. Œliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. **28**, 2387–2389 (2003). [CrossRef] [PubMed]

63. G. Brida, M. Genovese, M. Gramegna, M. G. A. Paris, E. Predazzi, and E. Cagliero, “On the reconstruction of diagonal elements of density matrix of quantum optical states by on/off detectors,” Open Syst. Inf. Dyn. **13**, 333–341 (2006). [CrossRef]

57. G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics **4**, 316–319 (2010). [CrossRef]

*s*= 0.1) and the on-off detectors (

*η*= 0.66). We see that our schemes yield a slightly better fidelity than the scissor scheme.

*F*

_{2}for the case of PNES up to two-photon correlation as a function of |

*C*

_{1}|

^{2}and |

*C*

_{2}|

^{2}. With the same parameters (

*η*= 0.66,

*s*

_{1}= 0.1 and

*C*

_{1}|

^{2}and |

*C*

_{2}|

^{2}. For both of the cases, the fidelity slightly increases with the vacuum-state probability |

*C*

_{0}|

^{2}, as the weak coupling (

*s*

_{1}= 0.1) of the NDPA makes a low photon-number state better controlled.

_{2}

*â*+ r

_{2}

*b*̂

^{†})(t

_{1}

*b*̂ + r

_{1}

*â*

^{†}) heralded by nonideal on-off detectors are sequentially applied to an arbitrary input state. This yields an output state where

*ρ*′

_{in}≡ |

*ψ*〉〈

*ψ*|

*⊗ |0〉〈0|*

_{ab}*and*

_{cdef}*Û*

_{2}

*≡ B*̂

*̂*

_{ef}B*̂*

_{ae}Ŝ_{bf}B*̂*

_{cd}B*. We investigate the fidelity*

_{bd}Ŝ_{ac}*N*= 1, 2), and the corresponding output state,

*N*= 1,2). In Fig. 5, we plot the fidelity

*F*′

_{1}(red square) as a function of |

*C*

_{0}|

^{2}with the detector efficiency

*η*= 0.66. With the parameters s

_{1}= s

_{2}= 0.1 and

*C*

_{0}|

^{2}. In Fig. 6, we investigate the fidelity

*F*′

_{2}as a function of |

*C*

_{1}|

^{2}and |

*C*

_{2}|

^{2}. With the same parameters (

*η*= 0.66, s

_{1}= s

_{2}= 0.1 and

*C*

_{1}|

^{2}and |

*C*

_{2}|

^{2}. Therefore, both of our schemes seem to make an output state at a very high fidelity even with nonideal on-off detectors used for heralding the conditional generation of the PNES.

*C*

_{0}|0〉

*|0〉*

_{a}*+*

_{b}*C*

_{1}|1〉

*|1〉*

_{a}*, the first scheme, with the condition of*

_{b}*s*

_{1}= 0.1,

*s*= 0.1 and

^{−6}(|

*C*

_{0}|

^{2}= 1/2) to 10

^{−4}, which increases with the coefficient |

*C*

_{0}|. On the other hand, the second scheme, with the condition of s

_{1}= s

_{2}= 0.1 and

^{−4}. The success probability can of course be made larger by using higher-squeezing NDPAs in each scheme at the expense of output fidelity to some extent.

64. A. Tipsmark, R. Dong, A. Laghaout, P. Marek, M. Jezek, and U. L. Andersen, “Experimental demonstration of a Hadamard gate for coherent state qubits,” Phys. Rev. A **84**, 050301(R) (2011). [CrossRef]

*t*of beam-splitter transmissivity. Compared with Fig. 6, it turns out that a high output fidelity is still achievable and that the second scheme is particularly insensitive to the beam-splitter error.

## 6. Summary

26. M. Allegra, P. Giorda, and M. G. A. Paris, “Role of initial entanglement and non-Gaussianity in the decoherence of photon-number entangled states evolving in a noisy channel,” Phys. Rev. Lett. **105**, 100503 (2010). [CrossRef] [PubMed]

28. K. K. Sabapathy, J. S. Ivan, and R. Simon, “Robustness of non-Gaussian entanglement against noisy amplifier and attenuator environments,” Phys. Rev. Lett. **107**, 130501 (2011). [CrossRef] [PubMed]

29. J. Lee, M. S. Kim, and H. Nha, “Comment on “Role of initial entanglement and non-Gaussianity in the decoherence of photon-number entangled states evolving in a noisy channel”,” Phys. Rev. Lett. **107**, 238901 (2011). [CrossRef] [PubMed]

## Acknowledgments

## References and links

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2. | J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics |

3. | J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. |

4. | M. Genovese, “Research on hidden variable theories: a review of recent progresses,” Phys. Rep. |

5. | P. W. Shor, “Algorithms for quantum computer computation: discrete logarithms and factoring,” in |

6. | C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. |

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10. | S. Takeda, H. Benichi, T. Mizuta, N. Lee, J. Yoshikawa, and A. Furusawa, “Quantum mode filtering of non-Gaussian states for teleportation-based quantum information processing,” arXiv:1202.2418. |

11. | T. Kiesel, W. Vogel, and B. Hage, “Entangled qubits in a non-Gaussian quantum state,” Phys. Rev. A |

12. | E. S. Gomez, W. A. T. Nogueira, C. H. Monken, and G. Lima, “Quantifying the non-Gaussianity of the state of spatially correlated down-converted photons,” Opt. Express |

13. | T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A |

14. | P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by conditional measurements on the two-mode squeezed vacuum,” Phys. Rev. A |

15. | S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A |

16. | A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A |

17. | Y. Yang and F.-L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A |

18. | F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A |

19. | F. Dell’Anno, S. De Siena, and F. Illuminati, “Realistic continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A |

20. | S.-Y. Lee, S.-W. Ji, H.-J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A |

21. | A. Kitagawa, M. Takeoka, K. Wakui, and M. Sasaki, “Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to the dense coding scheme,” Phys. Rev. A |

22. | S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. |

23. | S. D. Bartlett and B. C. Sanders, “Efficient classical simulation of optical quantum information circuits,” Phys. Rev. Lett. |

24. | H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. |

25. | R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. |

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27. | M. Allegra, P. Giorda, and M. G. A. Paris, “Decoherence of Gaussian and nonGaussian photon-number entangled states in a noisy channel,” Int. J. Quant. Inf. |

28. | K. K. Sabapathy, J. S. Ivan, and R. Simon, “Robustness of non-Gaussian entanglement against noisy amplifier and attenuator environments,” Phys. Rev. Lett. |

29. | J. Lee, M. S. Kim, and H. Nha, “Comment on “Role of initial entanglement and non-Gaussianity in the decoherence of photon-number entangled states evolving in a noisy channel”,” Phys. Rev. Lett. |

30. | H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett. |

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32. | G. S. Agarwal, “Generation of pair coherent states and squeezing via the competition of four-wave mixing and amplified spontaneous emission,” Phys. Rev. Lett. |

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34. | A. Gábris and G. S. Agarwal, “Quantuem teleportation with pair-coherent states,” Int. J. Quantum Inf. |

35. | C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A |

36. | A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett. |

37. | S. Daffer and P. L. Knight, “Generating optimal states for a homodyne Bell test,” Phys. Rev. A |

38. | W. J. Munro, “Optimal states for Bell-inequality violations using quadrature-phase homodyne measurements,” Phys. Rev. A |

39. | J. Wenger, M. Hafezi, F. Grosshans, R. Tualle-Brouri, and P. Grangier, “Maximal violation of Bell inequalities using continuous-variable measurements,” Phys. Rev. A |

40. | M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. |

41. | A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. |

42. | J. Park, S.-Y. Lee, H.-J. Kim, and H.-W. Lee, “Cavity-QED-based scheme for verification of the photon commutation relation,” New J. Phys. |

43. | H.-J. Kim, J. Park, and H.-W. Lee, “Cavity-QED based scheme for realization of photon annihilation and creation operations and their combinations,” J. Opt. Soc. Am. B |

44. | A. Zavatta, J. Fiurasek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics |

45. | S.-Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A |

46. | J. Park, S.-Y. Lee, H.-W. Lee, and H. Nha, “Enhanced Bell violation by a coherent superposition of photon subtraction and addition,” J. Opt. Soc. Am. B |

47. | K. Banaszek and K. Wódkiewicz, “Nonlocality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A |

48. | K. Banaszek and K. Wódkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. |

49. | A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. |

50. | A recent experiment achieved a higher-squeezing level ∼6.8dB of a pulsed light at the wavelength |

51. | R. Dong, J. Heersink, J. F. Corney, P. D. Drummond, U. L. Andersen, and G. Leuchs, “Experimental evidence for Raman-induced limits to efficient squeezing in optical fibers,” Opt. Lett. |

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53. | P. Marian and T. A. Marian, “Continuous-variable teleportation in the characteristic-function description,” Phys. Rev. A |

54. | H. Jeong, W. Son, M. S. Kim, D. Ahn, and C. Brukner, “Quantum nonlocality test for continuous-variable states with dichotomic observables,” Phys. Rev. A |

55. | S. M. Barnett and P. M. Radmore, |

56. | D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical state truncation by projection synthesis,” Phys. Rev. Lett. |

57. | G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics |

58. | D. Mogilevtsev, “Diagonal element inference by direct detection,” Opt. Commun. |

59. | D. Mogilevtsev, “Reconstruction of quantum states with binary detectors,” Acta Phys. Slov. |

60. | A. R. Rossi, S. Olivares, and M. G. A. Paris, “Photon statistics without counting photons,” Phys. Rev. A |

61. | D. Achilles, C. Silberhorn, C. Œliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. |

62. | 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 |

63. | G. Brida, M. Genovese, M. Gramegna, M. G. A. Paris, E. Predazzi, and E. Cagliero, “On the reconstruction of diagonal elements of density matrix of quantum optical states by on/off detectors,” Open Syst. Inf. Dyn. |

64. | A. Tipsmark, R. Dong, A. Laghaout, P. Marek, M. Jezek, and U. L. Andersen, “Experimental demonstration of a Hadamard gate for coherent state qubits,” Phys. Rev. A |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.6570) Quantum optics : Squeezed states

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: April 27, 2012

Revised Manuscript: May 31, 2012

Manuscript Accepted: May 31, 2012

Published: June 12, 2012

**Citation**

Su-Yong Lee, Jiyong Park, Hai-Woong Lee, and Hyunchul Nha, "Generating arbitrary photon-number entangled states for continuous-variable quantum informatics," Opt. Express **20**, 14221-14233 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14221

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### References

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- M. Allegra, P. Giorda, and M. G. A. Paris, “Decoherence of Gaussian and nonGaussian photon-number entangled states in a noisy channel,” Int. J. Quant. Inf.9, 27–38 (2011). [CrossRef]
- K. K. Sabapathy, J. S. Ivan, and R. Simon, “Robustness of non-Gaussian entanglement against noisy amplifier and attenuator environments,” Phys. Rev. Lett.107, 130501 (2011). [CrossRef] [PubMed]
- J. Lee, M. S. Kim, and H. Nha, “Comment on “Role of initial entanglement and non-Gaussianity in the decoherence of photon-number entangled states evolving in a noisy channel”,” Phys. Rev. Lett.107, 238901 (2011). [CrossRef] [PubMed]
- H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett.108, 030503 (2012). [CrossRef] [PubMed]
- H. Nha, G.-J. Milburn, and H. J. Carmichael, “Linear amplification and quantum cloning for non-Gaussian continuous variables,” New J. Phys.12, 103010 (2010). [CrossRef]
- G. S. Agarwal, “Generation of pair coherent states and squeezing via the competition of four-wave mixing and amplified spontaneous emission,” Phys. Rev. Lett.57, 827–830 (1986). [CrossRef] [PubMed]
- C. C. Gerry, J. Mimih, and R. Birrittella, “State-projective scheme for generating pair coherent states in traveling-wave optical fields,” Phys. Rev. A84, 023810 (2011). [CrossRef]
- A. Gábris and G. S. Agarwal, “Quantuem teleportation with pair-coherent states,” Int. J. Quantum Inf.5, 305–309 (2007). [CrossRef]
- C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A82, 013831 (2010). [CrossRef]
- A. Gilchrist, P. Deuar, and M. D. Reid, “Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements,” Phys. Rev. Lett.80, 3169–3172 (1998). [CrossRef]
- S. Daffer and P. L. Knight, “Generating optimal states for a homodyne Bell test,” Phys. Rev. A72, 034101 (2005). [CrossRef]
- W. J. Munro, “Optimal states for Bell-inequality violations using quadrature-phase homodyne measurements,” Phys. Rev. A59, 4197–4201 (1999). [CrossRef]
- J. Wenger, M. Hafezi, F. Grosshans, R. Tualle-Brouri, and P. Grangier, “Maximal violation of Bell inequalities using continuous-variable measurements,” Phys. Rev. A67, 012105 (2003). [CrossRef]
- M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett.101, 260401 (2008). [CrossRef]
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- H.-J. Kim, J. Park, and H.-W. Lee, “Cavity-QED based scheme for realization of photon annihilation and creation operations and their combinations,” J. Opt. Soc. Am. B27, 464–475 (2010). [CrossRef]
- A. Zavatta, J. Fiurasek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics5, 52 (2011) [CrossRef]
- S.-Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A82, 053812 (2010). [CrossRef]
- J. Park, S.-Y. Lee, H.-W. Lee, and H. Nha, “Enhanced Bell violation by a coherent superposition of photon subtraction and addition,” J. Opt. Soc. Am. B29, 906–911 (2012). [CrossRef]
- K. Banaszek and K. Wódkiewicz, “Nonlocality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A58, 4345–4347 (1998). [CrossRef]
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