## Remote state preparation of three-dimensional optical vortices |

Optics Express, Vol. 22, Issue 9, pp. 10898-10905 (2014)

http://dx.doi.org/10.1364/OE.22.010898

Acrobat PDF (1453 KB)

### Abstract

We propose a feasible scheme to remotely prepare three-dimensional (3D) optical vortex lines. Our scheme relies on the complete description of high-dimensional orbital angular momentum (OAM) entanglement in terms of the Laguerre-Gaussian modes. It is theoretically demonstrated that by simply changing the pump beam waist, we can remotely prepare the target photons in the vortex states of 3D interesting morphology, appearing as twisted vortex strands, separated vortex loops, and vortex link or knot. Furthermore, we employ the biphoton Klyshko picture to illustrate the conservation law of the OAM index *ℓ* and the spreading effect of the radial mode index *p*, where the Schmidt numbers are calculated to show the high-dimensional capacity of the quantum channels involved in the present remote state preparation.

© 2014 Optical Society of America

## 1. Introduction

*et al.*[1

1. 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. **70**, 1895–1899 (1993). [CrossRef] [PubMed]

2. C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and W. K. Wootters, “Remote state preparation,” Phys. Rev. Lett. **87**, 077902 (2001). [CrossRef] [PubMed]

3. H.-K. Lo, “Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity,” Phys. Rev. A **62**, 012313 (2000). [CrossRef]

5. S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. **92**, 047903 (2004). [CrossRef] [PubMed]

6. I. Devetak and T. Berger, “Low-entanglement remote state preparation,” Phys. Rev. Lett. **87**, 197901 (2001). [CrossRef] [PubMed]

7. D. W. Berry and B. C. Sanders, “Optimal remote state preparation,” Phys. Rev. Lett. **90**, 057901 (2003). [CrossRef] [PubMed]

8. B. Zeng and P. Zhang, “Remote-state preparation in higher dimension and the parallelizable manifold s^{n−1},” Phys. Rev. A **65**, 022316 (2002). [CrossRef]

9. Z. Kurucz, P. Adam, Z. Kis, and J. Janszky, “Continuous variable remote state preparation,” Phys. Rev. A **72**, 052315 (2005). [CrossRef]

10. N. A. Peters, J. T. Barreiro, M. E. Goggin, T.-C. Wei, and P. G. Kwiat, “Remote state preparation: Arbitrary remote control of photon polarization,” Phys. Rev. Lett. **94**, 150502 (2005). [CrossRef] [PubMed]

11. J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. **105**, 030407 (2010). [CrossRef]

12. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. **53**, 293 (2009). [CrossRef]

*iℓϕ*) possesses a well-defined orbital angular momentum (OAM) of

*ℓh̄*per photon, where

*ϕ*is the azimuthal angle and

*ℓ*is the OAM quantum number [13

13. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*π*. At the vortex center, the phase is singular and the intensity is zero. They occur at points in two-dimensional (2D) fields and along lines in three-dimensional (3D) space. It was theoretically discovered that the vortex lines can be manipulated to form links or knots [14

14. M. Berry and M. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Roy. Soc. A **457**, 2251–2263 (2001). [CrossRef]

15. M. Berry and M. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+ 1 spacetime,” J. Phys. A: Math. Gen. **34**, 8877 (2001). [CrossRef]

16. J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. **7**, 55 (2005). [CrossRef]

18. M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. **6**, 118–121 (2010). [CrossRef]

19. J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. **106**, 100407 (2011). [CrossRef] [PubMed]

## 2. RSP of 3D optical vortices

16. J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. **7**, 55 (2005). [CrossRef]

16. J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. **7**, 55 (2005). [CrossRef]

*a*

_{1}= 1 and

*w*= 1 mm. The link can be formed with

*ℓ*= 2, and the relative amplitudes:

*a*

_{0}= 0.35,

*a*

_{2}= −0.35,

*a*

_{3}= 0.36. In contrast, the knot can be formed with

*ℓ*= 3, and

*a*

_{0}= 0.25,

*a*

_{2}= −0.26,

*a*

_{3}= 0.29. According to Eqs. (2) and (4), we know the remotely prepared state of the target photons can be derived as follows,

*ℓ*= −

_{i}*ℓ*. In contrast, the correlation between

_{s}*p*and

_{s}*p*breaks down in a practical experiment, as the crystal size places an effective upper limit to pump beam waist [21

_{i}21. F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A **83**, 033816 (2011). [CrossRef]

*w*=

_{p}*w*= 1 mm in our case.

*ψ*

_{link}〉 or knot state |

*ψ*

_{knot}〉, if she has a spatial light modulator (SLM), a single-mode fiber (SMF) and a single-photon detector (APD) at hand. Based on the LabVIEW simulation according to Eq. (4), we show in the left panels of Fig. 2 the desired holograms addressed by SLM that are used to make vortex link and knot, and plot in the middle panels their 3D structures, respectively. Having been informed about which hologram, Fig. 2(a) or 2(d), has been used by Alice

*v*ia a classical link, Bob achieves the expected vortex lines encoded in his target photons. Let’s assume that Bob has an intensified CCD (ICCD) camera, which is triggered by the output from Alice’s APD, and can be translated along the beam axis to record the intensity of the beam at different planes. Besides, the ICCD camera has been over-saturated in order to measure the vortex positions within the beam cross section. Then the resulting image contains points of darkness corresponding to the vortices as they intersected the plane of the ICCD [16

**7**, 55 (2005). [CrossRef]

*w*= 1 mm, we present in the right panels of Fig. 2 the numerical simulations of 3D vortex lines. We have plotted the phase files at three transverse planes that the vortex lines intersect at

_{p}*z*= −10 mm,

*z*= 0 mm and

*z*= 10 mm, which are utilized to show conceptually the locations of the vortex points. Around each vortex point, there is a spiral phase change of 2

*π*, and, therefore, giving an individual topological charge

*Q*= 1. One can observe the differences between the topological singularities of the target and trigger photons, as are visualized by Figs. 2(b) and 2(c), Figs. 2(e) and 2(f). This is because the vortex link and knot states in Eq. (4) are the superpositions of only four specific LG modes. But the remotely prepared states of Eq. (5) are represented by the superpositions of many LG modes as a summation over all

*p*, resulted from the spreading effect of

_{i}*p*index. As a result, the vortex link and knot are erased by the superpositions of the spread LG modes for the idler photons. For an easy visualization, we also plotted the corresponding intensity patterns (

*w*= 1 mm) at these observing planes, as shown in Fig. 3. One can see that these vortex points are rotating clockwise along the direction of beam propagation, and the vortex lines are slightly twisted in the 3D space.

_{p}## 3. Biphoton Klyshko picture and the Schmidt number

23. H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A **61**, 062308 (2000). [CrossRef]

24. P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. **108**, 143603 (2012). [CrossRef] [PubMed]

*ρ*=

_{A}*Tr*(|Ψ〉 〈Ψ|) and

_{B}*ρ*=

_{B}*Tr*(|Ψ〉 〈Ψ|) being the reduced density matrices for the two subsystems, respectively. The Schmidt number also characterizes the number of effective entangled modes involved in the entangled state [25

_{A}25. C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. **92**, 127903 (2004). [CrossRef] [PubMed]

26. J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Angular phase-plate analyzers for measuring the dimensionality of multimode fields,” Phys. Rev. A **77**, 033845 (2008). [CrossRef]

*κ*different pump beam waists

*w*, as are listed in Table 1. One can see that in all cases the Schmidt numbers are obviously beyond the two-dimensional thinking with polarization entanglement, which therefore suggests that there may be many entangled spatial modes participating in process of our RSP protocol.

_{p}## 4. Conclusion

*ℓ*and the spreading effect of the radial index

*p*. We have illustrated the quantum channels involved in our RSP protocol with the biphoton Klyshko picture, and calculated the Schmidt numbers to show their high dimensions. Our work may raise the possibility of studying the initial-value problems through the RSP protocol: the non-trivial 3D topological information of optical vortices at Bob’s side are encrypted, non-locally, in the 2D section taken as the holograms at Alice’s side, which may also has the potential in understanding the theory of ghost imaging [27

27. M. D’Angelo and Y. Shih, “Quantum imaging,” Laser Phys. Lett. **2**, 567–596 (2005). [CrossRef]

## Acknowledgments

## References and links

1. | 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. |

2. | C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and W. K. Wootters, “Remote state preparation,” Phys. Rev. Lett. |

3. | H.-K. Lo, “Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity,” Phys. Rev. A |

4. | A. K. Pati, “Minimum classical bit for remote preparation and measurement of a qubit,” Phys. Rev. A |

5. | S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. |

6. | I. Devetak and T. Berger, “Low-entanglement remote state preparation,” Phys. Rev. Lett. |

7. | D. W. Berry and B. C. Sanders, “Optimal remote state preparation,” Phys. Rev. Lett. |

8. | B. Zeng and P. Zhang, “Remote-state preparation in higher dimension and the parallelizable manifold s |

9. | Z. Kurucz, P. Adam, Z. Kis, and J. Janszky, “Continuous variable remote state preparation,” Phys. Rev. A |

10. | N. A. Peters, J. T. Barreiro, M. E. Goggin, T.-C. Wei, and P. G. Kwiat, “Remote state preparation: Arbitrary remote control of photon polarization,” Phys. Rev. Lett. |

11. | J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. |

12. | M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. |

13. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A |

14. | M. Berry and M. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Roy. Soc. A |

15. | M. Berry and M. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+ 1 spacetime,” J. Phys. A: Math. Gen. |

16. | J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. |

17. | J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature |

18. | M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. |

19. | J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. |

20. | J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A |

21. | F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A |

22. | A. M. Yao, “Angular momentum decomposition of entangled photons with an arbitrary pump,” New J. Phys. |

23. | H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A |

24. | P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. |

25. | C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. |

26. | J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Angular phase-plate analyzers for measuring the dimensionality of multimode fields,” Phys. Rev. A |

27. | M. D’Angelo and Y. Shih, “Quantum imaging,” Laser Phys. Lett. |

28. | L. Chen, J. Leach, B. Jack, M. J. Padgett, S. Franke-Arnold, and W. She, “High-dimensional quantum nature of ghost angular Young’s diffraction,” Phys. Rev. A |

29. | L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(050.4865) Diffraction and gratings : Optical vortices

(270.5585) Quantum optics : Quantum information and processing

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: February 19, 2014

Revised Manuscript: April 8, 2014

Manuscript Accepted: April 23, 2014

Published: April 29, 2014

**Citation**

Ming Su and Lixiang Chen, "Remote state preparation of three-dimensional optical vortices," Opt. Express **22**, 10898-10905 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10898

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### References

- C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). [CrossRef] [PubMed]
- C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, W. K. Wootters, “Remote state preparation,” Phys. Rev. Lett. 87, 077902 (2001). [CrossRef] [PubMed]
- H.-K. Lo, “Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity,” Phys. Rev. A 62, 012313 (2000). [CrossRef]
- A. K. Pati, “Minimum classical bit for remote preparation and measurement of a qubit,” Phys. Rev. A 63, 014302 (2000). [CrossRef]
- S. A. Babichev, B. Brezger, A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004). [CrossRef] [PubMed]
- I. Devetak, T. Berger, “Low-entanglement remote state preparation,” Phys. Rev. Lett. 87, 197901 (2001). [CrossRef] [PubMed]
- D. W. Berry, B. C. Sanders, “Optimal remote state preparation,” Phys. Rev. Lett. 90, 057901 (2003). [CrossRef] [PubMed]
- B. Zeng, P. Zhang, “Remote-state preparation in higher dimension and the parallelizable manifold sn−1,” Phys. Rev. A 65, 022316 (2002). [CrossRef]
- Z. Kurucz, P. Adam, Z. Kis, J. Janszky, “Continuous variable remote state preparation,” Phys. Rev. A 72, 052315 (2005). [CrossRef]
- N. A. Peters, J. T. Barreiro, M. E. Goggin, T.-C. Wei, P. G. Kwiat, “Remote state preparation: Arbitrary remote control of photon polarization,” Phys. Rev. Lett. 94, 150502 (2005). [CrossRef] [PubMed]
- J. T. Barreiro, T.-C. Wei, P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105, 030407 (2010). [CrossRef]
- M. R. Dennis, K. O’Holleran, M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293 (2009). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]
- M. Berry, M. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Roy. Soc. A 457, 2251–2263 (2001). [CrossRef]
- M. Berry, M. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+ 1 spacetime,” J. Phys. A: Math. Gen. 34, 8877 (2001). [CrossRef]
- J. Leach, M. Dennis, J. Courtial, M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). [CrossRef]
- J. Leach, M. R. Dennis, J. Courtial, M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432, 165 (2004). [CrossRef] [PubMed]
- M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118–121 (2010). [CrossRef]
- J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106, 100407 (2011). [CrossRef] [PubMed]
- J. P. Torres, Y. Deyanova, L. Torner, G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003). [CrossRef]
- F. M. Miatto, A. M. Yao, S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011). [CrossRef]
- A. M. Yao, “Angular momentum decomposition of entangled photons with an arbitrary pump,” New J. Phys. 13, 053048 (2011). [CrossRef]
- H. Bechmann-Pasquinucci, W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000). [CrossRef]
- P. B. Dixon, G. A. Howland, J. Schneeloch, J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012). [CrossRef] [PubMed]
- C. K. Law, J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004). [CrossRef] [PubMed]
- J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, J. P. Woerdman, “Angular phase-plate analyzers for measuring the dimensionality of multimode fields,” Phys. Rev. A 77, 033845 (2008). [CrossRef]
- M. D’Angelo, Y. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005). [CrossRef]
- L. Chen, J. Leach, B. Jack, M. J. Padgett, S. Franke-Arnold, W. She, “High-dimensional quantum nature of ghost angular Young’s diffraction,” Phys. Rev. A 82, 033822 (2010). [CrossRef]
- L. Chen, J. Lei, J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3, e153 (2014). [CrossRef]

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