## Displacement-enhanced entanglement distillation of single-mode-squeezed entangled states |

Optics Express, Vol. 21, Issue 6, pp. 6670-6680 (2013)

http://dx.doi.org/10.1364/OE.21.006670

Acrobat PDF (1790 KB)

### Abstract

It has been shown that entanglement distillation of Gaussian entangled states by means of local photon subtraction can be improved by local Gaussian transformations. Here we show that a similar effect can be expected for the distillation of an asymmetric Gaussian entangled state that is produced by a single squeezed beam. We show that for low initial entanglement, our largely simplified protocol generates more entanglement than previous proposed protocols. Furthermore, we show that the distillation scheme also works efficiently on decohered entangled states as well as with a practical photon subtraction setup.

© 2013 OSA

1. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. **77**, 513–577 (2005) [CrossRef] .

3. C. Weedbrook, S. Pirandola, R. García-Patrón, N. Cerf, T. C. Ralph, J. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. **84**, 621–669 (2012) [CrossRef] .

4. D. E. Browne, J. Eisert, M. B. Plenio, and S. Scheel, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A **67**, 062320 (2003) [CrossRef] .

5. R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. **4**, 919–923 (2008) [CrossRef] .

6. B. Hage, A. Samblowski, J. DiGuglielmo, A. Franzen, J. Fiurášek, and R. Schnabel, “Preparation of distilled and purified continuous-variable entangled states,” Nat. Phys. **4**, 915–918 (2008) [CrossRef] .

7. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. **89**, 137903 (2002) [CrossRef] [PubMed] .

9. J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. **89**, 137904 (2002) [CrossRef] .

10. L.-M. Duan, G. Giedke, J. Cirac, and P. Zoller, “Entanglement purification of Gaussian continuous variable quantum states,” Phys. Rev. Lett. **84**, 4002–4005 (2000) [CrossRef] [PubMed] .

11. J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A **67**, 022304 (2003) [CrossRef] .

12. 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] .

14. S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A **67**, 032314 (2003) [CrossRef] .

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

16. 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 **80**, 022315 (2009) [CrossRef] .

19. C. Navarrete-Benlloch, R. García-Patrón, J. Shapiro, and N. Cerf, “Enhancing quantum entanglement by photon addition and subtraction,” Phys. Rev. A **86**, 012328 (2012) [CrossRef] .

12. 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] .

20. H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nature Photon. **4**, 178–181 (2010) [CrossRef] .

21. 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] .

22. S. Zhang and P. van Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A **84**, 062309 (2011) [CrossRef] .

23. J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A **84**, 012335 (2011) [CrossRef] .

20. H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nature Photon. **4**, 178–181 (2010) [CrossRef] .

25. O. Černotík and J. Fiurášek, “Displacement-enhanced continuous-variable entanglement concentration,” Phys. Rev. A **86**, 052339 (2012) [CrossRef] .

*U*is the beam splitter unitary transformation,

_{BS}*γ*

_{1}= tanh(

*s*

_{1}) and

*s*

_{1}is the squeezing parameter. In the weak squeezing limit (

*γ*≪ 1) the state can be truncated to The entanglement of a pure state like this is usually quantified by the entropy of entanglement; however, since we will later compare with mixed states for which the entropy is not a proper measure, we adopt instead as our entanglement measure the logarithmic negativity (LN). The LN is defined as the binary logarithm of the trace norm of the partially transposed density matrix (|

*ζ*

_{1}〉 〈

*ζ*

_{1}|)

^{TA}[26

26. G. Vidal and R. Werner, “Computable measure of entanglement,” Phys. Rev. A **65**, 032314 (2002) [CrossRef] .

*a*, the result is which is maximally entangled (in the two-dimensional subspace, thus neglecting higher order terms). The resulting entanglement is plotted in Fig. 2(a) with the black dash-dotted line, and it is shown that for very low squeezing degrees the LN reaches the maximum value of 1 for a 2D Hilbert space. However, the state should be described in a larger Hilbert space in which the state is not maximally entangled. We next consider the simultaneous subtraction of single photons at both sites as described by the operator

_{A}*â*⊗

_{A}*â*and denoted 2PS (2-Photon-Subtraction). The resulting LN is plotted in Fig. 2(a) by the black dashed line. In contrast to the previous protocol (with a single photon being subtracted, 1PS), this protocol is not very effective for low average photon numbers but for higher numbers (larger than about 0.21, corresponding to 3.9 dB of squeezing) it becomes more effective.

_{B}*α*and

*β*are the complex excitations of the displacements. By implementing these displacements in mode A and B prior to and after the two photon subtractions (see Fig. 1(b)), the state reads It is clear from this expression that the vacuum and single-photon contributions, which prevent the state from being strongly entangled, can be removed by setting which can be shown to give This state contains an entanglement of

*E*= 1.54 for low initial squeezing degrees. This is not maximally entangled in the 3-dimensional Hilbert space due to the unequal weights but it is close to - the maximally entangled state would have

_{N}*E*= 1.585.

_{N}*α*= −

*β*. The resulting optimized LN is plotted in Fig. 2(a) (both by black solid curves). Clearly, the pre-Gaussian processing improves the entanglement for all average photon numbers [24]. From Fig. 2(b) we furthermore see that the entanglement is highly sensitive to the exact value of the displacement amplitude for small initial photon numbers, but less so for increasing photon numbers.

23. J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A **84**, 012335 (2011) [CrossRef] .

*γ*

_{2}= tanh(

*s*

_{2}) and

*s*

_{2}is the two-mode squeezing parameter. This state, which has an average photon number

*n*〉,

*γ*

_{2}is smaller than

*γ*

_{1}. By displacing the state with excitations of

*γ*

_{2}≪ 1). This state is also maximally entangled in the two-dimensional sub space for low initial squeezing as was the case for the 1PS single mode squeezed state in Eq. (4). For larger initial squeezing levels, the optimal displacement is lower than

27. J. Johansson, P. Nation, and F. Nori, “QuTiP: An open-source Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. **183**, 1760–1772 (2012) [CrossRef] .

*adds*non-local photons to the final, heralded state. This is of course only possible because the photon subtraction is a highly probabilistic process – see the discussion on success probabilities later. With the two-mode squeezed vacuum as initial state, a delocalized photon is also obtained in the weak squeezing limit with displacement and two-photon subtraction, giving a large entanglement. For the single-photon subtraction, however, no entanglement is obtained – in stark contrast to the 1MSV case. The origin of this difference is the different form of the initial states: For 1MSV, the state in Eq. (2) is a delocalized 2-photon state (plus vacuum), which results in a delocalized single photon after subtraction. For 2MSV, Eq. (7) is a twin-photon state (plus vacuum), which turns into a

*localized*single photon upon detection in one of the modes.

28. R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. **84**, 2726–2729 (2000) [CrossRef] [PubMed] .

29. L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. **84**, 2722–2725 (2000) [CrossRef] [PubMed] .

*x*-quadrature difference and

*p*-quadrature sum of the two modes (the two-mode squeezing) is below the corresponding vacuum noise level,

*x*

_{−}=

*x*−

_{A}*x*,

_{B}*p*

_{+}=

*p*+

_{A}*p*. This criterion does not apply to non-Gaussian states, but it may still be relevant to see how the distilled states evaluate on this variance metric. For continuous variable quantum teleportation, for example, the performance is determined by the amount two-mode squeezing. We therefore plot the two-mode squeezing and its

_{B}*x*and

*p*components in Fig. 4. One can see the simpler structure of the single-mode squeezed state in that

4. D. E. Browne, J. Eisert, M. B. Plenio, and S. Scheel, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A **67**, 062320 (2003) [CrossRef] .

*n*〉 = 0.1 as a function of the channel attenuation. It is assumed that the two channels possess identical attenuations. Of course, the LN decrease with losses, but it appears that the displacement-distilled single-mode squeezed vacuum is more fragile to attenuation than the two-mode squeezing. While it is the most strongly entangled state in the ideal case, the exposure to losses soon makes it less entangled than the two-photon subtracted two-mode squeezing, both with and without displacement. Moreover, as the losses increase further, a point is reached where it is no longer advantageous to do displacement before the photon subtraction – the kink on the curve around 55% loss. This behaviour is not observed for the 2MSV input.

*α*for a range of different losses. The trends are qualitatively different for the 1MSV and 2MSV states. Whereas the 2MSV states have distinct optima that gradually tend towards zero as losses increase, the location of the peaks of the 1MSV curves go more slowly towards smaller amplitudes. On the other hand, for high losses the height of the peaks dip below the values for zero displacement – this is where the kink in the 1MSV D2PS curve in (a) comes from. Consequently, we discover that this curve in fact consists of two separate regimes: To the left of the kink on the low-loss side, the entangled states have been distilled

*with*displacement, while on the high-loss side, the distillation took place

*without*displacement. We would therefore expect the states to be of very different character, for example in terms of their quadrature variance – referring to Fig. 4(c), the two different regimes are, respectively, above and below the vacuum level – and this is indeed the case (not displayed here).

*increases*its average photon number, so the displacement (which is experimentally implemented by admixture of a coherent state) leads to a

*lower*increase in the photon number of mode B after the subtraction in mode A. Furthermore, the displaced photon subtraction results in a state which has a small displacement in phase space (as opposed to the zero-mean of the initial state). The subsequent displacement of mode B before the photon detection there is opposite in direction of the state’s displacement, leading to destructive interference in the detected mode. As a result of these two effects, the success probability of the second photon detection is considerably lower with displacement than without, outweighing the increased probability of the first detection.

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

32. N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light.” Science **332**, 330–3 (2011) [CrossRef] [PubMed] .

## Acknowledgments

## References and links

1. | S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. |

2. | U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. |

3. | C. Weedbrook, S. Pirandola, R. García-Patrón, N. Cerf, T. C. Ralph, J. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. |

4. | D. E. Browne, J. Eisert, M. B. Plenio, and S. Scheel, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A |

5. | R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. |

6. | B. Hage, A. Samblowski, J. DiGuglielmo, A. Franzen, J. Fiurášek, and R. Schnabel, “Preparation of distilled and purified continuous-variable entangled states,” Nat. Phys. |

7. | J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. |

8. | G. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A |

9. | J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. |

10. | L.-M. Duan, G. Giedke, J. Cirac, and P. Zoller, “Entanglement purification of Gaussian continuous variable quantum states,” Phys. Rev. Lett. |

11. | J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A |

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

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

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

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

16. | 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 |

17. | J. Fiurášek, “Distillation and purification of symmetric entangled Gaussian states,” Phys. Rev. A |

18. | 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 |

19. | C. Navarrete-Benlloch, R. García-Patrón, J. Shapiro, and N. Cerf, “Enhancing quantum entanglement by photon addition and subtraction,” Phys. Rev. A |

20. | H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nature Photon. |

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

22. | S. Zhang and P. van Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A |

23. | J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A |

24. | A. Tipsmark, “Generation of optical coherent state superpositions for quantum information processing,” Phd thesis, Technical University of Denmark (2012). |

25. | O. Černotík and J. Fiurášek, “Displacement-enhanced continuous-variable entanglement concentration,” Phys. Rev. A |

26. | G. Vidal and R. Werner, “Computable measure of entanglement,” Phys. Rev. A |

27. | J. Johansson, P. Nation, and F. Nori, “QuTiP: An open-source Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. |

28. | R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. |

29. | L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. |

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

31. | J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Optical continuous-variable qubit,” Phys. Rev. Lett. |

32. | N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light.” Science |

**OCIS Codes**

(000.6800) General : Theoretical physics

(270.5290) Quantum optics : Photon statistics

(270.6570) Quantum optics : Squeezed states

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: January 2, 2013

Revised Manuscript: March 1, 2013

Manuscript Accepted: March 2, 2013

Published: March 11, 2013

**Citation**

Anders Tipsmark, Jonas S. Neergaard-Nielsen, and Ulrik L. Andersen, "Displacement-enhanced entanglement distillation of single-mode-squeezed entangled states," Opt. Express **21**, 6670-6680 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-6670

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### References

- S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys.77, 513–577 (2005). [CrossRef]
- U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev.4, 337–354 (2010). [CrossRef]
- C. Weedbrook, S. Pirandola, R. García-Patrón, N. Cerf, T. C. Ralph, J. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys.84, 621–669 (2012). [CrossRef]
- D. E. Browne, J. Eisert, M. B. Plenio, and S. Scheel, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A67, 062320 (2003). [CrossRef]
- R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys.4, 919–923 (2008). [CrossRef]
- B. Hage, A. Samblowski, J. DiGuglielmo, A. Franzen, J. Fiurášek, and R. Schnabel, “Preparation of distilled and purified continuous-variable entangled states,” Nat. Phys.4, 915–918 (2008). [CrossRef]
- J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett.89, 137903 (2002). [CrossRef] [PubMed]
- G. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A66, 032316 (2002). [CrossRef]
- J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett.89, 137904 (2002). [CrossRef]
- L.-M. Duan, G. Giedke, J. Cirac, and P. Zoller, “Entanglement purification of Gaussian continuous variable quantum states,” Phys. Rev. Lett.84, 4002–4005 (2000). [CrossRef] [PubMed]
- J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A67, 022304 (2003). [CrossRef]
- T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A61, 032302 (2000). [CrossRef]
- P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by conditional measurements on the two-mode squeezed vacuum,” Phys. Rev. A65, 062306 (2002). [CrossRef]
- S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A67, 032314 (2003). [CrossRef]
- G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nature Photonics4, 316–319 (2010). [CrossRef]
- 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. A80, 022315 (2009). [CrossRef]
- J. Fiurášek, “Distillation and purification of symmetric entangled Gaussian states,” Phys. Rev. A82, 042331 (2010). [CrossRef]
- 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. A84, 012302 (2011). [CrossRef]
- C. Navarrete-Benlloch, R. García-Patrón, J. Shapiro, and N. Cerf, “Enhancing quantum entanglement by photon addition and subtraction,” Phys. Rev. A86, 012328 (2012). [CrossRef]
- H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nature Photon.4, 178–181 (2010). [CrossRef]
- 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]
- S. Zhang and P. van Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A84, 062309 (2011). [CrossRef]
- J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A84, 012335 (2011). [CrossRef]
- A. Tipsmark, “Generation of optical coherent state superpositions for quantum information processing,” Phd thesis, Technical University of Denmark (2012).
- O. Černotík and J. Fiurášek, “Displacement-enhanced continuous-variable entanglement concentration,” Phys. Rev. A86, 052339 (2012). [CrossRef]
- G. Vidal and R. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002). [CrossRef]
- J. Johansson, P. Nation, and F. Nori, “QuTiP: An open-source Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun.183, 1760–1772 (2012). [CrossRef]
- R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett.84, 2726–2729 (2000). [CrossRef] [PubMed]
- L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett.84, 2722–2725 (2000). [CrossRef] [PubMed]
- A. Tipsmark, R. Dong, A. Laghaout, P. Marek, M. Ježek, and U. Andersen, “Experimental demonstration of a Hadamard gate for coherent state qubits,” Phys. Rev. A84, 050301(R) (2011). [CrossRef]
- J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Optical continuous-variable qubit,” Phys. Rev. Lett.105, 053602 (2010). [CrossRef] [PubMed]
- N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light.” Science332, 330–3 (2011). [CrossRef] [PubMed]

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