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

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  1. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys.77, 513–577 (2005).
    [CrossRef]
  2. U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev.4, 337–354 (2010).
    [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. A67, 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]
  8. G. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A66, 032316 (2002).
    [CrossRef]
  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. A67, 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. A61, 032302 (2000).
    [CrossRef]
  13. 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]
  14. S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A67, 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 Photonics4, 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. A80, 022315 (2009).
    [CrossRef]
  17. J. Fiurášek, “Distillation and purification of symmetric entangled Gaussian states,” Phys. Rev. A82, 042331 (2010).
    [CrossRef]
  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. A84, 012302 (2011).
    [CrossRef]
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    [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. A84, 062309 (2011).
    [CrossRef]
  23. J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A84, 012335 (2011).
    [CrossRef]
  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. A86, 052339 (2012).
    [CrossRef]
  26. G. Vidal and R. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
    [CrossRef]
  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]
  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]
  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. A84, 050301(R) (2011).
    [CrossRef]
  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.105, 053602 (2010).
    [CrossRef] [PubMed]
  32. 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]

2012

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]

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]

O. Černotík and J. Fiurášek, “Displacement-enhanced continuous-variable entanglement concentration,” Phys. Rev. A86, 052339 (2012).
[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]

2011

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, 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]

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]

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]

2010

J. Fiurášek, “Distillation and purification of symmetric entangled Gaussian states,” Phys. Rev. A82, 042331 (2010).
[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]

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev.4, 337–354 (2010).
[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]

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]

2009

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]

2008

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]

2007

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]

2005

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

2003

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]

J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A67, 022304 (2003).
[CrossRef]

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A67, 032314 (2003).
[CrossRef]

2002

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]

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]

G. Vidal and R. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
[CrossRef]

2000

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]

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]

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]

Andersen, U.

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]

Andersen, U. L.

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev.4, 337–354 (2010).
[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]

Benichi, H.

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]

Bonifacio, R.

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A67, 032314 (2003).
[CrossRef]

Braunstein, S. L.

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

Browne, D. E.

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]

Cerf, N.

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]

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]

Cernotík, O.

O. Černotík and J. Fiurášek, “Displacement-enhanced continuous-variable entanglement concentration,” Phys. Rev. A86, 052339 (2012).
[CrossRef]

Cirac, J.

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]

Cirac, J. I.

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]

Cochrane, P. T.

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]

Dantan, A.

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]

DiGuglielmo, J.

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]

Dong, R.

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]

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]

Duan, L.-M.

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]

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]

Eisert, J.

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]

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

Filip, R.

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]

J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A67, 022304 (2003).
[CrossRef]

Fiurášek, J.

O. Černotík and J. Fiurášek, “Displacement-enhanced continuous-variable entanglement concentration,” Phys. Rev. A86, 052339 (2012).
[CrossRef]

J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A84, 012335 (2011).
[CrossRef]

J. Fiurášek, “Distillation and purification of symmetric entangled Gaussian states,” Phys. Rev. A82, 042331 (2010).
[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. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A67, 022304 (2003).
[CrossRef]

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

Franzen, A.

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]

Furusawa, A.

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]

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]

García-Patrón, R.

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]

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]

Giedke, G.

G. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A66, 032316 (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]

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]

Grangier, P.

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]

Hage, B.

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]

Hayasaka, K.

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]

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]

Heersink, J.

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]

Huntington, E.

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]

Ignacio Cirac, J.

G. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A66, 032316 (2002).
[CrossRef]

Ježek, M.

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]

Ji, S.-W.

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]

Johansson, J.

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]

Kim, H.-J.

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]

Kurizki, G.

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]

Laghaout, A.

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]

Lassen, M.

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]

Lee, N.

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]

Lee, S.-Y.

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]

Leuchs, G.

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev.4, 337–354 (2010).
[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]

Li, F.-L.

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]

Lloyd, S.

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]

Lund, A. P.

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]

Marek, P.

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]

Marquardt, C.

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]

Milburn, G. J.

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]

Mišta, L.

J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A67, 022304 (2003).
[CrossRef]

Nation, P.

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]

Navarrete-Benlloch, C.

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]

Neergaard-Nielsen, J. S.

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]

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]

Nha, H.

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]

Nori, F.

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]

Olivares, S.

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A67, 032314 (2003).
[CrossRef]

Opatrný, T.

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]

Ourjoumtsev, A.

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]

Paris, M. G. A.

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A67, 032314 (2003).
[CrossRef]

Pirandola, S.

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]

Plenio, M. B.

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]

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

Pryde, G. J.

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]

Ralph, T. C.

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]

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]

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]

Samblowski, A.

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]

Sasaki, M.

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]

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]

Scheel, S.

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]

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

Schnabel, R.

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]

Shapiro, J.

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]

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]

Silberhorn, C.

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev.4, 337–354 (2010).
[CrossRef]

Simon, R.

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

Takahashi, H.

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]

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]

Takeda, S.

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]

Takeno, Y.

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]

Takeoka, M.

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]

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]

Takeuchi, M.

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]

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]

Tipsmark, A.

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]

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

Tualle-Brouri, R.

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]

van Loock, P.

S. Zhang and P. van Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A84, 062309 (2011).
[CrossRef]

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

Vidal, G.

G. Vidal and R. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
[CrossRef]

Wakui, K.

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]

Walk, N.

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]

Webb, J.

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]

Weedbrook, C.

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]

Welsch, D.-G.

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]

Werner, R.

G. Vidal and R. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
[CrossRef]

Xiang, G. Y.

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]

Yang, Y.

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]

Zhang, S.

S. Zhang and P. van Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A84, 062309 (2011).
[CrossRef]

Zoller, P.

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]

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]

Comput. Phys. Commun.

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]

Laser Photon. Rev.

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev.4, 337–354 (2010).
[CrossRef]

Nat. Phys.

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]

Nature Photon.

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]

Nature Photonics

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]

Phys. Rev. A

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]

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. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A66, 032316 (2002).
[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]

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]

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]

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]

Phys. Rev. Lett.

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]

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. 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]

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

Rev. Mod. Phys.

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

Science

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]

Other

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

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Figures (7)

Fig. 1
Fig. 1

a) Schematic of the proposed setup. S denotes the single mode squeezed state and D(α), D(β) are displacement operations. b) Two equivalent setups for displacement controlled photon subtraction where β = 1 T β and T is the transmission of the beam splitter.

Fig. 2
Fig. 2

(a) Logarithmic negativity as a function of the average number of photons of the initial squeezed state. 1(2)MSV: Single (two) mode squeezed vacuum states used initially. 1(2)PS: Single (two) photon subtraction. D2PS: Displacement based two-photon subtraction with optimal displacement. (b) The logarithmic negativities obtained in the 1MSV D2PS setting for varying displacement amplitudes α = −β. The color scale goes from EN = 0 (blue) to EN = 2.1. The dashed black curves indicate the α values that optimize EN (which results in the entanglement curve in (a)), while the solid line follows the optimal value in the low-squeezing limit of Eq. (5). (c) As (b), but for the 2-mode squeezed vacuum.

Fig. 3
Fig. 3

Distribution of the different eigenstates of the entangled state that is produced from a single squeezed mode. The distributions are for the initial state (before distillation), the single photon subtracted state (1PS), the two photon subtracted state (2PS) and the displacement-enhanced protocol.

Fig. 4
Fig. 4

Two-mode squeezing: Variance of the x-quadrature difference (left), p-quadrature sum (center) and their sum (right) for the same states as studied in Fig. 2(a). See that figure for abbreviations. The gray line at 2 in the right graph indicates the Simon-Duan entanglement criterion for Gaussian states. In these plots, however, only the 1MSV and 2MSV states are Gaussian.

Fig. 5
Fig. 5

(a) The logarithmic negativity is plotted as a function of the loss of the two channels. The attenuation is identical in the two channels. See caption of Fig. 2 for abbreviations. (b) The logarithmic negativity as a function of the displacement amplitude for the two-photon subtracted single- and two-mode squeezed vacuum with losses varying from 0 (top curves) to 0.9 (bottom curves) in steps of 0.15. In both panels, the initial photon number is 0.1.

Fig. 6
Fig. 6

(a) Optimal displacement as a function of channel losses for varying losses. The losses are increasing for lower-lying curves. (b) Maximum attainable logarithmic negativity for varying losses. The losses are the same as for the solid curves in (a). The shaded areas in gray (yellow) designate the ranges of initial photon numbers for which the 1MSV (2MSV) states are superior.

Fig. 7
Fig. 7

Performance of the distillation protocol for realistic photon subtraction. Logarithmic negativity (a) and success probability (b) as a function of the initial average photon number. The tap-off beamsplitter reflectivity for photon subtraction is 5%. See caption of Fig. 2 for abbreviations.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

| ζ 1 = U B S ( 1 γ 1 2 ) 1 / 4 n = 0 ( 2 n ) ! n ! ( γ 1 2 ) n | 2 n | 0 ,
| ζ 1 | 00 + γ 1 2 ( 1 2 | 20 + 1 2 | 11 + 1 2 | 02 ) .
| ζ 1 a ^ A 𝕀 B | ζ 1
= γ 1 2 ( | 10 + | 01 ) ,
| Ψ = ( α β + γ 1 2 ) | 00 + γ 1 2 ( β + α ) | 10 + | 01 2 + γ 1 2 α β ( 1 2 | 20 + 1 2 | 11 + 1 2 | 02 ) + O ( γ 1 2 ) .
α = β = γ 1 2 ,
| Ψ = γ 1 2 2 2 ( 1 2 | 20 + 1 2 | 11 + 1 2 | 02 ) + O ( γ 1 3 ) .
| ζ 2 A B = 1 γ 2 2 n = 0 γ 2 n | n A | n B

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