Abstract

Photon subtraction (PS) is a widely used technique in the distillation of noise-damped continuous-variable Gaussian entangled states. In the literature, all distillation schemes with the PS technique are restricted to pure Gaussian entanglement or amplitude-damped entanglement, and it is not clear whether PS is applicable in overcoming the phase noise or even more general noise. In this paper, we show that the PS technique is versatile and can still be used to distill entanglement that is subject to phase noise and simultaneous phase and amplitude noise.

© 2013 Optical Society of America

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  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]
  2. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
    [CrossRef]
  3. J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (2002).
    [CrossRef]
  4. G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
    [CrossRef]
  5. 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]
  6. H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
    [CrossRef]
  7. S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleporation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314 (2003).
    [CrossRef]
  8. 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 73, 042310 (2006).
    [CrossRef]
  9. S. Zhang and P. van Loock, “Distillation of mixed-state continuous-variable entanglement by photon subtraction,” Phys. Rev. A 82, 062316 (2010).
    [CrossRef]
  10. S. Zhang and P. van Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A 84, 062309 (2011).
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  11. J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A 84, 012335 (2011).
    [CrossRef]
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  13. 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]
  14. J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
    [CrossRef]
  15. A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Experimental demonstration of continuous variable purification of squeezed states,” Phys. Rev. Lett. 97, 150505 (2006).
    [CrossRef]
  16. B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
    [CrossRef]
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  19. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
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  20. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).
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    [CrossRef]
  22. S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
    [CrossRef]
  23. S. Yang, M. Gong, C. Li, X. Zou, and G. Guo, “Optically pumping hole spins in coupled quantum dot molecules into a steady state of high concurrence entanglement,” Phys. Rev. B 80, 235322 (2009).
    [CrossRef]
  24. Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
    [CrossRef]
  25. Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (2008).
    [CrossRef]
  26. Solving partial differential equations using finite element methods, MATLAB, Partial Differential Equation toolbox, http://www.mathworks.cn/products/pde .

2011 (2)

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

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

2010 (2)

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

S. Zhang and P. van Loock, “Distillation of mixed-state continuous-variable entanglement by photon subtraction,” Phys. Rev. A 82, 062316 (2010).
[CrossRef]

2009 (3)

S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
[CrossRef]

S. Yang, M. Gong, C. Li, X. Zou, and G. Guo, “Optically pumping hole spins in coupled quantum dot molecules into a steady state of high concurrence entanglement,” Phys. Rev. B 80, 235322 (2009).
[CrossRef]

Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
[CrossRef]

2008 (2)

Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (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 (1)

B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
[CrossRef]

2006 (3)

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Experimental demonstration of continuous variable purification of squeezed states,” Phys. Rev. Lett. 97, 150505 (2006).
[CrossRef]

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 73, 042310 (2006).
[CrossRef]

2005 (1)

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

2003 (1)

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

2002 (4)

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

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

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

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

2001 (1)

T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
[CrossRef]

2000 (1)

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]

1998 (1)

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

1993 (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]

Andersen, U. L.

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

Bennett, C. H.

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]

Bonifacio, R.

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

Brassard, G.

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]

Braunstein, S. L.

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

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

Chefles, A.

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 73, 042310 (2006).
[CrossRef]

Cirac, J. I.

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

Crépeau, C.

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]

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]

B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
[CrossRef]

A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Experimental demonstration of continuous variable purification of squeezed states,” Phys. Rev. Lett. 97, 150505 (2006).
[CrossRef]

Dong, R.

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

Dong, Y.

Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
[CrossRef]

Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (2008).
[CrossRef]

Eisert, J.

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

Filip, R.

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

Fiurášek, J.

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

B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
[CrossRef]

A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Experimental demonstration of continuous variable purification of squeezed states,” Phys. Rev. Lett. 97, 150505 (2006).
[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]

B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
[CrossRef]

A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Experimental demonstration of continuous variable purification of squeezed states,” Phys. Rev. Lett. 97, 150505 (2006).
[CrossRef]

Furusawa, A.

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

Giedke, G.

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

Gong, M.

S. Yang, M. Gong, C. Li, X. Zou, and G. Guo, “Optically pumping hole spins in coupled quantum dot molecules into a steady state of high concurrence entanglement,” Phys. Rev. B 80, 235322 (2009).
[CrossRef]

Gong, Y.

Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (2008).
[CrossRef]

Guo, G.

S. Yang, M. Gong, C. Li, X. Zou, and G. Guo, “Optically pumping hole spins in coupled quantum dot molecules into a steady state of high concurrence entanglement,” Phys. Rev. B 80, 235322 (2009).
[CrossRef]

S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
[CrossRef]

Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
[CrossRef]

Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (2008).
[CrossRef]

S. Yang, X. Zou, S. Zhang, B.-S. Shi, P. van Loock, and G. Guo, “Multipartite continuous-variable entanglement distillation using local squeezing and only one photon-subtraction operation,” arXiv.org, arXiv:1106.1536.

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]

B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
[CrossRef]

A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Experimental demonstration of continuous variable purification of squeezed states,” Phys. Rev. Lett. 97, 150505 (2006).
[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,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Heersink, J.

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

Hiroshima, T.

T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
[CrossRef]

Huang, Y.

Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (2008).
[CrossRef]

Jin, C.

S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
[CrossRef]

Jozsa, R.

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]

Kimble, H. J.

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

Kitagawa, A.

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 73, 042310 (2006).
[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. A 61, 032302 (2000).
[CrossRef]

Leuchs, G.

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

Li, C.

S. Yang, M. Gong, C. Li, X. Zou, and G. Guo, “Optically pumping hole spins in coupled quantum dot molecules into a steady state of high concurrence entanglement,” Phys. Rev. B 80, 235322 (2009).
[CrossRef]

S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
[CrossRef]

Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
[CrossRef]

Lorenz, S.

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

Marek, P.

B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
[CrossRef]

Marquardt, Ch.

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

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,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Niu, X.

Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (2008).
[CrossRef]

Olivares, S.

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleporation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 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. A 61, 032302 (2000).
[CrossRef]

Paris, M. G. A.

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

Peres, A.

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]

Plenio, M. B.

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

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

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 73, 042310 (2006).
[CrossRef]

Scheel, S.

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

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]

B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
[CrossRef]

A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Experimental demonstration of continuous variable purification of squeezed states,” Phys. Rev. Lett. 97, 150505 (2006).
[CrossRef]

Shi, B.-S.

S. Yang, X. Zou, S. Zhang, B.-S. Shi, P. van Loock, and G. Guo, “Multipartite continuous-variable entanglement distillation using local squeezing and only one photon-subtraction operation,” arXiv.org, arXiv:1106.1536.

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,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Takeoka, M.

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

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 73, 042310 (2006).
[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,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

van Loock, P.

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

S. Zhang and P. van Loock, “Distillation of mixed-state continuous-variable entanglement by photon subtraction,” Phys. Rev. A 82, 062316 (2010).
[CrossRef]

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

S. Yang, X. Zou, S. Zhang, B.-S. Shi, P. van Loock, and G. Guo, “Multipartite continuous-variable entanglement distillation using local squeezing and only one photon-subtraction operation,” arXiv.org, arXiv:1106.1536.

Vidal, G.

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

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

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. A 61, 032302 (2000).
[CrossRef]

Werner, R. F.

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

Wootters, W. K.

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]

Yang, S.

S. Yang, M. Gong, C. Li, X. Zou, and G. Guo, “Optically pumping hole spins in coupled quantum dot molecules into a steady state of high concurrence entanglement,” Phys. Rev. B 80, 235322 (2009).
[CrossRef]

S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
[CrossRef]

Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
[CrossRef]

S. Yang, X. Zou, S. Zhang, B.-S. Shi, P. van Loock, and G. Guo, “Multipartite continuous-variable entanglement distillation using local squeezing and only one photon-subtraction operation,” arXiv.org, arXiv:1106.1536.

Zhang, S.

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

S. Zhang and P. van Loock, “Distillation of mixed-state continuous-variable entanglement by photon subtraction,” Phys. Rev. A 82, 062316 (2010).
[CrossRef]

Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
[CrossRef]

S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
[CrossRef]

S. Yang, X. Zou, S. Zhang, B.-S. Shi, P. van Loock, and G. Guo, “Multipartite continuous-variable entanglement distillation using local squeezing and only one photon-subtraction operation,” arXiv.org, arXiv:1106.1536.

Zhang, Y.

Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (2008).
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Zou, X.

S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
[CrossRef]

Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
[CrossRef]

S. Yang, M. Gong, C. Li, X. Zou, and G. Guo, “Optically pumping hole spins in coupled quantum dot molecules into a steady state of high concurrence entanglement,” Phys. Rev. B 80, 235322 (2009).
[CrossRef]

S. Yang, X. Zou, S. Zhang, B.-S. Shi, P. van Loock, and G. Guo, “Multipartite continuous-variable entanglement distillation using local squeezing and only one photon-subtraction operation,” arXiv.org, arXiv:1106.1536.

J. Mod. Opt. (1)

Y. Dong, X. Zou, S. Zhang, S. Yang, C. Li, and G. Guo, “Cavity-QED-based phase gate for photonic qubits,” J. Mod. Opt. 56, 1230–1233 (2009).
[CrossRef]

Nat. Photonics (1)

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

Nat. Phys. (1)

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).
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New J. Phys. (1)

B. Hage, A. Franzen, J. DiGuglielmo, P. Marek, J. Fiurášek, and R. Schnabel, “On the distillation and purification of phase-diffused squeezed states,” New J. Phys. 9, 227 (2007).
[CrossRef]

Phys. Rev. A (11)

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

T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
[CrossRef]

S. Zhang, X. Zou, S. Yang, C. Li, C. Jin, and G. Guo, “Steady atomic entanglement in cavity QED without state initialization,” Phys. Rev. A 80, 062320 (2009).
[CrossRef]

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

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 73, 042310 (2006).
[CrossRef]

S. Zhang and P. van Loock, “Distillation of mixed-state continuous-variable entanglement by photon subtraction,” Phys. Rev. A 82, 062316 (2010).
[CrossRef]

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

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

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

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]

Y. Gong, Y. Zhang, Y. Dong, X. Niu, Y. Huang, and G. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A 78, 042103 (2008).
[CrossRef]

Phys. Rev. B (1)

S. Yang, M. Gong, C. Li, X. Zou, and G. Guo, “Optically pumping hole spins in coupled quantum dot molecules into a steady state of high concurrence entanglement,” Phys. Rev. B 80, 235322 (2009).
[CrossRef]

Phys. Rev. Lett. (6)

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

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[CrossRef]

A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Experimental demonstration of continuous variable purification of squeezed states,” Phys. Rev. Lett. 97, 150505 (2006).
[CrossRef]

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]

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

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

Rev. Mod. Phys. (1)

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

Other (3)

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

S. Yang, X. Zou, S. Zhang, B.-S. Shi, P. van Loock, and G. Guo, “Multipartite continuous-variable entanglement distillation using local squeezing and only one photon-subtraction operation,” arXiv.org, arXiv:1106.1536.

Solving partial differential equations using finite element methods, MATLAB, Partial Differential Equation toolbox, http://www.mathworks.cn/products/pde .

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Figures (5)

Fig. 1.
Fig. 1.

Entanglement distillation scheme for noise-damped two-mode squeezed state ρ=|ψABψ|. L denotes the noise damping mechanism in each single-side noisy channel. The photon detectors are chosen to be the conventional on–off detectors, i.e., the ones that can only discriminate vacuum (corresponding to zero result) and nonvacuum (nonzero result).

Fig. 2.
Fig. 2.

(a) Entanglement of two-mode squeezing state: pure state |ψAB (line with circles), phase-damped state γp=0.1, t=3 (dotted curve) and PS-distilled state T=0.95 (solid curve). The photon number is truncated at D¯=7, which is sufficiently large for numerical convergence. (b) Entanglement distillation probability Psucc as a function of two-mode squeezing parameter r. (c) Average photon number in both modes A and B. The curve with circles and the dotted curve overlap because of the invariance of NA+NB during phase damping.

Fig. 3.
Fig. 3.

(a) Entanglement, (b) success probability, and (c) average number of photons as a function of time t. Again the line with circles land the dotted line overlap (see text for more information). Here we assume that the squeezing in the initial two-mode entanglement is fixed at r=0.4. Other parameters are phase damping γp=0.1 and photon number truncated at D¯=7.

Fig. 4.
Fig. 4.

(a) Entanglement of two-mode squeezing state: pure state |ψAB (curve with circles), both phase- and amplitude-damped state γa=γp=0.05, (γp+γa)t=0.3 (dotted curve), and PS-distilled state T=0.95 (solid curve). The photon number is truncated at D¯=7, which is sufficiently large for numerical convergence. (b) Entanglement distillation probability Psucc as a function of two-mode squeezing parameter r. (c) Average photon number in both modes A and B.

Fig. 5.
Fig. 5.

(a) Entanglement, (b) success probability, and (c) total average number of photons in both modes for PS-based distillation of a both phase- and amplitude-damped entangled state. Other parameters chosen are T=0.95, r=0.40, D¯=7, γp=γa=0.05.

Equations (18)

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

|ψAB=1cosh(r)n=0tanhn(r)|n|n.
dρdt=(L(γp/2aa)+L(γp/2bb))ρ,
ρmix=(1λ2)n1,n2=0λn1+n2eγpt|n1n2|2|n1,n1n2,n2|,
λ=tanh(r).
EN=log2(2n=0λ2ne4n2γpt[1+λe(4n+1)γpt]1).
ρ˜AB=TrCD[ρABCDIABConCon]/Psucc
Psucc=Tr[ρABCDIABConDon],
ρABCD=UACUBD[ρmix|00CD00|]UACUBD.
ρ˜ABΓA=K=0i,j=0KCij(K)|i,Kij,Kj|,
Cij(K)=(T1T)Kcosh2(r)Psuccp=p0[λ(1T)]2p+i+jeγpt(ij)2·(p+ii)(p+iKj)(p+jj)(p+jKi),
p0=max(Kij+1,1).
NA+NB=Tr[(aaI+Ibb)|ψABψ|]=sinh2(r).
ρmix=(1λ2)(|0000|+λeγpt|0011|+λeγpt|1100|+λ2|1111|+λ3eγpt|1122|+λ3eγpt|2211|+λ4|2222|+O(λ5)),
dρdt=(L(γp/2aa)+L(γp/2bb)+La)ρ,
ρHD¯HD¯ν⃗RD¯4.
Aρ(IA)ν⃗,
ρB(BτI)ν⃗,
AρB(BτI)(IA)ν⃗=(BτA)ν⃗.

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