Abstract

We show that the coherent superposition ta^+ra^ of photon subtraction and addition applied to each local mode of a two-mode entangled state can enhance the nonlocality manifested by the violation of a Bell inequality. A two-mode squeezed state is used as an input state for this demonstration with four different Bell inequalities employed: Bell inequalities adopting displaced parity operator, pseudospin operator, homodyne measurement, and conditional entropy, respectively. We find that the coherent operation significantly enhances the nonlocality remarkably in the weak squeezing limit, compared with other possible non-Gaussian operations. It can also give a maximal Bell violation with a very small squeezing for the inequalities with pseudospin operator and conditional entropy.

© 2012 Optical Society of America

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  1. M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001 (2008).
    [CrossRef]
  2. M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184 (1997).
    [CrossRef]
  3. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
    [CrossRef]
  4. See also P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
    [CrossRef]
  5. H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
    [CrossRef]
  6. R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and Ph. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
    [CrossRef]
  7. S. Olivares and M. G. A. Paris, “Enhancement of nonlocality in phase space,” Phys. Rev. A 70, 032112 (2004).
    [CrossRef]
  8. C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
    [CrossRef]
  9. H. Jeong, “Testing Bell inequalities with photon-subtracted Gaussian states,” Phys. Rev. A 78, 042101 (2008).
    [CrossRef]
  10. H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photon. 4, 178–181 (2010).
    [CrossRef]
  11. S. L. Zhang and P. van Loock, “Distillation of mixed-state continuous-variable entanglement by photon subtraction,” Phys. Rev. A 82, 062316 (2010).
    [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]
  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 65, 062306 (2002).
    [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. F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
    [CrossRef]
  16. G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
    [CrossRef]
  17. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
    [CrossRef]
  18. A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
    [CrossRef]
  19. 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]
  20. A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photon. 5, 52–60 (2011).
    [CrossRef]
  21. G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485 (1992).
    [CrossRef]
  22. V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing the quantum commutation rules through cavity QED,” Science 317, 1890 (2007).
    [CrossRef]
  23. Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Phys. Rev. A 78, 043801 (2008).
    [CrossRef]
  24. Y. Yang and F. L. Li, “Nonclassicality of photon-subtracted and photon-added-then-subtracted Gaussian states,” J. Opt. Soc. Am. B 26, 830–835 (2009).
    [CrossRef]
  25. J. Lee, J. Kim, and H. Nha, “Demonstrating higher-order nonclassical effects by photon-added classical states: realistic schemes,” J. Opt. Soc. Am. B 26, 1363–1369 (2009).
    [CrossRef]
  26. S.-Y. Lee, J. Park, S.-W. Ji, C. H. R. Ooi, and H.-W. Lee, “Nonclassicality generated by photon annihilation-then-creation and creation-then-annihilation operations,” J. Opt. Soc. Am. B 26, 1532–1537 (2009).
    [CrossRef]
  27. M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
    [CrossRef]
  28. J. Park, S.-Y. Lee, H.-J. Kim, and H.-W. Lee, “Cavity-QED-based scheme for verification of the photon commutation relation,” New J. Phys. 12, 033019 (2010).
    [CrossRef]
  29. H.-J. Kim, J. Park, and H.-W. Lee, “Cavity-QED based scheme for realization of photon annihilation and creation operations and their combinations,” J. Opt. Soc. Am. B 27, 464–475 (2010).
    [CrossRef]
  30. A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
    [CrossRef]
  31. S.-Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 82, 053812 (2010).
    [CrossRef]
  32. S.-Y. Lee, S.-W. Ji, H.-J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84, 012302 (2011).
    [CrossRef]
  33. K. Banaszek and K. Wódkiewicz, “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. 82, 2009–2013 (1999).
    [CrossRef]
  34. Z.-B. Chen, J.-W. Pan, G. Hou, and Y.-D. Zhang, “Maximal violation of Bells inequalities for continuous variable systems,” Phys. Rev. Lett. 88, 040406 (2002).
    [CrossRef]
  35. S. L. Braunstein and C. M. Caves, “Information-theoretic Bell inequalities,” Phys. Rev. Lett. 61, 662–665 (1988).
    [CrossRef]
  36. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
    [CrossRef]

2011

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photon. 5, 52–60 (2011).
[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. A 84, 012302 (2011).
[CrossRef]

2010

J. Park, S.-Y. Lee, H.-J. Kim, and H.-W. Lee, “Cavity-QED-based scheme for verification of the photon commutation relation,” New J. Phys. 12, 033019 (2010).
[CrossRef]

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

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

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

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

2009

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

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]

Y. Yang and F. L. Li, “Nonclassicality of photon-subtracted and photon-added-then-subtracted Gaussian states,” J. Opt. Soc. Am. B 26, 830–835 (2009).
[CrossRef]

J. Lee, J. Kim, and H. Nha, “Demonstrating higher-order nonclassical effects by photon-added classical states: realistic schemes,” J. Opt. Soc. Am. B 26, 1363–1369 (2009).
[CrossRef]

S.-Y. Lee, J. Park, S.-W. Ji, C. H. R. Ooi, and H.-W. Lee, “Nonclassicality generated by photon annihilation-then-creation and creation-then-annihilation operations,” J. Opt. Soc. Am. B 26, 1532–1537 (2009).
[CrossRef]

2008

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

H. Jeong, “Testing Bell inequalities with photon-subtracted Gaussian states,” Phys. Rev. A 78, 042101 (2008).
[CrossRef]

See also P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001 (2008).
[CrossRef]

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

2007

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing the quantum commutation rules through cavity QED,” Science 317, 1890 (2007).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

2006

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

2005

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

2004

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[CrossRef]

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

S. Olivares and M. G. A. Paris, “Enhancement of nonlocality in phase space,” Phys. Rev. A 70, 032112 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

2003

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 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. A 65, 062306 (2002).
[CrossRef]

Z.-B. Chen, J.-W. Pan, G. Hou, and Y.-D. Zhang, “Maximal violation of Bells inequalities for continuous variable systems,” Phys. Rev. Lett. 88, 040406 (2002).
[CrossRef]

2000

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]

1999

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

1997

M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184 (1997).
[CrossRef]

1992

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485 (1992).
[CrossRef]

1991

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

1988

S. L. Braunstein and C. M. Caves, “Information-theoretic Bell inequalities,” Phys. Rev. Lett. 61, 662–665 (1988).
[CrossRef]

1969

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485 (1992).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

Al-Amri, M.

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

Albano, L.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Anhut, T.

M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184 (1997).
[CrossRef]

Banaszek, K.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

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

Bellini, M.

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photon. 5, 52–60 (2011).
[CrossRef]

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

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

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing the quantum commutation rules through cavity QED,” Science 317, 1890 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Bonifacio, R.

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

Braunstein, S. L.

S. L. Braunstein and C. M. Caves, “Information-theoretic Bell inequalities,” Phys. Rev. Lett. 61, 662–665 (1988).
[CrossRef]

Carmichael, H. J.

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[CrossRef]

Caves, C. M.

S. L. Braunstein and C. M. Caves, “Information-theoretic Bell inequalities,” Phys. Rev. Lett. 61, 662–665 (1988).
[CrossRef]

Cerf, N. J.

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

Chen, Z.-B.

Z.-B. Chen, J.-W. Pan, G. Hou, and Y.-D. Zhang, “Maximal violation of Bells inequalities for continuous variable systems,” Phys. Rev. Lett. 88, 040406 (2002).
[CrossRef]

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

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. A 65, 062306 (2002).
[CrossRef]

Dakna, M.

M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184 (1997).
[CrossRef]

De Siena, S.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Dell’Anno, F.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Fiurášek, J.

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photon. 5, 52–60 (2011).
[CrossRef]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and Ph. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[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. Photon. 4, 178–181 (2010).
[CrossRef]

García-Patrón, R.

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

Grangier, P.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

Grangier, Ph.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and Ph. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[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. Photon. 4, 178–181 (2010).
[CrossRef]

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Hou, G.

Z.-B. Chen, J.-W. Pan, G. Hou, and Y.-D. Zhang, “Maximal violation of Bells inequalities for continuous variable systems,” Phys. Rev. Lett. 88, 040406 (2002).
[CrossRef]

Illuminati, F.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Invernizzi, C.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

Jeong, H.

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

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

H. Jeong, “Testing Bell inequalities with photon-subtracted Gaussian states,” Phys. Rev. A 78, 042101 (2008).
[CrossRef]

See also P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[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. A 84, 012302 (2011).
[CrossRef]

S.-Y. Lee, J. Park, S.-W. Ji, C. H. R. Ooi, and H.-W. Lee, “Nonclassicality generated by photon annihilation-then-creation and creation-then-annihilation operations,” J. Opt. Soc. Am. B 26, 1532–1537 (2009).
[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. A 84, 012302 (2011).
[CrossRef]

J. Park, S.-Y. Lee, H.-J. Kim, and H.-W. Lee, “Cavity-QED-based scheme for verification of the photon commutation relation,” New J. Phys. 12, 033019 (2010).
[CrossRef]

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

Kim, J.

Kim, M. S.

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

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

See also P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001 (2008).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing the quantum commutation rules through cavity QED,” Science 317, 1890 (2007).
[CrossRef]

Knöll, L.

M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184 (1997).
[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]

Laurat, J.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

Lee, H.-W.

Lee, J.

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. A 84, 012302 (2011).
[CrossRef]

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

J. Park, S.-Y. Lee, H.-J. Kim, and H.-W. Lee, “Cavity-QED-based scheme for verification of the photon commutation relation,” New J. Phys. 12, 033019 (2010).
[CrossRef]

S.-Y. Lee, J. Park, S.-W. Ji, C. H. R. Ooi, and H.-W. Lee, “Nonclassicality generated by photon annihilation-then-creation and creation-then-annihilation operations,” J. Opt. Soc. Am. B 26, 1532–1537 (2009).
[CrossRef]

Li, F. L.

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. A 80, 022315 (2009).
[CrossRef]

Marek, P.

See also P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (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. A 65, 062306 (2002).
[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,” Nat. Photon. 4, 178–181 (2010).
[CrossRef]

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. A 84, 012302 (2011).
[CrossRef]

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

J. Lee, J. Kim, and H. Nha, “Demonstrating higher-order nonclassical effects by photon-added classical states: realistic schemes,” J. Opt. Soc. Am. B 26, 1363–1369 (2009).
[CrossRef]

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[CrossRef]

Olivares, S.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

S. Olivares and M. G. A. Paris, “Enhancement of nonlocality in phase space,” Phys. Rev. A 70, 032112 (2004).
[CrossRef]

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

Ooi, C. H. R.

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]

M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184 (1997).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

Pan, J.-W.

Z.-B. Chen, J.-W. Pan, G. Hou, and Y.-D. Zhang, “Maximal violation of Bells inequalities for continuous variable systems,” Phys. Rev. Lett. 88, 040406 (2002).
[CrossRef]

Parigi, V.

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

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

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing the quantum commutation rules through cavity QED,” Science 317, 1890 (2007).
[CrossRef]

Paris, M. G. A.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

S. Olivares and M. G. A. Paris, “Enhancement of nonlocality in phase space,” Phys. Rev. A 70, 032112 (2004).
[CrossRef]

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

Park, J.

Ralph, T. C.

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by conditional measurements on the two-mode squeezed vacuum,” Phys. Rev. A 65, 062306 (2002).
[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. Photon. 4, 178–181 (2010).
[CrossRef]

Shimony, A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Sun, Q.

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

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

Tara, K.

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485 (1992).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

Tualle-Brouri, R.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

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

van Loock, P.

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

Viciani, S.

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[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. A 61, 032302 (2000).
[CrossRef]

M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184 (1997).
[CrossRef]

Wenger, J.

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

Wódkiewicz, K.

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

Yang, Y.

Y. Yang and F. L. Li, “Nonclassicality of photon-subtracted and photon-added-then-subtracted Gaussian states,” J. Opt. Soc. Am. B 26, 830–835 (2009).
[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. A 80, 022315 (2009).
[CrossRef]

Zavatta, A.

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photon. 5, 52–60 (2011).
[CrossRef]

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

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

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing the quantum commutation rules through cavity QED,” Science 317, 1890 (2007).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Zhang, S. L.

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

Zhang, Y.-D.

Z.-B. Chen, J.-W. Pan, G. Hou, and Y.-D. Zhang, “Maximal violation of Bells inequalities for continuous variable systems,” Phys. Rev. Lett. 88, 040406 (2002).
[CrossRef]

Zubairy, M. S.

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. B

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001 (2008).
[CrossRef]

Nat. Photon.

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

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photon. 5, 52–60 (2011).
[CrossRef]

New J. Phys.

J. Park, S.-Y. Lee, H.-J. Kim, and H.-W. Lee, “Cavity-QED-based scheme for verification of the photon commutation relation,” New J. Phys. 12, 033019 (2010).
[CrossRef]

Phys. Rev. A

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485 (1992).
[CrossRef]

Q. Sun, M. Al-Amri, and M. S. Zubairy, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Phys. Rev. A 78, 043801 (2008).
[CrossRef]

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

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

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

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by conditional measurements on the two-mode squeezed vacuum,” Phys. Rev. A 65, 062306 (2002).
[CrossRef]

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

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184 (1997).
[CrossRef]

See also P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

S. Olivares and M. G. A. Paris, “Enhancement of nonlocality in phase space,” Phys. Rev. A 70, 032112 (2004).
[CrossRef]

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

H. Jeong, “Testing Bell inequalities with photon-subtracted Gaussian states,” Phys. Rev. A 78, 042101 (2008).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[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. A 80, 022315 (2009).
[CrossRef]

Phys. Rev. Lett.

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[CrossRef]

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

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

Z.-B. Chen, J.-W. Pan, G. Hou, and Y.-D. Zhang, “Maximal violation of Bells inequalities for continuous variable systems,” Phys. Rev. Lett. 88, 040406 (2002).
[CrossRef]

S. L. Braunstein and C. M. Caves, “Information-theoretic Bell inequalities,” Phys. Rev. Lett. 61, 662–665 (1988).
[CrossRef]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

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

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

Science

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing the quantum commutation rules through cavity QED,” Science 317, 1890 (2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Bell parameter BBW in Eq. (4), optimized over the complex amplitudes {α,α,β,β} for each case, is plotted as a function of the squeezing parameter s for |TMSV (orange solid), (tAa^+rAa^)|TMSV (coherent operation on one local mode only, blue dot-dashed), a^|TMSV and a^|TMSV (gray long dot-dashed), (tAa^+rAa^)(tBb^+rBb^)|TMSV (coherent operation on both modes, thick red solid), a^b^|TMSV (black dashed), a^b^|TMSV (purple dotted), and a^b^|TMSV (brown long dashed). The ratio ti (i=A,B) in the coherent operations tAa^+rAa^ and tBb^+rBb^ are also optimized for each s.

Fig. 2.
Fig. 2.

Wigner function and its contour plot for the states (a), (b) |TMSV, (c), (d) a^b^|TMSV, (e), (f) a^b^|TMSV, and (g), (h) (tAa^+rAa^)(tBb^+rBb^)|TMSV with tA0.97 and tB0.88, for the same squeezing s=0.1. Black circles on the contour plot indicate the optimized parameters α, β, α, and β that maximize the Bell parameter BBW for each state.

Fig. 3.
Fig. 3.

Maximum value of Bell parameter BPS as a function of the squeezing parameter s for |TMSV (orange solid), (tAa^+rAa^)|TMSV (blue dot-dashed), (tAa^+rAa^)(tBb^+rBb^)|TMSV (thick red solid), a^b^|TMSV (black dashed), a^b^|TMSV (purple dotted), and a^b^|TMSV (brown long dashed). The ratio ti (i=A,B) in the coherent operations tAa^+rAa^ and tBb^+rBb^ are optimized for each s.

Fig. 4.
Fig. 4.

Maximum value of Bell parameter BHD as a function of the squeezing parameter s for (tAa^+rAa^)(tBb^+rBb^)|TMSV (thick red solid), a^b^|TMSV (black dashed). The ratios tA and tB in the coherent operation are optimized for each s.

Fig. 5.
Fig. 5.

Value of Bell parameter BBC within the pseudospin formalism, optimized numerically over the measurement angles for each case, as a function of the squeezing parameter s for |TMSV (orange solid), (tAa^+rAa^)|TMSV (blue dot-dashed), (tAa^+rAa^)(tBb^+rBb^)|TMSV (thick red solid), a^b^|TMSV (black dashed), a^b^|TMSV (purple dotted), and a^b^|TMSV (brown long dashed). The ratios tA and tB in the coherent operations are optimized for each s.

Equations (32)

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

|TMSV=1λ2n=0λn|n,n,
N(tAa^+rAa^)(tBb^+rBb^)|TMSV=N[n=0λn(λtAtB(n+1)|n,n+λtArB(n+1)(n+2)|n,n+2+λrAtB(n+1)(n+2)|n+2,n+rArB(n+1)|n+1,n+1)],
N=(1λ2)3λ2(1+|tArB*+rAtB*|2)+|tAtBλ2+rArB|2
BBW=|Π(α,β)+Π(α,β)+Π(α,β)Π(α,β)|2.
Π^(α,β)=D^a(α)D^b(β)(1)n^a+n^bD^a(α)D^b(β),
Π(α,β)=π24W(α,β).
W(α,β)=4e2(|α˜|2+|β˜|2)π2M{A2(14|α˜|2)(14|β˜|2)+B2+C2(18|β˜|2+8|β˜|4)+D2(18|α˜|2+8|α˜|4)+8AB(α˜β˜)82A(α˜β˜*)[C(12|β˜|2)+D(12|α˜|2)]+42B[C(β˜2)+D(α˜2)]+16(α˜2β˜*2)CD},
A=rArBcosh2s+tAtBsinh2s,B=(rArB+tAtB)coshssinhs,C=2tArBcoshssinhs,D=2rAtBcoshssinhs,M=A2+B2+C2+D2,
Wa^(α,β)=4π2e2(|α˜|2+|β˜|2)(4|β˜|21),
Wa^(α,β)=4π2e2(|α˜|2+|β˜|2)(4|α˜|21),
s^x=n=0(|2n2n+1|+|2n+12n|),
s^y=in=0(|2n2n+1||2n+12n|),
s^z=n=0(|2n+12n+1||2n2n|).
BPS=|E(a,b)+E(a,b)+E(a,b)E(a,b)|2.
E(a,b)=(a·s^)(b·s^)=i,jaibjs^is^j,
u=(ux,uy,uz)=(sinθucosϕu,sinθusinϕu,cosθu).
E(a,b)=Zcosθacosθb+Xsinθasinθbcos(ϕa±ϕb),
(BPS)max=2Z2+X2.
1λ2n=0λ2n|2n,2n,
1λ2n=0λ2n+1|2n+1,2n+1.
BHD=|E(ϕ1,ϕ2)+E(ϕ1,ϕ2)+E(ϕ1,ϕ2)E(ϕ1,ϕ2)|2,
E(ϕ1,ϕ2)=sgn(Xϕ1Xϕ2),
sgn(X)={+1ifX0,1ifX<0.
E(ϕ1,ϕ2)=d2kCE(λ1,λ2)Fs(k1)Fs(k2),
CE(λ1,λ2)=e(|Λ1|2+|Λ2|2)/2M[A2(1|Λ1|2)(1|Λ2|2)+B2+C22(24|Λ2|2+|Λ2|4)+D22(24|Λ1|2+|Λ1|4)+2AB(Λ1Λ2)+2A(Λ1Λ2*)[C(|Λ2|22)+D(|Λ1|22)]+2B[C(Λ22)+D(Λ12)]+CD(Λ12Λ2*2)},
BBC=H(A|B)H(A|B)H(B|A)H(A|B)0,
H(A|B)=aA,bBp(a,b)log2p(b)p(a,b).
E(a,b)=a,b{1,1}p(a,b)ab,E(a)=a,b{1,1}p(a,b)a,E(b)=a,b{1,1}p(a,b)b,
p(a,b)=14{1+aE(a)+bE(b)+abE(a,b)},p(a)=12{1+aE(a)},p(b)=12{1+bE(a)}.
E(a)=(a·s^)I=i=x,y,zais^iI,E(b)=I(b·s^)=j=x,y,zbjIs^j.
E(a)=azs^zI=cosθas^zI,E(b)=bzIs^z=cosθbIs^z.
E(ϕ1)=dk1CE(λ1,0)Fs(k1),E(ϕ2)=dk2CE(0,λ2)Fs(k2).

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