Distillability criterion for all bipartite Gaussian states

Quantum Information and Computation

Volumn 1, Issue 3, 2001, Pages 79-86

  Giedke, Geza ^a   Duan, Lu Ming ^a   Cirac, J Ignacio ^a   Zoller, Peter ^a  

^a UNIVERSITY OF INNSBRUCK   (Austria)

Author keywords

Distillation; Entanglement; Gaussian states

Indexed keywords

GAUSSIAN DISTRIBUTION; QUANTUM ENTANGLEMENT;

DISTILLABILITY; ENTANGLEMENT; GAUSSIAN STATE; LOCAL OPERATIONS; PARTIAL TRANSPOSE; PURE STATE;

DISTILLATION;

EID: 0002343677     PISSN: 15337146     EISSN: None     Source Type: Journal    
DOI: 10.26421/qic1.3-7     Document Type: Article

References (33)

1. 1. C.H. Bennett, G. Brassard, C. Crepeau, 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 (1993); A. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett. 67, 661 (1991).
2. C.H. Bennett, G. Brassard, C. Crepeau, 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 (1993); A. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett. 67, 661 (1991).
3. 2. C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels, Phys. Rev. Lett. 76, 722 (1996); quant-ph/9511027. C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54 3824 (1996); quant-ph/9604024.
4. C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels, Phys. Rev. Lett. 76, 722 (1996); quant-ph/9511027. C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54 3824 (1996); quant-ph/9604024.
5. H.-J. Briegel, W. Dür, J.I. Cirac, and P. Zoller, Quantum repeaters: The role of imperfect local operations in quantum communication, Phys. Rev. Lett. 81, 5932 (1998); quant-ph/9803056.
6. M. Horodecki, P. Horodecki, and R. Horodecki, Mixed-state entanglement and distillation: Is there a "bound" entanglement in nature?, Phys. Rev. Lett. 80, 5239 (1998).
7. P. Horodecki, Separability Criterion and inseparable mixed states with positive partial transpose, Phys. Lett. A 232, 333 (1997).
8. 6. W. Dür, J.I. Cirac, M. Lewenstein, D. Bruß, Distillability and partial transposition in bipartite systems, Phys. Rev. A 61, 062313 (2000), quant-ph/9910022. D. DiVincenzo, P.W. Shor, J.A. Smolin, B.M. Terhal, A.V. Thapliyal, Evidence for bound entangled states with negative partial transpose, Phys. Rev. A 61, 062312 (2000); quant-ph/9910026.
9. W. Dür, J.I. Cirac, M. Lewenstein, D. Bruß, Distillability and partial transposition in bipartite systems, Phys. Rev. A 61, 062313 (2000), quant-ph/9910022. D. DiVincenzo, P.W. Shor, J.A. Smolin, B.M. Terhal, A.V. Thapliyal, Evidence for bound entangled states with negative partial transpose, Phys. Rev. A 61, 062312 (2000); quant-ph/9910026.
10. P.W. Shor, J.A. Smolin, B.M. Terhal, Nonadditivity of Bipartite Distillable Entanglement follows from Conjecture on Bound Entangled Werner States, Phys. Rev. Lett. 86, 2681 (2001); quant-ph/0010054.
11. M. Horodecki, P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59, 4206 (1999).
12. M. Lewenstein, D. Bruß, J.I. Cirac, B. Kraus, M. Kuś, J. Samsonowicz, A. Sanpera, and R. Tarrach, Separability and distillability in composite quantum systems - a primer, J. Mod. Opt. 47, 2481 (2000), quant-ph/006064.
13. M. Horodecki, P. Horodecki, and R. Horodecki, Inseparable Two Spin-1/2 Density Matrices Can Be Distilled to a Singlet Form, Phys. Rev. Lett. 78, 574 (1997).
14. 11. L. Vaidman, Teleportation of quantum states, Phys. Rev. A 49, 1473 (1994). S.L. Braunstein and H.J. Kimble, Teleportation of Continuous Quantum Variables, Phys. Rev. Lett. 80, 869 (1998); A. Furusawa, J.L. Sørensen, S.L. Braunstein, C.A. Fuchs, H.J. Kimble, and E.S. Polzik, Unconditional Quantum Teleportation, Science 282, 706 (1998).
15. L. Vaidman, Teleportation of quantum states, Phys. Rev. A 49, 1473 (1994). S.L. Braunstein and H.J. Kimble, Teleportation of Continuous Quantum Variables, Phys. Rev. Lett. 80, 869 (1998); A. Furusawa, J.L. Sørensen, S.L. Braunstein, C.A. Fuchs, H.J. Kimble, and E.S. Polzik, Unconditional Quantum Teleportation, Science 282, 706 (1998).
16. L. Vaidman, Teleportation of quantum states, Phys. Rev. A 49, 1473 (1994). S.L. Braunstein and H.J. Kimble, Teleportation of Continuous Quantum Variables, Phys. Rev. Lett. 80, 869 (1998); A. Furusawa, J.L. Sørensen, S.L. Braunstein, C.A. Fuchs, H.J. Kimble, and E.S. Polzik, Unconditional Quantum Teleportation, Science 282, 706 (1998).
17. S. Braunstein and A. Pati (eds.), Quantum Information Theory with Continuous Variables, Kluwer Academic Publishers, Dodrecht 2001, in press.
18. 13. 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 (1999); quant-ph/9907048; S. Parker, S. Bose, and M.B. Plenio, Entanglement quantification and purification in continuous variable systems, Phys. Rev. A 61, 032305 (1999); quant-ph/9906098; L.-M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Entanglement Purification of Gaussian Continuous Variable Quantum States, Phys. Rev. Lett. 84, 4002 (2000); quant-ph/9912017.
19. 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 (1999); quant-ph/9907048; S. Parker, S. Bose, and M.B. Plenio, Entanglement quantification and purification in continuous variable systems, Phys. Rev. A 61, 032305 (1999); quant-ph/9906098; L.-M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Entanglement Purification of Gaussian Continuous Variable Quantum States, Phys. Rev. Lett. 84, 4002 (2000); quant-ph/9912017.
20. 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 (1999); quant-ph/9907048; S. Parker, S. Bose, and M.B. Plenio, Entanglement quantification and purification in continuous variable systems, Phys. Rev. A 61, 032305 (1999); quant-ph/9906098; L.-M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Entanglement Purification of Gaussian Continuous Variable Quantum States, Phys. Rev. Lett. 84, 4002 (2000); quant-ph/9912017.
21. P. Horodecki, M. Lewenstein, Bound Entanglement and Continuous Variables, Phys. Rev. Lett. 85, 2657 (2000); quant-ph/0001035.
22. R.F. Werner and M.M. Wolf, Bound entangled Gaussian States, Phys. Rev. Lett. 86, 3658 (2001); quant-ph/0009118.
23. L.-M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Inseparability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2722 (2000); quant-ph/9908056.
24. R. Simon, Peres-Horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2726 (2000); quant-ph/9909044.
25. C. Gardiner and P. Zoller, Quantum Noise 2nd ed., Springer-Verlag, Berlin (1999).
26. J. Manuceau and A. Verbeure, Comm. Math. Phys. 9, 293 (1968).
27. 2.
28. H. Scutaru, Transition Probabilities for Quasifree States, J. Math. Phys. 39, 6403 (1998).
29. H. Scutaru, The states with Gaussian Wigner function are quasi-free states, Phys. Lett. A 141, 223 (1989).
30. a/√ γ.
31. k is performed, the Wigner function of the final state of the remaining modes is calculated as follows: replace the phase space coordinate referring to the measured operator by the measured result and integrate out the conjugate phase space coordinate. Note that the CM of final state is independent of the measurement outcome.
32. i.
33. V.I. Arnold, Mathematical Methods of Classical Mechanics 2nd. ed., Springer Verlag, New York, 1989.