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Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

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Abstract

We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators. We prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement.

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References

  1. Alon, N., Lovász, L.: Unextendible product bases. J. Combinatorial Theory, Ser. A 95, 169–179 (2001)

    Google Scholar 

  2. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)

    MathSciNet  MATH  Google Scholar 

  3. Bennett, C.H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J.A., Wootters, W.K.: Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, 722–725 (1996)

    Article  Google Scholar 

  4. Bennett, C.H., DiVincenzo, D.P., Fuchs, C.A., Mor, T., Rains, E.M., Shor, P.W., Smolin, J.A., Wootters, W.K.: Quantum nonlocality without entanglement. Phys. Rev. A 59, 1070–1091 (1999), quant-ph/9804053

    Article  MathSciNet  Google Scholar 

  5. Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385–5388 (1999), quant-ph/9808030

    Article  MathSciNet  Google Scholar 

  6. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996)

    Article  MathSciNet  Google Scholar 

  7. Bennett, C.H.: Private communication

  8. Bollobás, B.: Graph Theory. New York: Springer, 1979

  9. Dür, W., Cirac, J.I., Lewenstein, M., Bruss, D.: Distillability and partial transposition in bipartite systems. Phys. Rev. A 61, 062313 (2000), quant-ph/9910022

    Article  Google Scholar 

  10. DiVincenzo, D.P., Shor, P.W., Smolin, J.A., Terhal, B.M., Thapliyal, A.V.: Evidence for bound entangled states with negative partial transpose. Phys. Rev. A 61, 062312 (2000), quant-ph/9910026

    Article  Google Scholar 

  11. DiVincenzo, D.P., Terhal, B.M.: Product bases in quantum information theory. Proceedings of the XIII International Congress on Mathematical Physics, London, July 2000

  12. DiVincenzo, D.P., Terhal, B.M., Thapliyal, A.V.: Optimal decompositions of barely separable states. J. Modern Optics 47(2/3), 377–385 (2000), quant-ph/9904005

    Google Scholar 

  13. Gottesman, D.: Stabilizer Codes and Quantum Error Correction. PhD thesis, CalTech, 1997, quant-ph/9705052

  14. Horodecki, M., Horodecki, P.: Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206–4216 (1999), quant-ph/9708015

    Article  Google Scholar 

  15. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: Necessary and sufficient conditions. Phys. Letts. A 223, 1–8 (1996), quant-ph/9605038

    Article  MATH  Google Scholar 

  16. Horodecki, M., Horodecki, P., Horodecki, R.: Mixed state entanglement and distillation: Is there a ``bound'' entanglement in nature? Phys. Rev. Lett. 80, 5239–5242 (1998), quant-ph/9801069

    Article  MathSciNet  MATH  Google Scholar 

  17. Horodecki, M., Horodecki, P., Horodecki, R.: General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A 60, 1888–1898 (1999), quant-ph/9807091

    Article  MathSciNet  MATH  Google Scholar 

  18. Horodecki, P., Horodecki, M., Horodecki, R.: Bound entanglement can be activated. Phys. Rev. Lett. 82, 1056–1059 (1999), quant-ph/9806058

    Article  MathSciNet  MATH  Google Scholar 

  19. Horodecki, P., Horodecki, M., Horodecki, R.: Binding entanglement channels. J. Modern Optics 47(2/3), 347–354 (2000), quant-ph/9904092

    Google Scholar 

  20. Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333–339 (1997), quant-ph/9703004

    Article  MathSciNet  MATH  Google Scholar 

  21. Horodecki, P., Smolin, J.A., Terhal, B.M., Thapliyal, A.V.: Rank two bound entangled states do not exist. J. Theor. Comp. Sci. 292(3), 589–596 (2003) See also [Ter99], quant-ph/9910122

    Article  Google Scholar 

  22. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. New York: Oxford University Press, Fifth edition, 1979

  23. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. on Inf. Theory 25, 1–7 (1979)

    Google Scholar 

  24. Newman, M.: On a theorem of Čebotarev. Linear and Multilinear Algebra 3, 259–262 (1976)

    MATH  Google Scholar 

  25. Peres, A.: Private communication

  26. Peres, A.: Quantum Theory: Concepts and Methods. Amsterdam: Kluwer Academic Publishers, 1993

  27. Rains, E.M.: Entanglement purification via separable superoperators. quant-ph/9707002

  28. Rains, E.M.: Rigorous treatment of distillable entanglement. Phys. Rev. A 60, 173–178 (1999)

    Article  Google Scholar 

  29. Terhal, B.M.: A family of indecomposable positive linear maps based on entangled quantum states. Lin. Alg. and Its Appl. 323, 61–73 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Terhal, B.M.: Quantum Algorithms and Quantum Entanglement. PhD thesis, University of Amsterdam, 1999

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Authors and Affiliations

  1. IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA

    David P. DiVincenzo, John A. Smolin & Barbara M. Terhal

  2. Dept. of Electrical Engineering, UCLA, Los Angeles, CA, 90095-1594, USA

    Tal Mor

  3. AT&T Research, Florham Park, NJ, 07932, USA

    Peter W. Shor

Authors
  1. David P. DiVincenzo
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  2. Tal Mor
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  3. Peter W. Shor
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  4. John A. Smolin
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  5. Barbara M. Terhal
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R.H. Dijkgraaf

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DiVincenzo, D., Mor, T., Shor, P. et al. Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement. Commun. Math. Phys. 238, 379–410 (2003). https://doi.org/10.1007/s00220-003-0877-6

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  • DOI: https://doi.org/10.1007/s00220-003-0877-6

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