Advertisement

Minimum distance of the boundary of the set of PPT states from the maximally mixed state using the geometry of the positive semidefinite cone

  • Shreya BanerjeeEmail author
  • Aryaman A. Patel
  • Prasanta K. Panigrahi
Article
  • 21 Downloads

Abstract

Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices (Patel and Panigrahi in Geometric measure of entanglement based on local measurement, 2016. arXiv:1608.06145), we obtain the minimum distance between the set of bipartite n-qudit density matrices with a positive partial transpose and the maximally mixed state. This minimum distance is obtained as \(\frac{1}{\sqrt{d^n(d^n-1)}}\), which is also the minimum distance within which all quantum states are separable. An idea of the interior of the set of all positive semidefinite matrices has also been provided. A particular class of Werner states has been identified for which the PPT criterion is necessary and sufficient for separability in dimensions greater than six.

Keywords

Entanglement Separability Partial transpose Werner states PPT criterion Positive semidefinite cone 

Notes

Acknowledgements

The authors want to acknowledge valuable inputs from Prof. Somshubhro Bandyopadhyay (Bose Institute, Kolkata).

References

  1. 1.
    Bruß, D.: Characterizing entanglement. J. Math. Phys. 43, 4237–4251 (2002)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Xu, J.-S., Li, C.-F., Xu, X.-Y., Shi, C., Zou, X.-B., Guo, G.-C.: Experimental characterization of entanglement dynamics in noisy channels. Phys. Rev. Lett. 103, 240502 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Jaeger, G.: Entanglement, Information, and the Interpretation of Quantum Mechanics. Springer, Berlin (2009)CrossRefGoogle Scholar
  4. 4.
    Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 046–2052 (1996)CrossRefGoogle Scholar
  5. 5.
    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)ADSCrossRefGoogle Scholar
  6. 6.
    Fortes, R., Rigolin, G.: Probabilistic quantum teleportation via thermal entanglement. Phys. Rev. A 96, 022315 (2017)ADSCrossRefGoogle Scholar
  7. 7.
    Imai, H., Hanaoka, G., Maurer, U., Zheng, Y., Naor, M., Segev, G., Smith, A., Safavi-Naini, R., Wild, P.R., Channels, Broadcast, et al.: Special issue on information theoretic security. IEEE Trans. Inf. Theory 52, 4348 (2006)CrossRefGoogle Scholar
  8. 8.
    Alonso, J.G., Brun, T.A.: Error correction with orbital angular momentum of multiple photons propagating in a turbulent atmosphere. Phys. Rev. A 95, 032320 (2017)ADSCrossRefGoogle Scholar
  9. 9.
    Boileau, J.-C., Tamaki, K., Batuwantudawe, J., Laflamme, R., Renes, J.M.: Unconditional security of a three state quantum key distribution protocol. Phys. Rev. Lett. 94, 040503 (2005)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Wang, J., Li, L., Peng, H., Yang, Y.: Quantum-secret-sharing scheme based on local distinguishability of orthogonal multiqudit entangled states. Phys. Rev. A 95, 022320 (2017)ADSCrossRefGoogle Scholar
  11. 11.
    Gao, G., Wang, Y.: Comment on “proactive quantum secret sharing”. Quantum Inf. Process. 16, 74 (2017)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Moroder, T., Gittsovich, O., Huber, M., Gühne, O.: Steering bound entangled states: a counterexample to the stronger peres conjecture. Phys. Rev. Lett. 113, 050404 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    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)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Braunstein, S.L., Caves, C.M.: Geometry of quantum states. In: Belavkin, V.P., Hirota, O., Hudson, R.L. (eds.) Quantum Communications and Measurement, pp. 21–30. Springer, Berlin (1995)CrossRefGoogle Scholar
  15. 15.
    Zyczkowski, K., Sommers, H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A Math. Gen. 34, 7111 (2001) ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Zyczkowski, K., Slomczynski, W.: The monge metric on the sphere and geometry of quantum states. J. Phys. A Math. Gen. 34, 6689 (2001)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Patel, A.A., Panigrahi, P.K.: Geometric measure of entanglement based on local measurement (2016). arXiv preprint arXiv:1608.06145
  18. 18.
    Boyer, M., Liss, R., Mor, T.: Geometry of entanglement in the bloch sphere. Phys. Rev. A 95, 032308 (2017)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Ozawa, M.: Entanglement measures and the Hilbert–Schmidt distance. Phys. Lett. A 268, 58–160 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Heydari, H., Björk, G.: Entanglement measure for general pure multipartite quantum states. J. Phys. A Math. Gen. 37, 9251 (2004)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Goswami, A.K., Panigrahi, P.K.: Uncertainty relation and inseparability criterion. Found. Phys. 47, 229–235 (2017)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    La Guardia, G.G., Pereira, F.F.: Good and asymptotically good quantum codes derived from algebraic geometry. Quantum Inf. Process. 16, 165 (2017)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Holik, F., Sergioli, G., Freytes, H., Giuntini, R., Plastino, A.: Toffoli gate and quantum correlations: a geometrical approach. Quantum Inf. Process. 16(2), 55 (2017)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Bhaskara, V.S., Panigrahi, P.K.: Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange’s identity and wedge product. Quantum Inf. Process. 16, 118 (2017)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhou, Lan, Sheng, Yu-Bo: Concurrence measurement for the two-qubit optical and atomic states. Entropy 17(6), 4293–4322 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    Banerjee, S., Panigrahi, P.K.: Parallelism of Vectors and Tangle as an Inequality in Area (2019).  https://doi.org/10.13140/RG.2.2.31620.48002
  28. 28.
    Zhu, X.-N., Fei, S.-M.: Entanglement monogamy relations of qubit systems. Phys. Rev. A 90, 024304 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    Zhou, L., Sheng, Y.-B.: Detection of nonlocal atomic entanglement assisted by single photons. Phys. Rev. A 90, 024301 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    Zhang, M., Zhou, L., Zhong, W., Sheng, Y.-B.: Direct measurement of the concurrence of hybrid entangled state based on parity check measurements. Chin. Phys. B 28, 010301 (2019)ADSCrossRefGoogle Scholar
  31. 31.
    Sheng, Y.-B., Guo, R., Pan, J., Zhou, L., Wang, X.-F.: Two-step measurement of the concurrence for hyperentangled state. Quantum Inf. Process. 14(3), 963–978 (2015)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Gurvits, L., Barnum, H.: Largest separable balls around the maximally mixed bipartite quantum state. Phys. Rev. A 66, 062311 (2002)ADSCrossRefGoogle Scholar
  33. 33.
    Đoković, D.Ž.: On two-distillable Werner states. Entropy 18, 216 (2016)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Bandyopadhyay, S., Roychowdhury, V.: Maximally disordered distillable quantum states. Phys. Rev. A 69, 040302 (2004)ADSCrossRefGoogle Scholar
  35. 35.
    Lewenstein, M., Sanpera, A.: Separability and entanglement of composite quantum systems. Phys. Rev. Lett. 80, 2261–2264 (1998) ADSCrossRefGoogle Scholar
  36. 36.
    Lasserre, J.B.: A trace inequality for matrix product. Trans. IEEE Autom. Control 40, 1500–1501 (1995)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Życzkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A 58, 883–892 (1998)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Rana, S.: Negative eigenvalues of partial transposition of arbitrary bipartite states. Phys. Rev. A 87, 054301 (2013)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Shreya Banerjee
    • 1
    Email author
  • Aryaman A. Patel
    • 2
  • Prasanta K. Panigrahi
    • 1
  1. 1.Indian Institute of Science Education and Research KolkataMohanpurIndia
  2. 2.National Institute of Technology KarnatakaSurathkal, MangaloreIndia

Personalised recommendations