Skip to main content
SpringerLink
Log in

The decompositions of Werner and isotropic states

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

The decompositions of separable Werner states and isotropic states are well-known tough issues in quantum information theory. In this work, we investigate them in the Bloch vector representation, exploring the symmetric informationally complete positive operator-valued measure (SIC-POVM) in the Hilbert space. In terms of regular simplexes, we successfully get the decomposition for arbitrary Werner state in \({\mathbb {C}}^N\otimes {\mathbb {C}}^N\), and the explicit separable decompositions are constructed based on the SIC-POVM. Meanwhile, the decomposition of isotropic states is found related to the decomposition of Werner states via partial transposition. It is interesting to note that when dimension N approaches to infinity, the Werner states are either separable or non-steerably entangled, and most of the isotropic states tend to be steerable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  2. Aspect, A., Grangier, P., Roger, G.: Experimental tests of realistic local theories via Bell’s theorem. Phys. Rev. Lett. 47, 460–463 (1981)

  3. Hensen, B., Bernien, H., Dréau, A.E., Reiserer, A., Kalb, N., Blok, M.S., Ruitenberg, J., Vermeulen, R.F.L., Schouten, R.N., Abellán, C., Amaya, W., Pruneri, V., Mitchell, M.W., Markham, M., Twitchen, D.J., Elkouss, D., Wehner, S., Taminiau, T.H., Hanson, R.: Loophole-cc separated by 1.3 kilometres, Nature 526, 682-686 (2015)

  4. Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)

    Article  ADS  Google Scholar 

  5. Wiseman, H.M., Jones, S.J., Doherty, A.C.: Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox. Phys. Rev. Lett. 98, 140402 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  6. Chitambar, E.: Quantum correlations in high-dimensional states of high symmetry. Phys. Rev. A 86, 4134–4139 (2012)

    Article  Google Scholar 

  7. 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  ADS  MathSciNet  Google Scholar 

  8. 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)

    Article  ADS  MathSciNet  Google Scholar 

  9. Dür, W., Cirac, J.I., Lewenstein, M., Bruß, D.: Distillability and partial transposition in bipartite systems. Phys. Rev. A 61, 062313 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  10. Unanyan, R.G., Kampermann, H., Bruß, D.: A decomposition of separable Werner states. J. Phys. A Math. Theor. 40, F483–F490 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  11. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)

    Article  ADS  Google Scholar 

  12. Azuma, H., Ban, M.: Another convex combination of product states for the separable Werner state. Phys. Rev. A 73, 032315 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  13. Graydon, M.A., Appleby, D.M.: Quantum conical designs. J. Phys. A Math. Theor. 49, 085301 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  14. Li, J.-L., Qiao, C.-F.: A necessary and sufficient criterion for the separability of quantum state. Sci. Rep. 8, 1442 (2018)

    Article  ADS  Google Scholar 

  15. Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171–2180 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  16. Kimura, G.: The Bloch vector for \(N\)-level systems. Phys. Lett. A 314, 339–349 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  17. Byrd, M.S., Khaneja, N.: Characterization of the positivity of the density matrix in terms of the coherence vector representation. Phys. Rev. A 68, 062322 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  18. de Vicente, J.I.: Separability criteria based on the Bloch representation of density matrices. Quantum Inf. Comput. 7, 624–638 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Horodecki, M., Horodecki, P.: Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206–4216 (1999)

    Article  ADS  Google Scholar 

  20. Thomas, P., Bohmann, M., Vogel, W.: Verifying bound entanglement of dephased Werner states. Phys. Rev. A 96, 042321 (2017)

    Article  ADS  Google Scholar 

  21. Salazar, R., Goyeneche, D., Delgado, A., Saavedra, C.: Constructing symmetric informationally complete positive-operator-valued measures in Bloch space. Phys. Lett. A 376, 325–329 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  22. Appleby, D.M.: Symmetric informationally complete measurements of arbitrary rank. Opt. Spectrosc. 103, 416–428 (2007)

    Article  ADS  Google Scholar 

  23. Grassl, M., Scott, A.J.: Fibonacci-Lucas SIC-POVMs. J. Math. Phys. 58, 122201 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  24. Fuchs, C.A., Hoang, M.C., Stacey, B.C.: The SIC question: history and state of play. Axioms 6, 21 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (NSFC) under the Grants 11975236 and 11635009 and by the University of Chinese Academy of Sciences.

Author information

Authors and Affiliations

  1. School of Physical Sciences, University of Chinese Academy of Sciences, YuQuan Road 19A, Beijing, 100049, China

    Ma-Cheng Yang & Cong-Feng Qiao

  2. Center of Materials Science and Optoelectronics Engineering & CMSOT, University of Chinese Academy of Sciences, YuQuan Road 19A, Beijing, 100049, China

    Jun-Li Li

  3. Key Laboratory of Vacuum Physics, University of Chinese Academy of Sciences, YuQuan Road 19A, Beijing, 100049, China

    Jun-Li Li & Cong-Feng Qiao

Authors
  1. Ma-Cheng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-Li Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cong-Feng Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cong-Feng Qiao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, MC., Li, JL. & Qiao, CF. The decompositions of Werner and isotropic states. Quantum Inf Process 20, 255 (2021). https://doi.org/10.1007/s11128-021-03193-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-021-03193-y

Keywords

Search

Navigation