Skip to main content
SpringerLink
Log in

Dimension formula for induced maximal faces of separable states and genuine entanglement

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

The normalized separable states of a finite-dimensional multipartite quantum system, represented by its Hilbert space \(\mathcal {H}\), form a closed convex set \(\mathcal {S}_1\). The set \(\mathcal {S}_1\) has two kinds of faces, induced and non-induced. An induced face, F, has the form \(F=\Gamma (F_V)\), where V is a subspace of \(\mathcal {H}\), \(F_V\) is the set of \(\rho \in \mathcal {S}_1\) whose range is contained in V, and \(\Gamma \) is a partial transposition operator. Such F is a maximal face if and only if V is a hyperplane. We give a simple formula for the dimension of any induced maximal face. We also prove that the maximum dimension of induced maximal faces is equal to \(d(d-2)\) where d is the dimension of \(\mathcal {H}\). The equality \(\mathrm{Dim\,}\Gamma (F_V)=d(d-2)\) holds if and only if \(V^\perp \) is spanned by a genuinely entangled vector.

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

Similar content being viewed by others

References

  1. Alfsen, E., Shultz, F.: Unique decompositions, faces, and automorphisms of separable states. J. Math. Phys. 51, 052201 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  2. Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, New York, vol. 175, p. 8 (1984)

  3. Bennett, C.H., Brassard, G., Crepeau, 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 (1993)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Chen, L., Đoković, D. Ž.: Boundary of the set of separable states, arXiv:1404.0738v3 [quant-ph] 20 Dec (2014)

  5. Chen, L., Gittsovich, O., Modi, K., Piani, M.: Role of correlations in the two-body-marginal problem. Phys. Rev. A 90, 042314 (2014)

    Article  ADS  Google Scholar 

  6. Choi, H.-S., Kye, S.-H.: Facial structure for separable states. J. Korean Math. Soc. 49, 623–639 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)

    Article  ADS  Google Scholar 

  8. Gour, G., Wallach, N.R.: All maximally entangled four-qubit states. J. Math. Phys. 51, 112201 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  9. Goyeneche, D., Zyczkowski, K.: Genuinely multipartite entangled states and orthogonal arrays. Phys. Rev. A 90, 022316 (2014)

    Article  ADS  Google Scholar 

  10. Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem. arXiv:0712.0921 [quant-ph] (2007)

  11. Ha, K.-C., Kye, S.-H.: Entanglement witnesses arising from exposed positive linear maps, Open Syst. Inf. Dyn. 18, 323–337 (2011). arXiv:1108.0130v2 [quant-ph] 7 Jan 2012

  12. Ha, K.-C., Kye, S.-H.: Separable states with unique decompositions. CMP 328, 131–153 (2014)

    MATH  MathSciNet  Google Scholar 

  13. Ha, K.-C., Kye, S.-H.: Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose. arXiv:1402.5813 [quant-ph] (2014)

  14. Ha, K.-C., Kye, S.-H.: Construction of exposed indecomposable positive linear maps between matrix algebras, arXiv:1410.5545v1 [math.OA] 21 Oct 2014

  15. Haffner, H., Hansel, W., Roos, C.F., et al.: Scalable multiparticle entanglement of trapped ions. Nature (London) 438, 643 (2005)

    Article  ADS  Google Scholar 

  16. Horodecki, M., Horodecki, P., Horodecki, R.: Phys. Lett. A 223, 1 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Kye, S.-H.: Facial structures for various notions of positivity and applications to the theory of entanglement. Rev. Math. Phys. 25, 1330002 (2013). arXiv:1202.4255v2 [quant-ph]

    Article  MathSciNet  Google Scholar 

  18. Kye, S.-H.: Private communication (2014)

  19. Marciniak, M.: Rank properties of exposed positive maps. Linear Multilinear Algebra 61, 970–975 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Toth, G., Wieczorek, W., Krischek, R., Kiesel, N., Michelberger, P., Weinfurther, H.: Practical methods for witnessing genuine multi-qubit entanglement in the vicinity of symmetric states. New J. Phys. 11, 083002 (2009)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

We thank an anonymous referee for pointing out that \(d(d-2)\) is the expected dimension for maximal faces of \(\mathcal {S}_1\). LC was partially supported by the Fundamental Research Funds for the Central Universities (Grant Nos. 30426401 and 30458601). The work started when LC was also supported in part by the Singapore National Research Foundation under NRF Grant No. NRF-NRFF2013-01. DD was supported in part by an NSERC Discovery Grant.

Author information

Authors and Affiliations

  1. School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China

    Lin Chen

  2. International Research Institute for Multidisciplinary Science, Beihang University, Beijing, 100191, China

    Lin Chen

  3. Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada

    Dragomir Ž. Ɖoković

  4. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada

    Dragomir Ž. Ɖoković

Authors
  1. Lin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dragomir Ž. Ɖoković
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lin Chen.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, L., Ɖoković, D.Ž. Dimension formula for induced maximal faces of separable states and genuine entanglement. Quantum Inf Process 14, 3335–3350 (2015). https://doi.org/10.1007/s11128-015-1051-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-015-1051-8

Keywords

Search

Navigation