Skip to main content
SpringerLink
Log in

Duality of graph invariants

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We study a new set of duality relations between weighted, combinatoric invariants of a graph G. The dualities arise from a non-linear transform \(\mathcal{B}\), acting on the weight function p. We define \(\mathcal{B}\) on a space of real-valued functions \(\mathcal{O}\) and investigate its properties. We show that three invariants (the weighted independence number, the weighted Lovász number, and the weighted fractional packing number) are fixed points of \(\mathcal{B}^2\), but the weighted Shannon capacity is not. We interpret these invariants in the study of quantum non-locality.

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. Abramsky S, Brandenburger A. The sheaf-theoretic structure of non-locality and contextuality. New J Phys, 2011, 13: 113036

    Article  Google Scholar 

  2. Acín A, Brunner N, Gisin N, et al. Device-independent security of quantum cryptography against collective attacks. Phys Rev Lett, 2007, 98: 230501

    Article  Google Scholar 

  3. Acín A, Duan R, Roberson D E, et al. A new property of the Lovász number and duality relations between graph parameters. Discrete Appl Math, 2017, 216: 489–501

    Article  MathSciNet  Google Scholar 

  4. Acín A, Fritz T, Leverrier A, et al. A combinatorial approach to nonlocality and contextuality. Comm Math Phys, 2015, 334: 533–628

    Article  MathSciNet  Google Scholar 

  5. Anders J, Browne D E. Computational power of correlations. Phys Rev Lett, 2009, 102: 050502

    Article  MathSciNet  Google Scholar 

  6. Berge C. Graphs and Hypergraphs. Amsterdam: Elsevier, 1973

    MATH  Google Scholar 

  7. Bermejo-Vega J, Delfosse N, Browne D E, et al. Contextuality as a resource for models of quantum computation with qubits. Phys Rev Lett, 2017, 119: 120505

    Article  MathSciNet  Google Scholar 

  8. Brandão F G S L, Datta N. One-shot rates for entanglement manipulation under non-entangling maps. IEEE Trans Inform Theory, 2011, 57: 1754–1760

    Article  MathSciNet  Google Scholar 

  9. Brunner N, Cavalcanti D, Pironio S, et al. Bell nonlocality. Rev Mod Phys, 2014, 86: 419–478

    Article  Google Scholar 

  10. Bu K F, Singh U, Fei S-M, et al. Maximum relative entropy of coherence: An operational coherence measure. Phys Rev Lett, 2017, 119: 150405

    Article  MathSciNet  Google Scholar 

  11. Buhrman H, Cleve R, Massar S, et al. Nonlocality and communication complexity. Rev Modern Phys, 2010, 82: 665–698

    Article  Google Scholar 

  12. Cabello A, Severini S, Winter A. Graph-theoretic approach to quantum correlations. Phys Rev Lett, 2014, 112: 040401

    Article  Google Scholar 

  13. Dantzig G B, Thapa M N. Linear Programming 2: Theory and Extensions. Berlin: Springer-Verlag, 2003

    MATH  Google Scholar 

  14. Datta N. Min- and max-relative entropies and a new entanglement monotone. IEEE Trans Inform Theory, 2009, 55: 2816–2826

    Article  MathSciNet  Google Scholar 

  15. Datta N. Max-relative entropy of entanglement, alias log robustness. Int J Quant Inform, 2009, 7: 475–491

    Article  Google Scholar 

  16. Delfosse N, Guerin P A, Bian J, et al. Wigner function negativity and contextuality in quantum computation on rebits. Phys Rev X, 2015, 5: 021003

    Google Scholar 

  17. Fenchel W. On conjugate convex functions. Canad J Math, 1949, 1: 73–77

    Article  MathSciNet  Google Scholar 

  18. Howard M, Wallman J, Veitch V, et al. Contextuality supplies the magic for quantum computation. Nature, 2014, 510: 351–355

    Article  Google Scholar 

  19. Imrich W, Klavzar S. Product Graphs: Structure and Recognition. New York: Wiley-Intersciene, 2000

    MATH  Google Scholar 

  20. Knuth D E. The sandwich theorem. Electron J Combin, 1994, 1: A1

    Article  MathSciNet  Google Scholar 

  21. Kochen S B, Specker E. The problem of hidden variables in quantum mechanics. J Math Mech, 1967, 17: 59–87

    MathSciNet  MATH  Google Scholar 

  22. Koenig R, Renner R, Schaffner C. The operational meaning of min- and max-entropy. IEEE Trans Inform Theory, 2009, 55: 4337–4347

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Raussendorf R. Contextuality in measurement-based quantum computation. Phys Rev A (3), 2013, 88: 022322

    Article  Google Scholar 

  25. Shannon C E. The zero error capacity of a noisy channel. IRE Trans Inform Theory, 1956, 2: 8–19

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the Templeton Religion Trust (Grant No. TRT 0159). The third author was supported by USA Army Research Office (ARO) (Grant No. W911NF1910302). The first author is grateful for the support of the Academic Award for Outstanding Doctoral Candidates from Zhejiang University. The authors thank Zhengwei Liu for his support, help, and comments. The second author thanks Liming Ge and Boqing Xue for discussion on related material.

Author information

Authors and Affiliations

  1. Harvard University, Cambridge, MA, 02138, USA

    Kaifeng Bu & Arthur Jaffe

  2. School of Mathematical Science, Zhejiang University, Hangzhou, 310027, China

    Kaifeng Bu

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, China

    Weichen Gu

Authors
  1. Kaifeng Bu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weichen Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arthur Jaffe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arthur Jaffe.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bu, K., Gu, W. & Jaffe, A. Duality of graph invariants. Sci. China Math. 63, 1613–1626 (2020). https://doi.org/10.1007/s11425-018-9563-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-018-9563-3

Keywords

MSC(2010)

Search

Navigation