Bounds on Entanglement Assisted Source-channel Coding Via the Lovász Theta Number and Its Variants

Authors Toby Cubitt, Laura Mancinska, David Roberson, Simone Severini, Dan Stahlke, Andreas Winter

Thumbnail PDF


  • Filesize: 340 kB
  • 4 pages

Document Identifiers

Author Details

Toby Cubitt
Laura Mancinska
David Roberson
Simone Severini
Dan Stahlke
Andreas Winter

Cite AsGet BibTex

Toby Cubitt, Laura Mancinska, David Roberson, Simone Severini, Dan Stahlke, and Andreas Winter. Bounds on Entanglement Assisted Source-channel Coding Via the Lovász Theta Number and Its Variants. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 48-51, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs G and H. Such vectors exist if and only if theta(G) <= theta(H) where theta represents the Lovász number. We also obtain similar inequalities for the related Schrijver theta^- and Szegedy theta^+ numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: alpha^*(G) <= theta^-(G). Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovász number. Beigi introduced a quantity beta as an upper bound on alpha^* and posed the question of whether beta(G) = \lfloor theta(G) \rfloor. We answer this in the affirmative and show that a related quantity is equal to \lceil theta(G) \rceil. We show that a quantity chi_{vect}(G) recently introduced in the context of Tsirelson's conjecture is equal to \lceil theta^+(G) \rceil.
  • source-channel coding
  • zero-error capacity
  • Lovász theta


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Noga Alon. The shannon capacity of a union. Combinatorica, 18(3):301-310, 1998. Google Scholar
  2. Salman Beigi. Entanglement-assisted zero-error capacity is upper-bounded by the Lovász ϑ function. Physical Review A, 82:010303, July 2010. Google Scholar
  3. Jop Briët, Harry Buhrman, Monique Laurent, Teresa Piovesan, and Giannicola Scarpa. Zero-error source-channel coding with entanglement, 2013. Google Scholar
  4. Runyao Duan, S. Severini, and A. Winter. Zero-error communication via quantum channels and a quantum Lovász ϑ-function. In Proc. IEEE International Symposium on Information Theory (ISIT), 2011, pages 64-68, August 2011. Google Scholar
  5. Viktor Galliard and Stefan Wolf. Pseudo-telepathy, entanglement, and graph colorings. In Proc. IEEE International Symposium on Information Theory (ISIT), 2002, page 101, 2002. Google Scholar
  6. Willem H. Haemers. An upper bound for the Shannon capacity of a graph. Colloquia Mathematica Societatis Janos Bolyai, 25:267-272, 1978. Google Scholar
  7. Willem H. Haemers. On some problems of Lovász concerning the Shannon capacity of a graph. IEEE Transactions on Information Theory, 25:231-232, 1979. Google Scholar
  8. Geňa Hahn and Claude Tardif. Graph homomorphisms: structure and symmetry. In Graph symmetry, pages 107-166. Springer, 1997. Google Scholar
  9. Pavol Hell and Jaroslav Nešetřil. Graphs and Homomorphisms (Oxford Lecture Series in Mathematics and Its Applications). Oxford University Press, USA, 9 2004. Google Scholar
  10. Debbie Leung, Laura Mančinska, William Matthews, Maris Ozols, and Aidan Roy. Entanglement can increase asymptotic rates of zero-error classical communication over classical channels. Communications in Mathematical Physics, 311(1):97-111, 2012. Google Scholar
  11. Jayanth Nayak, Ertem Tuncel, and Kenneth Rose. Zero-Error Source-Channel Coding With Side Information. IEEE Transactions on Information Theory, 52(10):4626-4629, 2006. Google Scholar
  12. René Peeters. Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica, 16(3):417-431, 1996. Google Scholar
  13. David E. Roberson. Variations on a Theme: Graph Homomorphisms. PhD thesis, University of Waterloo, 2013. Google Scholar
  14. David E. Roberson and Laura Mančinska. Graph Homomorphisms for Quantum Players, 2012. Google Scholar