Bounds on entanglement-assisted source-channel coding via the lovász number and its variants

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{G}}) ≤\H , where represents the Lovász number. We also obtain similar inequalities for the related Schrijver {^- and Szegedy (⁺ 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: α(G) \le {^- (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 α) and posed the question of whether β (G) = \lfloor (G) \rfloor \). We answer this in the affirmative and show that a related quantity is equal to \(\lceil (G) \rceil \). We show that a quantity χ G)recently introduced in the context of Tsirelson's problem is equal to \(lceil ⁺} {G}}) \rceil \). In an appendix, we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.