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) ≤ -(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 β as an upper bound on α∗ and posed the question of whether β(G) = ⌊(G)⌋. We answer this in the affirmative and show that a related quantity is equal to ⌈(G)⌉. We show that a quantity χ_vect(G) recently introduced in the context of Tsirelson's conjecture is equal to ⌈+(G)⌉.