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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 &#x03D1;(G&#x0305;) &#x2264; &#x03D1;(H&#x0305;), where &#x03D1; represents the Lova&#x0301;sz number. We also obtain similar inequalities for the related Schrijver &#x03D1;- and Szegedy &#x03D1;+ 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: &#x03B1;*(G) &#x2264; &#x03D1;-(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 Lova&#x0301;sz number. Beigi introduced a quantity &#x03B2; as an upper bound on &#x03B1;* and posed the question of whether &#x03B2;(G) = &#x230A;&#x03D1;(G)&#x230B;. We answer this in the affirmative and show that a related quantity is equal to &#x230A;&#x03D1;(G)&#x230B;. We show that a quantity &#x03C7;vect(G) recently introduced in the context of Tsirelson's problem is equal to &#x230A;&#x03D1;+(G)&#x230B;. In an appendix, we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.

ieee transactions on information theory

Bounds on Entanglement-Assisted Source-Channel Coding via the Lov&#x00E1;sz <inline-formula> <tex-math notation="TeX">$\vartheta $ </tex-math></inline-formula> Number and Its Variants

Bounds on Entanglement-Assisted Source-Channel Coding via the Lovász $\vartheta $ Number and Its Variants