Quantum asymptotic spectra of graphs and non-commutative graphs, and quantum Shannon capacities
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We study several quantum versions of the Shannon capacity of graphs and non-commutative graphs. We introduce the asymptotic spectrum of graphs with respect to quantum homomorphisms and entanglement-assisted homomorphisms, and we introduce the asymptotic spectrum of non- commutative graphs with respect to entanglement-assisted homomorphisms. We prove that the preorders in these scenarios are Strassen preorders. This allows us to apply Strassen's spectral theorem (J. Reine Angew. Math., 1988) and obtain a dual characterization of the corresponding Shannon capacities and asymptotic preorders in terms of the asymptotic spectra. This work extends the study of the asymptotic spectrum of graphs initiated by Zuiddam in (arXiv:1807.00169, 2018) to the quantum domain. We study the relations among the three new quantum asymptotic spectra and the asymptotic spectrum of graphs. The bounds on the several Shannon capacities that have appeared in the literature we fit into the corresponding asymptotic spectra. We find a new element in the asymptotic spectrum of graphs with respect to quantum homomorphisms, namely the fractional Haemers bound over the complex numbers.
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