Quantum Asymptotic Spectra of Graphs and Non-Commutative Graphs, and Quantum Shannon Capacities

Abstract

We study quantum versions of the Shannon capacity of graphs and non-commutative graphs. We introduce the asymptotic spectrum of graphs with respect to quantum and entanglement-assisted homomorphisms, and we introduce the asymptotic spectrum of non-commutative graphs with respect to entanglement-assisted homomorphisms. We apply Strassen’s spectral theorem (J. Reine Angew. Math., 1988) in order to obtain dual characterizations of the corresponding Shannon capacities and asymptotic preorders in terms of their asymptotic spectra. This work extends the study of the asymptotic spectrum of graphs initiated by Zuiddam (Combinatorica, 2019) to the quantum domain. We then exhibit spectral points in the new quantum asymptotic spectra and discuss their relations with the asymptotic spectrum of graphs. In particular, we prove that the (fractional) real and complex Haemers bounds upper bound the quantum Shannon capacity, which is defined as the regularization of the quantum independence number (Mancinska and Roberson, J. Combin. Theory Ser. B, 2016), and that the fractional real and complex Haemers bounds are elements in the quantum asymptotic spectrum of graphs. This is in contrast to the Haemers bounds defined over certain finite fields, which can be strictly smaller than the quantum Shannon capacity. Moreover, since the Haemers bound can be strictly smaller than the Lovasz theta function (Haemers, IEEE Trans. Inf. Theory, 1979), we find that the quantum Shannon capacity and the Lovasz theta function do not coincide. As a consequence, two well-known conjectures in quantum information theory, namely: 1) the entanglement-assisted zero-error capacity of a classical channel is equal to the Lovasz theta function and 2) maximally entangled states and projective measurements are sufficient to achieve the entanglement-assisted zero-error capacity, cannot both be true.