Sandwich theorems and capacity bounds for non-commutative graphs

We define non-commutative versions of the vertex packing polytope, the theta convex body and the fractional vertex packing polytope of a graph, and establish a quantum version of the Sandwich Theorem of Gr\"{o}tschel, Lov\'{a}sz and Schrijver. We define new non-commutative versions of the Lov\'{a}sz number of a graph which lead to an upper bound of the zero-error capacity of the corresponding quantum channel that can be genuinely better than the one established previously by Duan, Severini and Winter. We define non-commutative counterparts of widely used classical graph parameters and establish their interrelation.