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 Grötschel, Lovász and Schrijver (1986) [7]. We define new non-commutative versions of the Lovász 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 by Duan, Severini and Winter (2013) [5]. We define non-commutative counterparts of widely used classical graph parameters and establish their interrelation.