  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.
