# Sandwich theorems and capacity bounds for non-commutative graphs

## TL;DR

This paper introduces non-commutative analogs of classical graph parameters and theorems, providing new bounds on quantum channel capacities and extending classical combinatorial concepts into the quantum domain.

## Contribution

It develops non-commutative versions of key graph invariants and the Sandwich Theorem, offering improved bounds on quantum zero-error capacities.

## Key findings

- Quantum versions of the vertex packing polytope and theta body are established.
- A quantum Sandwich Theorem analogous to the classical one is proved.
- New upper bounds on quantum channel capacities surpass previous results.

## Abstract

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.