Exclusivity structures and graph representatives of local complementation orbits

TL;DR

This paper introduces a graph construction linked to local complementation orbits that connects quantum non-locality, Bell inequalities, and zero-error information theory, revealing new insights into quantum graph states and capacities.

Contribution

It presents a novel graph construction H(G) representing local complementation orbits, connecting quantum physics with combinatorial graph properties and information theory.

Findings

H(G) has independence number less than its Lovász number

H(G) corresponds to the orbit of G under local complementation

The construction achieves maximum entanglement-assisted capacity

Abstract

We describe a construction that maps any connected graph G on three or more vertices into a larger graph, H(G), whose independence number is strictly smaller than its Lov\'asz number which is equal to its fractional packing number. The vertices of H(G) represent all possible events consistent with the stabilizer group of the graph state associated with G, and exclusive events are adjacent. Mathematically, the graph H(G) corresponds to the orbit of G under local complementation. Physically, the construction translates into graph-theoretic terms the connection between a graph state and a Bell inequality maximally violated by quantum mechanics. In the context of zero-error information theory, the construction suggests a protocol achieving the maximum rate of entanglement-assisted capacity, a quantum mechanical analogue of the Shannon capacity, for each H(G). The violation of the Bell inequality is expressed by the one-shot version of this capacity being strictly larger than the independence number. Finally, given the correspondence between graphs and exclusivity structures, we are able to compute the independence number for certain infinite families of graphs with the use of quantum non-locality, therefore highlighting an application of quantum theory in the proof of a purely combinatorial statement.