# Condition for zero and non-zero discord in graph Laplacian quantum states

## TL;DR

This paper explores the conditions under which quantum discord is zero or non-zero in graph Laplacian quantum states derived from weighted directed graphs, extending previous simple graph analyses.

## Contribution

It generalizes the conditions for quantum discord in graph states to weighted directed graphs, broadening the scope of graph-theoretic quantum correlation analysis.

## Key findings

- Derived graph-theoretic conditions for zero and non-zero discord.
- Applied conditions to important quantum states like Werner, Isotropic, and X-states.
- Extended previous simple graph results to weighted directed graphs.

## Abstract

This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph-theoretic conditions for zero and non-zero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and $X$-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.