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Borrowing ideas from the relation between simply laced Lie algebras andDynkin diagrams, a weighted graph theory representation of quantum informationis addressed. In this way, the density matrix of a quantum state can beinterpreted as a signless Laplacian matrix of an associated graph. Usingsimilarities with root systems of simply laced Lie algebras, one-qubit theoryis analyzed in some details and is found to be linked to a non-orientedweighted graph having two vertices. Moreover, this one-qubit theory isgeneralized to n-qubits. In this representation, quantum gates correspond tograph weight operations preserving the probability condition. A speculationfrom string theory, via D-brane quivers, is also given.

Weighted Graph Theory Representation of Quantum Information Inspired by Lie Algebras