The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States

By Samuel L. Braunstein, Sibasish Ghosh, Simone Severini. 2004

187 citations recorded by the source corpus.

Abstract.We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graphs on four vertices representing entangled states. It turns out that for these graphs the value of the concurrence is exactly fractional.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/765290186ba02424f4ee84ef23bf8a81513fd27e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. M. A. Porter
2. Alex Arenas
3. Manlio De Domenico
4. Yamir Moreno
5. Sergio Gómez
6. Andrea Gabrielli
7. Guido Caldarelli
8. Mikko Kivelä

In Atomic and Molecular Physics, and Optics

1. Alex Arenas
2. Manlio De Domenico
3. M. A. Porter
4. Simone Severini

In Geometry and Topology

1. Manlio De Domenico
2. M. A. Porter
3. Alex Arenas
4. Albert Solé-Ribalta
5. Yamir Moreno
6. Sergio Gómez
7. Mikko Kivelä
8. Emanuele Cozzo

Citations in the corpus

• Multilayer Analysis and Visualization of Networks (2014, methodology): “This estimates the similarity between two networks based on their Von Neumann entropy [66].”
• Mathematical Formulation of Multilayer Networks (2013, methodology): “As an example, let us consider the von Neumann entropy of a monoplex network [69].”
• The von Neumann Entropy of Networks (2008, background): “Precisely, the quantum entropy is an extension of the Gibbs entropy to the quantum realm and it may be viewed as the average information the experimenter obtains in the repeated observations of many copies of an identically prepared mixed state.”
• The statistical physics of real-world networks (2018, background): “[176] S.”