Combinatorial approach to multipartite quantum systems:basic formulation

Abstract

In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study the related issues like classification of pure and mixed states, Von-Neumann entropy, separability of multipartite quantum states and quantum operations in terms of the graphs associated with quantum states. In order to address the separability and entanglement questions using graphs, we introduce a modified tensor product of weighted graphs, and establish its algebraic properties. In particular, we show that Werner's definition [1] of a separable state can be written in terms of a graphs, for the states in a real or complex Hilbert space. We generalize the separability criterion (degree criterion) due to S.L. Braunstein, S. Ghosh, T. Mansour, S. Severini, R.C. Wilson [2], to a class of weighted graphs with real weights. We have given some criteria for the Laplacian associated with a weighted graph to be positive semidefinite.

55 pages,24 figures. Comments are welcome

References in corpus (2)

• The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states
• Some families of density matrices for which separability is easily tested

Cited by in corpus (6)

• On the degree conjecture for separability of multipartite quantum states
• Condition for zero and non-zero discord in graph Laplacian quantum states
• Laplacian matrices of weighted digraphs represented as quantum states
• Quantum Entanglement & Purity Testing: A Graph Zeta Function Perspective
• Quantum state representation based on combinatorial Laplacian matrix of star-relevant graph
• The tripartite separability of density matrices of graphs