516 research outputs found
On graphs whose Laplacian matrix's multipartite separability is invariant under graph isomorphism
Normalized Laplacian matrices of graphs have recently been studied in the
context of quantum mechanics as density matrices of quantum systems. Of
particular interest is the relationship between quantum physical properties of
the density matrix and the graph theoretical properties of the underlying
graph. One important aspect of density matrices is their entanglement
properties, which are responsible for many nonintuitive physical phenomena. The
entanglement property of normalized Laplacian matrices is in general not
invariant under graph isomorphism. In recent papers, graphs were identified
whose entanglement and separability properties are invariant under isomorphism.
The purpose of this note is to characterize the set of graphs whose
separability is invariant under graph isomorphism. In particular, we show that
this set consists of , and all complete graphs.Comment: 5 page
Combinatorial laplacians and positivity under partial transpose
Density matrices of graphs are combinatorial laplacians normalized to have
trace one (Braunstein \emph{et al.} \emph{Phys. Rev. A,} \textbf{73}:1, 012320
(2006)). If the vertices of a graph are arranged as an array, then its density
matrix carries a block structure with respect to which properties such as
separability can be considered. We prove that the so-called degree-criterion,
which was conjectured to be necessary and sufficient for separability of
density matrices of graphs, is equivalent to the PPT-criterion. As such it is
not sufficient for testing the separability of density matrices of graphs (we
provide an explicit example). Nonetheless, we prove the sufficiency when one of
the array dimensions has length two (for an alternative proof see Wu,
\emph{Phys. Lett. A}\textbf{351} (2006), no. 1-2, 18--22).
Finally we derive a rational upper bound on the concurrence of density
matrices of graphs and show that this bound is exact for graphs on four
vertices.Comment: 19 pages, 7 eps figures, final version accepted for publication in
Math. Struct. in Comp. Sc
The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states
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
graph on four vertices representing entangled states. It turns out that for
some of these graphs the value of the concurrence is exactly fractional.Comment: 20 pages, 11 figure
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